Algebraic construction of the Darboux matrix revisited

Algebraic construction of the Darb oux matrix revisited Jan L. Cie ´ sli ´ nski ∗ Uniw ersytet w Bia lymstoku, Wydzia l Fizyki, 15-424 Bia lystok, ul. Lipow a 41, Poland Abstract W e presen t algebraic co nstruction of Darb o ux m atrices for 1+1- dimensional in tegrable systems of nonlinear partial diﬀeren tial equa- tions with a sp ecial stress on the non isosp ectral case. W e d iscuss diﬀeren t approac h es to the Darb oux-B¨ ac klund transformation, based on diﬀeren t λ -dep endencies of th e Darb oux matrix: polynomial, sum of p artial fractions, or the transfer matrix form. W e derive symmetric N -soliton form ulas in the general c ase. Th e matrix sp ectral parameter and dr essing actions in lo op group s are also discu ssed. W e describ e reductions to twisted lo op group s, unitary reductions, the matrix Lax pair for the K dV equation and reductions of c hiral mo dels (harmonic maps) to S U ( n ) and to Grassmann spaces. W e sh o w that in the KdV case the nilp oten t D arb oux matrix generates the binary Dar- b oux tr ansformation. The p ap er is in tended as a revie w of kno wn results (usually presented in a nov el con text) bu t some new results are included as w ell, e.g., general compact form ulas for N -soliton sur- faces and linear and bilinear constrain ts on the nonisosp ectral Lax pair matrice s whic h are preserv ed b y Darb oux transformations. P ACS Numb ers : 02.30.Ik, 03.50.Kk, 05.4 5Yv. MSC 2000 : 3 7K35, 37 K 30, 37K25, 35 Q 53, 22E67. Keywor ds : integrable systems, D arb oux-B¨ ac klund transformation, Darb oux matrix, dressing me tho d, loo p g roups, reduction gro up, nonisosp ectral linear problems, inv ariants of D arb oux transfor ma t io ns, N -soliton surfaces, c hiral mo dels, KdV equation ∗ e-mail: jan ek @ al pha.u wb.ed u.pl 1 1 In tro duc t ion A 1 +1-dimensional in tegrable system can b e considered as in tegrability con- ditions for a line ar pr oblem (a system of linear partial diﬀeren tial equations deﬁned b y t wo matr ices containing the sp e ctr al p ar ame ter ), see fo r instance [56]. T he Darb oux-B¨ ac klund transform is a gauge-lik e transformation (de- ﬁned b y the Darb oux matrix ) whic h preserv es the form of the linear problem [14, 22, 27, 40, 66]. All approac hes to the construction of Darb oux matrices originate in the dressing metho d [56, 68, 81, 82]. The pap er is in tended as a presen tation of Darb oux-B¨ ac klund t r ansforma- tions f r om a uniﬁed persp ectiv e, ﬁrst presen ted in [13, 14]. The construction of the Darb oux ma t r ix is divided in to t wo stages. First, w e uniquely c har- acterize the considered linear problem in terms of algebraic constrain ts (the divisor of p oles, lo op group reductions and ot her algebraic prop erties, e.g., linear and biblinear constraints ). Then, w e construct the Darb oux matrix preserving all these constrain ts. Using general theorems, including those from the presen t pap er, one may construct t he D arb oux matr ix in a w ay whic h is almost algorithmic. The pap er is in tended as a review of kno wn results but some new results are a lso included. W e discuss in detail elemen tary Darb oux transforma t ion (Darb o ux mat r ix which has a single simple zero), symme tric formulas for Darb oux matrices and soliton surfaces ( in the general case), and lo op group reductions for p o lynomial Darb oux matrices. Tw o ex amples are discussed in detail: the Kortewe g-de V ries equation and chiral mo dels (harmonic maps). The part whic h seems to b e most original con tains the description of linear and bilinear inv a r ia n ts of Darb oux tra nsformations. W e pro v e that m ultilinear constraints in tr o duced in [14] a re in v arian t with resp ect to the p olynomial D arb oux transforma t io n (also in the nonisosp ectral case). T aking them into accoun t w e can av oid some cum b ersome calculations, our construc- tion assumes a more elegan t form and, last but not least, we do not need any assumptions concerning b oundary conditions. Another imp ortan t aim of this pa p er is to sho w similarities and ev en an equiv alence b et w een diﬀeren t algebraic approa c hes to the construction o f the Darb oux matrix. This is a no v elty in itself b ecause sometimes it is diﬃcult to not ice connections b etw een diﬀeren t metho ds. The existing monographs, ev en the recen t ones, fo cus on a c hosen single approa c h, compare [24, 27, 46, 47, 56, 6 0 ]. 2 W e consider a nonlinear system of partia l diﬀerential equations which is equiv alen t to the compat ibility conditio ns U µ , ν − U ν , µ +[ U µ , U ν ] = 0 , (1 6 µ < ν 6 m ) , (1.1) for the following system of linear equations (kno wn as t he L a x pair, at least in the case of tw o indep enden t v ariables) Ψ , ν = U ν Ψ , ( ν = 1 , . . . , m ) , (1.2) where n × n mat rices U ν dep end on x 1 , . . . , x m and on the so called sp ectral parameter λ (and, as usual, Ψ , ν = ∂ Ψ /∂ x ν , etc.). W e assume that Ψ is also a matrix (the fundamen tal solution of the linear system (1.2 ) ) . W e ﬁx our atten t io n on t he case m = 2 (although most results hold for a n y m ) a nd shortly denote b y x the set of all v ariables, i.e., x = ( x 1 , . . . , x m ). 1.1 Isosp ectral and nonisosp ectral Lax pairs Let us recall that the most imp o rtan t c hara cteristic of the matrices U 1 , U 2 is their dep endence on the sp ectral parameter λ . In the t ypical case U ν are rational with resp ect to λ . Actually w e will consider a more general situatio n. W e assume that the Lax pair is rational with resp ect to λ , and • “ isosp ectral case”: λ is a constant parameter, • “ non-isosp ectral” case: λ, ν = L ν ( x, λ ) , ( ν = 1 , . . . , m ) , (1.3) where L ν are giv en functions, rational with res p ect to λ (this case reduces to the isosp ectral one for L ν ( x, λ ) ≡ 0). Remark 1.1. T h e diﬀer ential e quations (1.3) ar e of the ﬁrst or der, so their solution λ = Λ( x, ζ ) dep en ds on a c ons tant of inte gr ation ζ which p l a ys the r ole of the c on s tant s p e ctr al p ar ameter. The solution of the system (1.3) exists pro vided that compatibility con- ditions hold, for more details see [14]. In general Λ = Λ( x, ζ ) is an implicit function, although in man y sp ecial cases explicit expre ssion for Λ can b e found, compare [11, 14, 6 9 ]). 3 1.2 The Darb oux-B¨ ac klund transformation The application of the dressing metho d to generate new solutions of no nlinear equations “co ded” in (1.1) consists in the follow ing (see [56, 79, 82]). Supp ose that we are able to construct a g auge-lik e transforma t ion ˜ Ψ = D Ψ (where D = D ( x, λ ) will b e called the Darb oux matrix) suc h that the structure of matrices ˜ U ν , ˜ U ν = D , ν D − 1 + D U ν D − 1 , ( ν = 1 , . . . , m ) , (1.4) is iden tical with the structure of the matrices U ν . The soliton ﬁelds en tering U ν are replaced b y some new ﬁelds whic h, ob viously , hav e to satisfy the nonlinear system ( 1.1) as w ell. Remark 1.2. T h e Da rb oux tr ansformation should pr es e rve divisors o f p oles (i.e., p oles and their multiplicities) of matric es U ν . T his is the most im p or- tant structur al pr op erty of U ν to b e pr eserve d. The se c ond imp ortant pr op erty is the so c al le d r e duction gr oup, se e Se ction 6. F or any pair of solutions of (1.1 ) one can “compute” D := ˜ ΨΨ − 1 . The crucial p oin t is, how ev er, to express D solely by the w av e function Ψ b ecause only then one can use D t o construct new solutions. Suc h D is kno wn as the Darb o ux mat r ix [40, 45, 46]. The Darb o ux matrix deﬁnes an explicit map S 7→ S , where S is the set of solutions o f the linear pro blem (1.2). The construction o f the D arb oux matrix is based on the imp o rtan t observ ation: Remark 1.3. The Darb o ux matrix c an b e expr es se d in an algebr aic way by the original wave func tion Ψ . By the “ original w a v e function” w e mean one b efore the transformatio n. In f a ct, it is rather diﬃcult to ﬁnd sp ecial solutions of the linear problem. Usually v ery limited n um b er of cases is av ailable. Ho wev er, know ing any solution Ψ = Ψ( x, λ ) and the Darb o ux matrix one can generate a sequence of explicit solutions. Starting from the trivial bac kgro und ( x -indep enden t and m utually commuting U ν ) we usually get the so called soliton solutions. 1.3 Equiv alen t Darb ou x matrices It is quite natural to consider as equiv alen t Darb oux matrices whic h pro - duce exactly the same transformation (1.4) of matrices U ν of a giv en linear problem. 4 Remark 1.4. The line ar pr oblem (1.2) is invariant under tr ansformations Ψ 7→ Ψ C 0 (for any c onstant nonde gen er ate matrix C 0 = C 0 ( λ ) ). Therefore Darb oux matrices D a nd D ′ are equiv alen t if there exists a matrix C suc h that D Ψ = D ′ Ψ C (for an y Ψ). Thus C should comm ute with Ψ what, in practice, means that C = f ( λ ) ∈ C . Remark 1.5. The matrix D ′ = f ( λ ) D , wher e f is a c o m plex function o f λ only, is e quivalent to D . 1.4 Soliton surfaces approac h Giv en a solution Ψ = Ψ( x, λ ), where λ dep ends on x and ζ , we deﬁne a new ob ject F b y the so called Sym-T afel (or Sym) form ula: F = Ψ − 1 Ψ , ζ , (1.5) If Ψ assumes v alues in a matrix Lie gro up G , t han (f o r any ﬁxed ζ ) F describ es an immersion (a “ soliton surface”) into the corresp onding L ie algebra [71, 74]. Soliton surfaces are a natural frame to unify a v ariety of diﬀerent phy sical mo dels lik e soliton ﬁelds, strings, v ortices, c hiral mo dels and spin mo dels [72]. In the framew ork o f the soliton surfaces approac h one can reconstruct man y in tegra ble cases known f r o m the classical diﬀeren tial geometry [7, 15, 16, 74]. The Darb oux-B¨ ac klund tra nsformation for solito n surfaces r eads ˜ F = F + Ψ − 1 D − 1 D , ζ Ψ (1.6) where D , ζ = λ, ζ D , λ . The equiv alen t Darb o ux matrices yield the same soliton surfaces. Indeed, if we tak e D ′ = f D , then ˜ F = F + Ψ − 1 D ′− 1 D ′ , ζ Ψ = F + f , ζ f + Ψ − 1 D − 1 D , ζ Ψ , (1.7) i.e., surfaces correspo nding to D and D ′ diﬀer by the constan t (ln f ) , ζ . In o r der to illustrate usefulness of the geometric approac h we presen t the follo wing theorem [18]. Theorem 1.6. We assume that U 1 , U 2 ar e line ar c ombin a tion s o f 1 , λ and λ − 1 , wi th x -dep end ent su (2) -value d c o e ﬃcients, and U ν ( − λ ) = E 0 U ν ( λ ) E − 1 0 (wher e E 0 ∈ su (2) is a c onstant m a trix). Then F given by the Sym formula (1.5) is (in the isosp e ctr al c a se) a pseudosph eric al (i.e ., of n e gative Gaussia n curvatur e) surfac e immerse d in su ( 2) ≃ R 3 . I n the nonis osp e ctr al c ase the same assumptions yield the so c al le d Bianch i surfac es. 5 W e p oint out that surprisingly few assumptions (restrictions) on the sp ec- tral problem leads to the very imp ortan t class of pseudospherical surfaces. It is easy to assure the preserv ation of these restrictions b y the Darb oux transformation. Darb oux transformations usually preserv e man y other constraints (e.g., linear and bilinear inv ar ia n ts discuss ed in Section 8) what leads to the preser- v ation of some g eometric c hara cteristics (e.g, curv ature lines) and to a sp eciﬁc c hoice of co ordinates and other auxiliary parameters. 2 Binary Darb oux matrix In this pap er by the binary D arb oux matrix w e mean one p ole matrix with non-degenerate normalization D = N  I + λ 1 − µ 1 λ − λ 1 P  , P 2 = P , det N 6 = 0 , (2.1) suc h that its in ve rse has the same form: D − 1 =  I + µ 1 − λ 1 λ − µ 1 P  N − 1 . (2.2) Here λ 1 , µ 1 are complex parameters (whic h can dep end on x in the non- isosp ectral case), P = P ( x ) is a pro jector matrix ( P 2 = P ), and N = N ( x ) is the so called normalizatio n matrix. 2.1 Binary or elemen tary? The name “binary” for Darb oux matrices of the form (2.1) is ra ther ten tativ e, b ecause binary Da rb oux transformations we re in tro duced in another context (compare [46, 83]). The “classical” binary transformation corresp onds to the degenerate case of (2.1) when µ 1 → λ 1 (see Section 4.4), i.e., D = N  I + M λ − λ 1  , D − 1 =  I − M λ − λ 1  N − 1 , (2.3) where M 2 = 0 (the so called nilp oten t case, see [14]). Therefore, we use this notion in an extended sense. How ev er, it seems to b e compatible with under- standing binary D arb oux transformation as a comp osition of a n elemen tary 6 Darb oux transformation and a Darb oux transformation of the adjoint linear problem [46, 57, 67]. In the case of the Zakharov -Shabat sp ectral problems (1.2) the adjoin t sp ectral problem is given b y − Φ , ν = Φ U ν , (2.4) and one can easily c hec k that Φ = Ψ − 1 solv es the adjoin t sp ectral problem. The general solution of (2.4) is Φ = Ψ − 1 C , where C is a constant (i.e., x - indep enden t) matrix. W e will see in Section 5.1 that the binar y D arb oux matrix can be expressed b y a pair of solutions: one solv es the spectral prob- lem (1.2), a nd the second one solv es the adjoint problem (2.4). The matrix (2.1) is equiv alent to the linear in λ matr ix ˆ D ˆ D = N ( λ − λ 1 + ( λ 1 − µ 1 ) P ) . (2.5) Darb oux matrices linear in λ are sometimes referred to as “elemen tary”, see [60]. Indeed, iterating suc h transformations w e can get any Darb oux transformation with nondegenerate normalization. How ev er, w e reserv e the name “elemen tary” f o r matrices whic h a re no t only linear in λ but hav e a single zero (see Section 4 ), or ev en a single simple zero. The p olynomial form (2.5) of the binary D arb oux matrix has tw o zeros: λ 1 , µ 1 . The sum of their m ultiplicities is n . Therefore these zeros are simple o nly in the case n = 2. 2.2 Suﬃcien t conditions for the pro jector Assuming that U ν are regular (holomorphic) at λ = λ 1 and λ = µ 1 , and demanding that ˜ U ν (expresse d b y (1.4 )) ha ve no p oles at λ = λ 1 and λ = µ 1 as w ell, w e get t he follo wing conditions (for v anishing the corresp onding residues), compare [14]: P ◦ ( − ∂ ν + U ν ( λ 1 )) ◦ ( I − P ) = 0 , ( I − P ) ◦ ( − ∂ ν + U ν ( µ 1 )) ◦ P = 0 , (2.6) λ 1 , ν = L ν ( x, λ 1 ) , µ 1 , ν = L ν ( x, µ 1 ) , (2.7) where the circles mean composition o f linear op erators and L ν are deﬁned b y (1.3). Note that for an y op erat o rs A, B w e ha ve : A ◦ B = 0 ⇒ im B ⊂ k er A . (2.8) Indeed, ( A ◦ B ) ϕ = 0 for an y v ector ϕ , i.e. A ( B ϕ ) = 0 what means exactly that B ϕ ∈ ke r A . On the ot her hand, an y elemen t of im B is of the form B ϕ . 7 Remark 2.1. The assumption that U ν ar e r e gular a t λ = λ 1 and λ = µ 1 (assume d thr oughout this p ap er) is essential. R elaxing this r e quir emen t we c an get solutions d iﬀer ent f r om those obtaine d b y the standar d Darb oux-B¨ ack- lund tr ansformation. Solutions of this kind ( unitons) have b e en found in the c ase of harmo n ic ma p s into Lie gr oups [77 ] , se e also [27]. If the system (1.3) has the g eneral solution λ = Λ( x, ζ ), then the equations (2.7) can b e solv ed in terms of the function Λ : λ 1 = Λ ( x, ζ 1 ) , µ 1 = Λ( x, ζ ′ 1 ) , (2.9) where ζ 1 , ζ ′ 1 are constan t parameters, compare [4, 5 1]. T aking into accoun t k er P = im( I − P ), w e easily sho w that the system (2.6) is equiv alen t to: ( − ∂ ν + U ν ( λ 1 )) k er P ⊂ ker P , ( − ∂ ν + U ν ( µ 1 )) im P ⊂ im P . (2.10) No w w e easily see that the conditions (2.10) are satisﬁed by the pro j ector deﬁned by the Zakharov -Shabat form ula s (compare [56 , 79, 82]): k er P = Ψ( λ 1 ) V k er , im P = Ψ( µ 1 ) V im , (2.11) where Ψ( λ 1 ) = Ψ( x, λ 1 ), Ψ( µ 1 ) = Ψ( x, µ 1 ) and V k er and V im are constant v ector spaces such that V k er ⊕ V im = V . Indeed, in this case, by virtue of (1.2), the left-hand sides of (2.10) are simply equal to zero. T aking in to account that an y pro jector P can b e expressed explicitly b y its k ernel and image, P = (im P , 0)(im P , k er P ) − 1 , w e can summarize the ab ov e discussion as follows . Prop osition 2.2. Th e tr ansfo rmation (1.4) wi th D given by (2.1), w h e r e P = (Ψ( µ 1 ) V im , 0)(Ψ( µ 1 ) V im , Ψ( λ 1 ) V k er ) − 1 , (2.12) pr eserves the diviso rs of p oles of matric es U ν . The form ula (2.12) yields a suﬃcien t condition for P to generate the Darb oux matrix. It is in teresting to ﬁnd also necessary conditions. Therefore, w e will try to obtain t he most general solution of (2.10). 8 2.3 The general form of the binary Darb oux matrix It is con v enien t to represen t v ector spaces in a matrix form. Namely , if w 1 , . . . , w k span a v ector space V , then w e can identify V with the matrix V = ( w 1 , . . . , w k ) . (2.13) This matrix has k columns ( w 1 , . . . , w k ) and n ro ws ( n = dim V ). Note that b ecause of the freedom in c ho osing a basis in the v ector space there a r e many matrices represen ting the same v ector space. If a ij are co ef- ﬁcien ts of a k × k non-degenerate matrix A , then the vec tors w ′ j = k X i =1 w i a ij , form another basis in V whic h can b e represen ted b y the matrix V ′ = ( w ′ 1 , . . . , w ′ k ) = V A . The matrices V and V ′ (for any non-degenerate A ) represen t the same vec tor space and, in this contex t, are considered as equiv alen t ones. The space of k - dimensional subspaces of an n -dimensional v ector space o v er C is kno wn as Grassmannian G k ,n ( C ) (of course , considering real vec tor spaces w e hav e real Grassmannian G k ,n ( R )). The ele men ts of the Grassman- nian are classes of equiv alence of k × n matrices with resp ect to the equiv alence relation: V ≃ V ′ if there exists k × k matrix A (det A 6 = 0) suc h tha t V ′ = V A . Therefore using the same notation for the ve ctor space and the matrix represen ting it, one should remem b er a b out this equiv alence. In particular, in order to sho w tha t some v ector spaces W and V are identical, one has to consider the equation W = V A with a n arbitrary non-degenerate A . In the similar wa y one can c hec k whether W is a subspace of V (of course a necessary requiremen t is dim V ≥ dim W ). Prop osition 2.3. L et V , W b e ve ctor sp a c es, k ′ = dim W ≤ dim V = k . Then W ⊂ V if and on ly if ther e exists k × k matrix B such that W = V B . Pro of: If W ⊂ V , then there exists a basis of V suc h that its ﬁrst k ′ v ectors span W . W e represen t V b y v ectors of this basis, i.e., w e c ho ose A suc h that w ′ 1 , . . . , w ′ k ′ span W . Finally , w e put B = A diag (1 , . . . , 1 , 0 , . . . , 0). ✷ 9 Note that formally W and V b elong (in general) to diﬀeren t Gra ssman- nians. But if det B = 0, then the columns of V B are linearly dependen t and V B can b e treated as an elemen t of a Grassmannian o f lo wer dimension. W e pro ceed to solving the system (2.10). Assuming det Ψ( λ 1 ) 6 = 0 and det Ψ( µ 1 ) 6 = 0, w e can alw ays put k er P = Ψ( λ 1 ) V , im P = Ψ( µ 1 ) W , where W , V are some v ector spaces (in general x -dep enden t). Substituting to (2.10 ) w e ha ve: Ψ( λ 1 ) V , ν ⊂ Ψ( λ 1 ) V , Ψ( µ 1 ) W , ν ⊂ Ψ( µ 1 ) W . Hence, V , ν ⊂ V , W , ν ⊂ W . By Prop osition 2.3, w e r ewrite V , ν ⊂ V as V , ν = V B ν for some B ν (whic h ha ve to satisfy appropriate compatibility conditions), and ana logical equations for W . T aking in to accoun t the freedom of changing the basis when c hanging x : V ′ = V A (det A 6 = 0), w e obtain: ( V ′ A − 1 ) , ν = V ′ A − 1 B ν Therefore, choosing A suc h that ( A − 1 ) , ν = A − 1 B ν , (2.14) w e obtain V ′ , ν = 0, i.e. there exists an x -indep enden t basis in V (the same conclusion holds for W ). The solution of ( 2 .14) exists b ecause B ν satisfy the compatibilit y conditions men tioned a b ov e. Th us w e hav e show n tha t the form ulas (2.11) giv e the most g eneral solution of (2.10). 3 P olynomial Darb oux matrice s: g eneral c ase In this pap er we consider only rational Darb oux ma t r ices ( n × n matrices with co eﬃcien ts whic h are rationa l functions of λ ). Remark 3.1. In the isosp e ctr al c ase, every r ational Darb oux ma trix is e quiv- alent to a p olynom i a l Darb oux matrix ˆ D = N X k =0 T k ( x ) λ N − k . (3.1) Inde e d, it is enough to multiply given D by the le ast c om mon multiple of al l denominators. The obtaine d p olynom ial wil l b e deno te d by ˆ D . 10 Another equiv alen t form of D is a p olynomial in λ − 1 , obtained from ˆ D ( λ ) b y dividing it b y λ N . In some cases this p olynomial is more con v enien t that ˆ D b ecause it is analytic at λ = ∞ . In the nonisospectral case the least common m ultiple of all denominators dep ends on x . Therefore, an y rational Darb oux matrix is equiv alen t to some p olynomial matrix up t o a scalar x -dep enden t factor. 3.1 The determinan t of the Darb oux matrix The trace of a qu adratic matrix is deﬁne d a s the sum of diagonal elemen ts of this matrix. Bo th the trace and the determinan t are in v arian t with resp ect to similarit y tr ansformations: T r( B AB − 1 ) = T r A , det( B AB − 1 ) = det A . Theorem 3.2 (L iouville) . If Ψ , ν = U ν Ψ , whe r e ν is ﬁxe d and U ν = U ν ( x ) is given, then (det Ψ) , ν = T r U ν det Ψ . (3.2) This theorem is w ell kno wn as the Liouville theorem on W ronskians, see, for instance, [2]. Applying the Liouville theorem to the Darb oux transform ˜ Ψ ≡ D Ψ w e get (det D det Ψ) , ν = T r ˜ U ν det D det Ψ. Hence, using once more (3.2), w e obtain: (det D ) , ν det D = T r( ˜ U ν ) − T r( U ν ) . (3.3) Remark 3.3. We usual ly c onsider tr ac eless line ar pr oblems ( T r U ν = 0 for ν = 1 , . . . , m ). In such c ase det D has to b e c onstant (i.e., det D do es not dep end on x ). T her efor e, in the isosp e ctr al (and tr ac eless) c ase det D c an dep end only on λ and al l its zer os ar e c onstants. In the nonisosp ectral case the situation is more complicated b ecause λ dep ends o n x . Ho we v er, it is still p o ssible to o btain a strong general result c hara cterizing zeros o f det D . Theorem 3.4. We c onsider a p olynomial Da rb o ux matrix ˆ D for a non- isosp e ctr al li n e ar pr obl e m (1.2) with λ satisfying (1.3). If det ˆ D ( λ k ) = 0 and matric es U ν ar e r e gular at λ k , then λ k , ν = L ν ( x, λ k ) , (3.4) i.e., λ k = Λ( x, ζ k ) , wher e ζ k = const . 11 Pro of: The determinan t of the p olynomial ˆ D ( λ ) has a ﬁnite n umber of ro ots ( x -dep enden t, in general). W e denote them b y λ k , k = 1 , . . . , K , and their m ultiplicities by m k . Note that m 1 + m 2 + . . . m K = nN , where N is the degree of the p olynomial ˆ D ( λ ) and n is the order of the matrix ˆ D . Th us det ˆ D ( λ ) = h K Y k =1 ( λ − λ k ) m k , (3.5) where h = h ( x ) and λ k = λ k ( x ). T aking into accoun t (1.3) we compute (det D ) , ν det D = h, ν h + K X k =1  m k L ν ( x, λ ) − λ k , ν λ − λ k  . (3.6) The equation (3.3 ) with U ν regular at λ k implies that the righ t- hand side o f (3.6) should hav e no po les. Therefore residua of (3.6 ) at λ = λ k v anish wh at implies (3.4). The x -dep endence of λ k follo ws from Remark 1.1. ✷ The regularity of U ν at λ = λ k is assumed thro ug hout this pap er. If we allo w that some λ k coincides with a singularity of U ν , then the x -dependence of λ k in principle can b e diﬀeren t from (3.4) and w e get an additional freedom. 3.2 Neugebauer’s approac h A simple but quite general metho d to construct p olynomial Darb oux-B¨ ac klund transformations has b een prop osed by Neugebauer and his collab orators [47, 51, 52], see also [34, 6 0]. W e are going to ﬁnd conditions on p olynomial ˆ D implying that divisors of p oles of ˜ U ν and U ν coincide (compare Remark 1.2). F rom ˜ Ψ , ν = ˜ U ν ˜ Ψ w e get ˜ U ν = ˜ Ψ , ν ( λ ) ˜ Ψ c ( λ ) det ˜ Ψ( λ ) = 1 det ˆ D ( ˆ D , ν ˆ D c + ˆ D U ν ˆ D c ) , (3.7) where b y ˆ D c w e denote the matrix of cofactors of ˆ D . Ob viously ˆ D c is also a p olynomial in λ . If U ν are rational functions of λ , then ˜ U ν giv en by (3.7) are rational as w ell (b ecause ˆ D and ˆ D − 1 are ra tional). Therefore the only candidates for p oles of ˜ U ν are p oles of U ν and zeros of det ˆ D (i.e., λ k ). The necessary condition for the r egularit y of ˜ U ν at λ = λ k is ˜ Ψ , ν ( λ k ) ˜ Ψ c ( λ k ) = 0 . (3.8) 12 If λ k is a simple zero of det ˆ D ( λ ), then the condition (3.8) is a lso suﬃcien t. F ollow ing [47], w e will ﬁnd anot her, mor e constructiv e, ch aracterization of the condition (3.8). If det ˆ D ( λ k ) = 0 , then w e ha ve also det ˜ Ψ( λ k ) = 0 (3.9) (b ecause ˜ Ψ( λ ) = ˆ D ( λ )Ψ( λ )). W e assume that the function Ψ( λ ) (kno wn as a “bac kground solution” or a “seed solution”) is non-degenerate at λ = λ k . As a consequenc e of (3.9), t he equation ˜ Ψ( λ k ) p k = 0 has a non-zero solu- tion p k ∈ C n (where, in principle, p k can dep end on x ). Then, w e compute: ˜ Ψ , ν ( λ k ) p k = ˜ U ν ( λ k ) ˜ Ψ( λ k ) p k = 0 , where w e to ok in to accoun t that ˜ Ψ( λ ) satisﬁes (1 .2). Thus w e hav e: ˜ Ψ( λ k ) p k = ˜ Ψ , ν ( λ k ) p k = 0 , (3.10) what implies (3.8), as o ne can see fro m t he following fact of linear algebra ([47], see also [34]). Lemma 3.5. L e t us c onside r two de gener ate d matric es X and Y . Supp ose that ther e exists a v e ctor p such that X p = 0 and Y p = 0 . Then: Y X c = 0 . Pro of: Let us p erform computations in a basis ( e 1 , . . . , e n ) suc h that e 1 ≡ p . Then all elemen ts of the ﬁrst column of matrices X , Y are equal to zero. Th us, using the deﬁnition of the cofactor, w e easily see that t he ro ws o f Y c (except the ﬁrst ro w) hav e all en tries equal to zero. Hence, X Y c ob viously yields zero. ✷ Lemma 3.6. The ve ctor p k such that ˜ Ψ( λ k ) p k = 0 is deﬁne d up to a sc alar factor. If λ k is a simple zer o, then we c an cho ose this multiplier in such a way that p k = const . Pro of: W e diﬀeren tiate the equation deﬁning p k : ˜ Ψ , ν ( λ k ) p k + ˜ Ψ( λ k ) p k , ν = 0. Hence, ˜ Ψ( λ k ) p k , ν = 0, whic h means that p k , ν is prop ortional to p k (pro vided that λ k is a simple zero of det ˆ D ( λ )). Th us p k , ν = f k ν p k , where f k ν are some scalar functions. F rom the iden tity p k , ν µ ≡ p k , µν it follo ws that f k ν , µ = f k µ , ν . Therefore, there exists ϕ k suc h t hat f k ν = ϕ k , ν . Henc e, p k e − ϕ k do es not dep end on x . ✷ 13 Corollary 3.7. Polynomi a l Da rb o ux matrix (3.1) c an b e c onstructe d as fol- lows. In the i sosp e ctr al c ase w e cho ose N n p airwise diﬀer ent c o mplex num- b ers λ 1 , λ 2 , . . . , λ N n and N n c onstant C n -ve ctors p 1 , p 2 , . . . , p N n . We also cho ose the ma trix T 0 (“normalization matrix”), det T 0 6 = 0 . Matrix c o e ﬃ- cients T 1 , . . . , T N ar e c ompute d fr om ˆ D ( λ k )Ψ( λ k ) p k = 0 , ( k = 1 , . . . , nN ) , (3.11) wher e Ψ( λ ) is g i v en (“se e d solution ”). In the nonisosp e ctr al c ase we cho ose c onstants ζ 1 , . . . , ζ N n and use (3.4). F or a ﬁxed k the equation (3 .11) consists of n scalar equations. Th us w e hav e a system of n 2 N equations fo r N unkno wn matr ices n × n . In the generic case suc h system should ha v e a unique solution. The freedom in c ho osing T 0 corresp onds to a gauge transformation. No t e that an iden tical situation ha s place in the case of the binary Darb o ux matrix, where N is, in general, undetermined. Usually it is suﬃcien t to put T 0 = I (“canonical normalization”). If this c hoice leads to a c on tradiction ( i.e., the Da rb oux matrix with the canonical normalization do es not exist), then w e may relax this assumption and searc h for Darb o ux matrices with more general normalization. The case det T 0 = 0 can b e treated in a similar wa y but with one excep- tion: the t otal num b er of zeros is smaller than N n . As an example of suc h situation we will presen t elemen tary Darb oux matrices, see Section 4. 3.3 Explicit m ultisoliton form ulas Let us intro duce the no t a tion ϕ k := Ψ( λ k ) p k , (3.12) where ϕ k ∈ R n are column v ectors. W e assume det T 0 6 = 0 and denote θ j := T − 1 0 T j , ( j = 1 , . . . , N ) , (3.13) where T j are deﬁned by (3.1). The equations (3.11) read: λ N k + N X j =1 λ N − j k θ j ! ϕ k = 0 , ( k = 1 , . . . , M ) , (3.14) 14 where M = nN . After the transp osition w e get: N X j =1 ϕ T k θ T j λ N − j k = − ϕ T k λ N k . (3.15) It is con v enien t t o solv e these equations in the matrix form:     θ T 1 θ T 2 · · · θ T N     = −     λ N − 1 1 ϕ T 1 . . . λ 1 ϕ T 1 ϕ T 1 λ N − 1 2 ϕ T 2 . . . λ 2 ϕ T 2 ϕ T 2 . . . . . . . . . . . . . . . . . . . . . . . . . . λ N − 1 M ϕ T M . . . λ M ϕ T M ϕ T M     − 1     ϕ T 1 λ N 1 ϕ T 2 λ N 2 . . . ϕ T M λ N M     . (3.16) Usually , in practical applications, one uses Cramer’s rule to express θ k in terms of determinan ts, compare [52 , 58, 70]. Ha ving co eﬃcien ts T k w e can a pply the Darb oux transformation to Lax pairs of pr escrib ed f orm. As an illustrative example w e presen t the simplest but v ery imp ortan t case (linear in λ ) : U 1 = u 0 λ + u 1 , (3.17) The equation (1.4) for ν = 1, i.e., ˜ U 1 D = DU 1 + D , 1 , yields: ( ˜ u 0 λ + ˜ u 1 ) N X k =0 λ N − k T k = N X k =0 λ N − k T k ( u 0 λ + u 1 ) + N X k =0 λ N − k T k , 1 . (3.18) Considering co eﬃcien ts b y λ N +1 and λ N , w e get explicit formu las for the transformed ﬁelds ˜ u 1 and ˜ u 0 : ˜ u 0 = T 0 u 0 T − 1 0 , ˜ u 1 = T 0 u 1 T − 1 0 + [ T 1 T − 1 0 , ˜ u 0 ] + T 0 , 1 T − 1 0 . (3.19) In the classical AKNS case u 0 = iσ 3 ≡ diag( i, − i ) a nd it is suﬃcien t to tak e the canonical normalization T 0 = I . Therefore, we get ˜ u 0 = u 0 , ˜ u 1 = u 1 + [ T 1 , iσ 3 ] , (3.20) where T 1 = θ 1 can b e explicitly computed from (3.16) , compare [52]. 15 4 Elemen t ary Darb oux matrix The elemen ta r y Darb oux matrix is linear in λ and its determinan t has just a single simple zero. This case is men tioned b y Its [35] and discussed in mor e detail in, for instance, [24, 38]. An obvious w ay to pro duce matrices o f this t yp e is to tak e matrices with a single entry linear in λ and all other entries λ -indep enden t. In this pap er w e conﬁne ourselv es to elemen tary Da r b oux matrices fo r n = 2. They can b e represen ted in the form D = N  λ − λ 1 0 − α 1  M (4.1) where N , M do not dep end on λ . As a simple exercise (compare Coro l- lary 3.7) w e can express the co eﬃcien t α b y Ψ ev aluated a t λ 1 , namely: α = η 1 ξ 1 ,  ξ 1 η 1  = M Ψ( λ 1 ) p 1 , (4.2) where p 1 is a constant v ector. 4.1 Binary Darb oux matrix as a sup erp osition of ele- men tary transformations Theorem 4.1. I n the c ase n = 2 any binary Darb oux tr ansformation is a sup erp osition of two elemen tary Darb oux tr ansformations. Pro of: W e will sho w that D = N 2  1 − β 0 λ − λ 2  N − 1 1 N 1  λ − λ 1 0 − α 1  M , (4.3) is a binary Darb oux matrix ( N 1 , N 2 , M are non-degenerate matrices whic h do not depend on λ ). First, p erforming the m ultiplication in (4.3), we get D = N ( λ − λ 1 + ( λ 1 − λ 2 ) P ) (4.4) where N = N 2  1 0 − α 1  M , P = 1 ∆ λ M − 1  αβ − β α ( α β − ∆ λ ) ∆ λ − αβ  M , (4.5) 16 and ∆ λ = λ 1 − λ 2 . Then, we easily c hec k that P 2 = P . The co eﬃcien ts α , β can b e expressed by Ψ ev aluated at λ 1 , λ 2 . Indeed, denoting Ψ( λ k ) p k =  ξ k η k  , (4.6) and using equations (3.1 1), we obtain α = η 1 ξ 1 , β = ξ 1 ξ 2 ∆ λ ξ 2 η 1 − η 2 ξ 1 . (4.7) The pro jector P r eads P = 1 ξ 1 η 2 − η 1 ξ 2 M − 1  − η 1 ξ 2 ξ 1 ξ 2 − η 1 η 2 η 2 ξ 1  M . (4.8) If M = I , then Ψ( λ 1 ) p 1 ∈ ker P and Ψ( λ 2 ) p 2 ∈ im P . Therefore, the bi- nary Darb oux matrix with P given by (4.8) is a sup erp osition of elemen tary transformations ( 4 .3) with M = I and α , β giv en by (4.7). ✷ 4.2 KdV equation The Darb o ux transformation for the fa mous Kortew eg-de V ries equation is almost alw a ys presen ted in the scalar case, see [4 6]. The matrix a pproac h is less con venie n t. How ev er, having in mind a p edag ogical mo t iv atio n, we are g o ing t o show in detail that the matrix construction w orks also in that case. It is interesting, that in this pap er w e do not need the “KdV reality condition” (usually used in earlier pap ers, compare [14, 27, 76]). The standard scalar Lax pair for KdV equation consists of the Sturm- Liouville-Sc hr¨ odinger sp ectral problem and the se cond equation deﬁning t he time ev o lution of the wa v e function: − ψ , 11 + uψ = λψ , ψ , 2 = − 4 ψ , 111 +6 uψ , 1 +3 u, 1 ψ . (4.9) The compatibilit y conditions ψ , 112 = ψ , 211 yield the KdV equation u, 2 − 6 uu, 1 + u, 111 = 0 . (4.10) 17 The Lax pa ir (4.9) can b e tr a nsformed, in a standard w a y , to the matrix form Ψ , 1 =  0 1 u − λ 0  Ψ , Ψ , 2 =  − u, 1 2 u + 4 λ − 4 λ 2 + 2 u λ + 2 u 2 − u, 11 u, 1  Ψ , (4.11) where Ψ =  ~ ψ , ~ φ  , ~ ψ =  ψ ψ , 1  , ~ φ =  φ φ, 1  , (4.12) and ψ , φ are linearly indep enden t solutions of (4.9). Lemma 4.2. Supp ose that U =  0 1 u − λ 0  , V =  0 4 λ 2 λu − 4 λ 2 0  +  − a b c a  , (4.13) wher e u, a, b, c do not dep end on λ . Then, the c omp atibility c onditions U, 2 − V , 1 +[ U, V ] = 0 uniquely yield: a = u , 1 , b = 2 u , c = 2 u 2 − u, 11 . (4.14) i.e., U, V given by (4.1 3) ar e i d e ntic al w ith the L ax p ai r (4.11) for the KdV e quation. Pro of is straig htforw ard: compatibilit y conditions reduce to (4.10) and ( 4 .14). 4.3 Elemen tary Darb oux matrix and the classical Dar- b oux transformation W e will compute the action of the elemen tary Darb oux tr ansformation in the KdV case, compare [27]. W e assume D = N  λ − λ 1 0 − α 1  (4.15) where N ( det N 6 = 0) do es not de p end on λ and α is a function to b e expresse d b y Ψ( λ 1 ), namely D ( λ 1 )Ψ( λ 1 ) p 1 = 0 , (4.16) 18 where p 1 is a constant v ector. W e denote  ξ 1 η 1  = Ψ( λ 1 ) p 1 (4.17) The constrain t (4.16) (with D giv en by (4.15)) is equiv alen t to α = η 1 ξ 1 = ˆ ψ 1 , 1 ˆ ψ 1 , (4.18) where ˆ ψ 1 satisﬁes (4.9) with λ = λ 1 (i.e., ˆ ψ 1 is a linear com binat ion of ψ 1 and φ 1 ). The function α satisﬁes the follo wing system of Riccati equations: α, 1 = u − λ 1 − α 2 , α, 2 = (2 u 2 − u, 11 +2 uλ 1 − 4 λ 2 1 ) + 2 u, 1 α − (2 u + 4 λ 1 ) α 2 , (4.19) whic h can b e obtained directly from (4 .11). The elemen ta r y Darb oux transformation fo r U, V (i.e., the form ulas (1 .4) with D given b y (4.15)) reads: ˜ U = M 1 λ − λ 1 + λ N  0 1 0 0  N − 1 − N  − α λ 1 1 α  N − 1 + N , 1 N − 1 , ˜ V = M 2 λ − λ 1 +  4 λ 2 + bλ  N  0 1 0 0  N − 1 − 4 λ N  − α λ 1 1 α  N − 1 + ˜ V 0 , (4.20) where M 1 =  u − λ 1 − α 2 − α , 1  N  0 0 1 0  N − 1 , M 2 =  − α, 2 + c + 2 aα − bα 2 − 4 λ 2 1 + 2 λ 1 ( u − 2 α 2 )  N  0 0 1 0  N − 1 , (4.21) and ˜ V 0 do es not dep end on λ (its explicit form follo ws from Lemma 4.2 a nd, therefore, is a utomatically preserv ed b y the D arb oux tra nsformation). The necessary condition for the Darb oux transformation is v anishing of residua M 1 , M 2 (what is equiv alen t to (4.16) and, a s a consequence, to the Riccati equations (4.19)). 19 In o rder t o assure the Darb oux in v ariance of the co eﬃcien t s b y λ in U and by λ 2 in V w e ha v e to imp ose some constraints on the normalization matrix N (compare [41]), namely N  0 1 0 0  N − 1 =  0 0 − 1 0  , (4.22) what implies the following form of N : N = f  0 1 − 1 − γ  , (4.23) where f , γ a re functions of x . No w, the transformation (4.2 0) b ecomes ˜ U =  γ − α 1 ˜ u − λ α − γ  + f , 1 f  1 0 0 1  , ˜ V = 4 λ  γ − α 1 ˜ v − λ α − γ  + f , 2 f  1 0 0 1  + ˜ V 0 , (4.24) where ˜ u = λ 1 + 2 γ α − γ 2 − γ , 1 , ˜ v = λ 1 + 2 γ α − γ 2 − 1 4 b . (4.25) Comparing (4.24) with (4.13) w e ﬁnd the remaining constrain ts on the form of the D arb oux matrix: f = const , γ = α , 2 ˜ v = ˜ u . (4.26) W e assume f = − 1. By virtue of (4.1 4) b = 2 u , and w e easily v erify that the constrain t 2 ˜ v = ˜ u coincides with the ﬁrst Riccati equation (4.19). Corollary 4.3. The elementary Darb oux matrix for the KdV e quation is given by D =  0 1 − 1 − α   λ − λ 1 0 − α 1  =  − α 1 α 2 − λ + λ 1 − α  , (4.27) wher e α i s c o m pute d fr om (4.16), se e als o (4 .18). 20 The t r a nsformation of u can b e obtained f rom (4.19) and (4.25), see (4.28). T aking into accoun t (4.12) w e g et the transformation for ψ . Corollary 4.4. The elemen tary D arb oux matrix (4.27) gen er ates the cla ssi- c al Darb oux tr ansformation: ˜ ψ = ψ , 1 − αψ ≡ ψ , 1 − (ln ˆ ψ 1 ) , 1 ψ , ˜ u = u − 2 α, 1 ≡ u − 2(ln ˆ ψ 1 ) , 11 , (4.28) wher e ˆ ψ 1 = ˆ ψ ( x, λ 1 ) satisﬁes (4.9) . F ormu las (4.28) were ﬁrst obtained b y Gaston Darb oux [23], see also [46]. Prop osition 4.5. D ≡ D α,λ 1 given by (4.27) h as the fol lo w ing pr op erties: D − 1 α,λ 1 is e quivalent to D − α,λ 1 and D β ,λ 2 D α,λ 1 = N ( λ − λ 1 + M ) , wher e M = ( λ 1 − λ 2 ) P (and P 2 = P ) for λ 2 6 = λ 1 and M 2 = 0 for λ 2 = λ 1 . Pro of: b y straightforw ard computation. First, D − α,λ 1 = ( λ 1 − λ ) D − 1 α,λ 1 . Then, D β ,λ 2 D α,λ 1 =  − 1 0 α + β − 1  ( λ − λ 1 + M ) (4.29) where M =  − α ( α + β ) α + β α ( λ 1 − λ 2 ) − α 2 ( α + β ) α ( α + β ) − ( λ 1 − λ 2 )  , (4.30) and w e easily v erify that M 2 = ( λ 1 − λ 2 ) M , whic h means that (for λ 2 6 = λ 1 ) M = ( λ 1 − λ 2 ) P (where P 2 = P ), compare (2.5). ✷ 4.4 Nilp oten t Darb oux matri x and c lassical b inary Dar- b oux transformation Let us consider the Darb oux matrix of the form (2.3). In the case n = 2 the nilp oten t matrix M ( M 2 = 0) can b e parameterized as M = g  − σ 1 − σ 2 σ  (4.31) where g , σ are some functions. 21 Considering the tra nsfor mat ion (1.4) w e hav e to demand that ˜ U ν are regular at λ 1 b y cancelling the p ole of second order at λ = λ 1 . W e get t wo conditions: M , ν +[ M , U ν ( λ 1 )] − M U ′ ν ( λ 1 ) M = 0 , M , ν M + M U ν ( λ 1 ) M = 0 , (4.32) where the pr ime denotes diﬀeren tiation with resp ect to λ . The second set of equations turns out to b e a consequence of the ﬁrst equations (it is enough to m ultiply them b y M from the righ t). The classical binary Darb oux transformation is usually deﬁned only for the time-independent sp ectral problem [46]. Therefore, in order to sho w that the considered transformation (2.3) coincides with the classical binary transformation it is suﬃcien t to conﬁne ourselv es to ν = 1. The equations (4.32) (fo r ν = 1) can b e rewriten in terms of g , σ : g , 1 − 2 σ g + g 2 = 0 , g , 1 σ + g σ, 1 + σ g 2 − g ( u − λ 1 + σ 2 ) = 0 , g , 1 σ 2 + 2 g σ σ , 1 + σ 2 g 2 − 2 g σ ( u − λ 1 ) = 0 , (4.33) Using the ﬁrst equation we can reduce the last t wo equations to: σ , 1 + σ 2 − u + λ 1 = 0 . (4.34) Therefore σ = ˆ ψ 1 , 1 ˆ ψ 1 , (4.35) where ˆ ψ 1 satisﬁes the ﬁrst equation o f (4.9) for λ = λ 1 , compare (4.18), (4.19). T aking in to accoun t (4.3 2) we rewrite (1.4) for U 1 ≡ U = u 0 λ + u 1 as ˜ u 0 = N u 0 N − 1 , ˜ u 1 = N , 1 N − 1 + N ( u 1 + [ M , u 0 ]) N − 1 . (4.36) In the KdV case, see (4.13), the ﬁrst equation o f (4.36) is satisﬁed fo r N =  1 0 γ 1  . (4.37) 22 Then, the second equation of (4.36) r educes to: ˜ u = u + γ , 1 +2 σ γ + γ 2 , g = − γ . (4.38) T aking into accoun t the ﬁrst equation o f (4.33) w e ﬁnally get g et ˜ u = u + 2 γ , 1 , γ , 1 − 2 σ γ − γ 2 = 0 , (4.39) where σ is giv en by (4.35). Therefore, the last equation is equiv alen t to: ∂ ∂ x ˆ ψ 2 1 γ ! = − ˆ ψ 2 1 , (4.40) whic h means that γ = ˆ ψ 2 1 c 0 − R ˆ ψ 2 1 , ˜ u = u − ∂ 2 ∂ x 2 ln     c 0 − Z ˆ ψ 2 1     , (4.41) where c 0 is a constan t of in t egration. The last form ula coincides with t he classical binary Darb oux tra nsforma t ion fo r the Sturm-Liouville-Sc hr¨ odinger sp ectral problem [46]. Corollary 4.6. The nilp otent Darb oux matrix (2.3) gener ates the cl a ssic al binary Darb oux tr ansforma tion . The “second” binary Darb o ux transformat io n, intro duced in [83], corre- sp onds to the choice c 0 = 1. 5 F ractional form o f the Darb oux matrix Another p o pular repres en tatio n of the Darb oux matrix (with nondegenerate normalization) is decomp osition into partial fractions [56, 79, 80, 82 ]: D = N ( I + A 1 λ − λ 1 + . . . + A N λ − λ N ) , D − 1 = ( I + B 1 λ − µ 1 + . . . + B N λ − µ N ) N − 1 . (5.1) In principle the num b ers of p oles of D a nd D − 1 could b e diﬀeren t but here, follo wing other pap ers, we assume the “symmetric” case (5.1 ) . 23 W e will denote by D 0 the Darb oux matrix in the fractional fo rm with the canonical normalization (in other w ords, D = N D 0 ). The form (5.1) of D and D − 1 imp oses restrictions on A k and B k implied b y equations D D − 1 = I and D − 1 D = I , see ( 5 .2). Multiplying D b y the least common m ultiple of the denominators w e obtain the equiv alent p olynomial form ˆ D ( λ ) (a p olynomial of N th degree). The determinant det ˆ D ( λ ) is a p olynomial of degree N n v anishing at p oles of D and D − 1 , i.e., at λ = λ k and λ = µ k ( k = 1 , . . . , N ). The sum of m ultiplicities of all zeros of det ˆ D ( λ ) equals N n . Therefore, for n = 2 all zeros are simple, while for n > 2 some o f them hav e to b e multiple zeros. The fra ctio na l for m is con v enien t in the case of some reductions (e.g., orthogonal or unitary), w hen the eigen v alues λ k ( k = 1 , . . . , nN ) can b e naturally divided in t o pairs λ k , µ k . 5.1 Zakharo v-Mikhailo v’s appr oac h W e start fro m fra ctional represen tation of the Darb oux matrix (5.1), where A k , B k ha ve to satisfy constrain ts resulting from the condition D D − 1 = I : A k I + N X j =1 B j λ k − µ j ! = 0 , I + N X j =1 A j µ k − λ j ! B k = 0 , I + N X j =1 B j λ k − µ j ! A k = 0 , B k I + N X j =1 A j µ k − λ j ! = 0 , (5.2) ( k = 1 , . . . , N ). W e assume the nonisosp ectral case and demand tha t ˜ U ν deﬁned by ( 1 .4) hav e the same form as U ν . In particular, it means tha t the righ t-hand sides of (1.4) hav e no p oles. Equating to zero the residua a t λ = λ j and at λ = µ k , w e get ( A j , ν + A j U ( λ j )) I + N X i =1 B i λ j − µ i ! + ( L ν ( λ j ) − λ j , ν ) N X i =1 A j B i ( λ j − µ i ) 2 = 0 , I + N X i =1 A i µ k − λ i ! ( U ( µ k ) B k − B k , ν ) − ( L ν ( µ k ) − µ k , ν ) N X i =1 A i B k ( µ k − λ i ) 2 = 0 , (5.3) 24 for j, k = 1 , . . . , N . Multiplying ﬁrst equations b y A j from the righ t and the second equations by B k from the left, and t hen using (5.2), w e obtain ( L ν ( λ j ) − λ j , ν ) N X i =1 A j B i ( λ j − µ i ) 2 = 0 , ( L ν ( µ k ) − µ k , ν ) N X i =1 A i B k ( µ k − λ i ) 2 = 0 , (5.4) whic h is satisﬁed when (3.4) (and similar equations f o r µ k ) hold. Note that w e deriv ed here a prop osition analogical to Theorem 3.4. In order to solv e the system (5.2) , (5.3) w e assume (3.4) and represen t A k , B k as follows: A k = | s k ih a k | , B k = | b k ih q k | (5.5) where | s k i , | b k i are matrices built o f linearly indep enden t n -comp onent column v ectors and h q k | , h a k | are matrices built of linearly indep endent n -comp onen t ro w v ectors. In o ther words, a ll these matrices ha ve maximal rank. In particular, | s k i and h a k | ha v e the same rank (denoted b y rk A k ) but (in general) diﬀerent than the rank of | b k i and h q k | (denoted b y rk B k ). Using the no tation (5.5) w e rewrite equations (5.2) and (5.3) as follo ws: | s k ih a k | D − 1 0 ( λ k ) = 0 , D 0 ( µ k ) | b k ih p k | = 0 , D − 1 0 ( λ k ) | s k ih a k | = 0 , | b k ih p k | D 0 ( µ k ) = 0 , (5.6)  | s k i , ν h a k | + | s k ih a k | , ν + | s k ih a k | U ν ( λ k )  D − 1 0 ( λ k ) = 0 , D 0 ( µ k )  − | b k i , ν h q k | − | b k ih q k | , ν + | b k ih q k | U ν ( µ k )  = 0 , (5.7) where k = 1 , . . . , N a nd ν = 1 , . . . , m . Moreo v er, D 0 ( µ k ) = I + N X j =1 | s j ih a j | µ k − λ j ! , D − 1 0 ( λ k ) = I + N X j =1 | b j ih q j | λ k − µ j ! . (5.8) Lemma 5.1. If | a i and h b | have the maximal r ank, then: | a ih b | = 0 ⇐ ⇒ | a i = 0 or h b | = 0 . (5.9) 25 Pro of: immediately follo ws fro m the deﬁnition of the maximal ra nk. All columns of | a i (and all ro ws of h b | ) hav e to b e linearly indp enden t. ✷ Using (5.6) and applying Lemma 5.1 to equations (5.7), w e g et t he fol- lo wing linear system: h a k | , ν = −h a k | U ν ( λ k ) , | b k i , ν = U ν ( µ k ) | b k i , (5.10) whic h is satisﬁed b y: h a k | = h a k 0 | Ψ − 1 ( λ k ) , | b k i = Ψ( µ k ) | b k 0 i , (5.11) where h a k 0 | and | b k 0 i are constant. If Ψ is regular at λ k and µ k , then t he solution given b y (5.11) is general (compare Section 2 .3). 5.2 Symmetric r epr esen tation of the Darb oux matrix W e pro ceed to deriv e compact form ulas for the remaining ingredien ts of D , namely fo r h q k | and | p k i . T aking into accoun t Lemma 5 .1 we can simplify equations (5 .6): h a k | + N X j =1 M k j h q j | = 0 , | b k i − N X j =1 | s j i M j k = 0 , h q k | + N X j =1 K k j h a k | = 0 , | s k i − N X j =1 | b j i K j k = 0 , (5.12) where M k j = h a k | b j i λ k − µ j , K j k = h q j | s k i µ j − λ k . (5.13) The expression h a k | b j i denotes matrix multiplication: h a k | b j i = h a k || b j i (for an y ﬁxed j, k ). The resulting matrix is not necessarily quadratic. The n umber of its columns is rk( B j ) and the n um b er o f its ro ws is rk( A k ). Simi- larly , h q j | s k i is also a matrix (for any ﬁxed j, k ). Remark 5.2. Matric es M j k form the so c al le d “soliton c orr elation matrix” ˆ M which ha s P N j =1 rk( B j ) c olumns an d P N k =1 rk( A k ) r ows. F r om (5.12) it fol lows that ˆ K = ˆ M − 1 . (5.14) 26 Ther efor e, ˆ M and ˆ K have to b e quadr atic matric es, i.e., N X k =1 rk( A k ) = N X j =1 rk( B j ) . (5.15) The soliton correlation matr ix ˆ M is a Cauc hy-lik e matrix (compare [53]) whic h has b een reobtained sev eral times in v arious particular cases (see, for instance, [12, 32, 62, 76 ]) . Corollary 5.3. The symmetric form of the m ultip ole Darb oux matrix is given by D ( λ ) = N I + N X k =1 N X j =1 | b j i K j k h a k | λ − λ k ! , D − 1 ( λ ) = I − N X k =1 N X j =1 | b j i K j k h a k | λ − µ j ! N − 1 , (5.16) wher e N is a norm a l i z ation ma trix (we assume det N 6 = 0 ), ˆ K = ˆ M − 1 , ˆ M is given by (5.13), a n d | b j i , h a j | ( j = 1 , . . . , N ) ar e expr esse d by (5.11). 5.3 Ho w to represen t N -soliton surfaces? Iterated Darb oux matrix is a comp osition of N binary Darb oux transforma- tions (see, for instance, [40, 56]): D = N  I + λ N − µ N λ − λ N P N  . . .  I + λ 2 − µ 2 λ − λ 2 P 2   I + λ 1 − µ 1 λ − λ 1 P 1  , (5.17) where pro jectors P k are deﬁned b y k er P k = Ψ k − 1 ( λ k ) , im P k = Ψ k − 1 ( µ k ) , (5.18) where Ψ k are deﬁned by: Ψ 0 ( x, λ ) = Ψ( x, λ ) and (fo r k > 1): Ψ k ( λ ) :=  I + λ k − µ k λ − λ k P k  Ψ k − 1 ( λ ) . (5 .1 9) 27 In this case (1.6) yields ˜ F = F + N X k =1 ( µ k − λ k ) λ, ζ ( λ − λ k )( λ − µ k ) Ψ − 1 k − 1 ( λ ) P k Ψ k − 1 ( λ ) (5.20) Note that the formula (5.20) do es not contain N . Indeed, gauge equiv a lent linear problems ha v e iden t ical soliton surfaces, see [7 3]. Remark 5.4. The determin ant of the iter ate d Darb oux matrix (5.17) c an b e e asily c ompute d (c omp ar e [14]): det D = det N N Y k =1  λ − λ k λ − µ k  dim im P k (5.21) Usually the f o rm ula (5.20) is used in the isosp ectral S U ( n ) case, when λ, ζ ≡ 1, µ k = ¯ λ k (see Section 6), and T r F = T r ˜ F = 0 (what can b e atta ined b y m ultiplying D b y an a ppropriate factor f , see (1.7)): ˜ F = F + N X k =1 2Im λ k | λ − λ k | Ψ − 1 k − 1 ( λ ) i  dim im P k n I − P k  Ψ k − 1 ( λ ) , (5.22) (in this case pro jectors ar e orthogonal, P † k = P k , see for example [14, 39, 74]). The sum on the right-hand side o f (5.22) consists of traceless components o f constan t length (using the Killing- Cartan form a · b = − n T r( ab ) as a scalar pro duct in su ( n )) , see [74]. Thus the form ula (5.22) generalizes the classical Bianc hi- L ie transformation for pseudospherical surfaces [72, 74]. The formula (5.20) is not manifestly symmetric with resp ect to p erm uta - tions of λ k . The symmetric f orm ula for the D arb oux-B¨ ac klund transforma- tion for soliton surfaces can b e obtained b y substituting (5.16) into (1.6). Theorem 5.5. The symmetric r epr esentation for N -soliton surfac es has the form: ˜ F = F − λ, ζ N X j =1 N X k =1 Ψ − 1 ( λ ) | b j i K j k h a k | Ψ( λ ) ( λ − λ k )( λ − µ j ) , (5.23) (the notation is ex plaine d in Cor ol lary 5.3, se e also Se ction 1.4). 28 Pro of. W e compute D − 1 D , λ where D is giv en by (5.16): − D − 1 D , λ = N X j,k =1 | b j i K j k h a k | ( λ − λ k ) 2 − N X i,j,k ,l =1 | b j i K j i h a i | b l i K lk h a k | ( λ − λ k ) 2 ( λ − µ j ) . W e use (5.13) and perform the summation o ver i, l in the sec ond comp onent: N X i,l =1 K j i ( λ i − µ l ) M il K lk = N X i =1 K j i λ i ˆ δ A ik − N X l =1 ˆ δ B j l µ l K lk = K j k ( λ k − µ j ) , ( 5 .24) where ˆ δ A ik , ˆ δ B j l are natural g eneralizations of Kronec ke r’s delta (e.g., δ A k k is unit matrix of order rk( A k ) and δ B j j is unit matrix of order rk( B j )). Finally , by virtue of an o b vious iden tit y ( λ − µ j ) − ( λ k − µ j ) = λ − λ k , w e get (5.23). ✷ The expression (5.2 3) is a generalizatio n of the symmetric formulas fo r N -soliton surfaces whic h has b een earlier obtained in t he su (2)- AKNS case ([12], see also [16]). Prop osition 5.6. The Da rb oux matrix f D , w i th D given by (5.16) an d f given by f = n v u u t N Y k =1 ( λ − λ k ) rk A k ( λ − µ k ) rk B k (5.25) tr ansforms tr ac eless F into tr ac eless ˜ F . What is mor e, det( f D ) = det N . Pro of: If D is giv en b y (5.16), then, using (5.23), we compute: T r( F − ˜ F ) λ, ζ = T r N X j,k =1 h a k | b j i K j k ( λ − λ k )( λ − µ j ) ! = N X k =1  rk( A k ) λ − λ k − rk( B k ) λ − µ k  , (5.26) where w e to ok in to accoun t that P N j =1 M k j K j k is the unit matr ix of order rk( A k ) and P N k =1 K j k M k j is the unit matrix of order rk( B j ). Multiplying D b y a λ -dep enden t function w e can c hange ˜ F − F , b y virtue of (1.7). In order to get T r ˜ F = T r F , we ha ve to tak e f such that the righ t- ha nd side of (5.26) equals n (ln f ) , λ . Hence w e get (5.25 ) . 29 Surprisingly enough, in this wa y w e can compute also the determinan t of D giv en by (5.16). Indeed, fro m (1.6) we ha v e T r(( f D ) − 1 ( f D ) , ζ ) = 0 (pro vided that T r ˜ F = T r F ) and then Theorem 3.2 implies that det( f D ) do es not dep end on ζ (and is λ -indep enden t, as w ell). Therefore we can ev aluate det( f D ) at λ = ∞ . Th us we obtain det( f D ) = det N . ✷ Let ˜ Ψ = f D Ψ, where D is giv en b y (5.16) and f is giv en by (5.25). F or simplicit y w e assume also rk( A k ) = r k( B k ) = r k . Then: ˜ F = F + λ, ζ Ψ − 1 ( λ ) N X j,k =1  ( λ k − µ k ) r k δ j k − | b j i K j k h a k | ( λ − λ k )( λ − µ j )  Ψ( λ ) , (5.27) where δ j k is K ronec ke r’s delta. Not e that the symmetric form o f (5 .2 2) is giv en b y the specialization of the formula (5.27) to the case µ k = ¯ λ k . Another represen ta t ion for m ultisoliton surfaces can b e deriv ed from the p olynomial represen t ation o f D . In order to compute N -soliton addition to the surface F := Ψ − 1 Ψ , λ | λ = λ 0 w e assume the Da rb oux matrix in a general form D = N X k =0 T k ( λ − λ 0 ) k . The matrices T 1 , . . . , T N are computed f rom the follo wing linear system: N X k =0 T k ( λ ν − λ 0 ) k Ψ( λ ν ) p ν = 0 , ( ν = 1 , . . . , N n ) , where λ ν ∈ C and p ν ∈ C n are constan t, and T 0 is a giv en normalization matrix. Of course, o ne should take care of reductions what can result in some constrain ts on λ ν , p ν and also on T 0 , see Section 6. The form ula (1 .6) assumes the fo rm: ˜ F = F + Ψ − 1 ( λ ) T − 1 0 T 1 Ψ( λ ) ≡ F + Ψ − 1 ( λ ) θ 1 Ψ( λ ) , (5.28) where θ 1 is giv en b y (3.16). Not e that also in this case ˜ F do es not dep end on the normalization matr ix T 0 (a c hange o f T 0 implies suc h change of T 1 that θ 1 remains unchanged, compare Section 3.3). 30 6 Group r e ductio n s The so called reduction group w as in tro duced b y Mikhailo v [4 9] and detailed description of v arious reductions is giv en in [5 0, 80], see also [1 4]. Group reductions (under a diﬀeren t name) f ound a r igorous treatmen t in the fra me- w ork of the lo op group theory [29, 59, 76], compare Section 7.3 . In this section w e describe sev eral imp o rtan t t yp es of r eduction gr o ups. W e consider only the case of non-degenerate no rmalization det N 6 = 0 (whic h means that det ˆ D ( λ ) is a p olynomial of degree N n ). The most conv enien t form o f the Darb o ux matrix dep ends on the r eduction. The p olynomial fo rm (3.1) is v ery go o d fo r reductions to twis ted g r o ups, the f r a ctional form (5.1 ) (and esp ecially its symmetric ve rsion (5.16)) is a ppro priate f o r unita ry and orthogonal reductions. Then, as an example, w e presen t in more detail the principal chiral mo del (sigma mo del) and its reductions. The symmetric form (5.16) is of great adv antage in this case. 6.1 Reductions to t wisted lo op groups Twisted lo op groups are deﬁned b y Ψ( ω λ ) = Q Ψ( λ ) Q − 1 where ω = exp 2 π i K (hence ω K = 1) and, necessarily , Q K = I (we assum e also Q = const), compare [29]. An imp ortan t example, tw o dimensional T o da c hain (then K = n ), is discussed in detail in [50], using the fractional represen tation of D (5.1). Here w e presen t a diﬀerent a ppro ac h, based on the p olynomial represen tation. Usually it is b etter to consider some natural extensions of lo op groups whic h follo w f rom the for m of the linear problem. The starting p oin t is the assumption ab out the form of the linear problem (i.e., U ν are constrained to the corresp onding Lie algebra): U ν ( ω λ ) = QU ν ( λ ) Q − 1 , (6.1) whic h implies (Ψ ( ω λ )) , ν = U ν ( ω λ )Ψ( ω λ ) = QU ν ( λ ) Q − 1 Ψ( ω λ ). Hence  Q − 1 Ψ( ω λ )  , ν = U ν ( λ )  Q − 1 Ψ( ω λ )  , whic h means (see Remark 1.4) that Q − 1 Ψ( ω λ ) = Ψ( λ ) C 0 ( λ ), where the matrix C 0 ( λ ) do es not dep end on x . Therefore Ψ( ω λ ) = Q Ψ( λ ) C 0 ( λ ) , (6 .2) 31 and similar equation for ˜ Ψ = ˆ D Ψ (with a diﬀeren t ˜ C 0 ( λ ), in general). There- fore: ˆ D ( ω λ ) Q Ψ( λ ) C 0 ( λ ) = Q ˆ D ( λ )Ψ( λ ) ˜ C 0 ( λ ). In order to eliminate Ψ( λ ) w e ha ve to assume tha t ˜ C 0 ( λ ) = γ 0 ( λ ) C 0 ( λ ), where γ 0 : λ → γ 0 ( λ ) ∈ C is a rational complex function of λ . Then ˆ D ( ω λ ) = γ 0 ( λ ) Q ˆ D ( λ ) Q − 1 , det ˆ D ( ω λ ) = ( γ 0 ( λ )) n det ˆ D ( λ ) . (6.3) Remark 6.1. Computing ˆ D ( ω 2 λ ) ,. . . , ˆ D ( ω K λ ) we obtain a ne c es s a ry c on- str aint for γ 0 : γ 0 ( λ ) γ 0 ( ω λ ) . . . γ 0 ( ω K − 1 λ ) = 1 . (6.4) This c onstr aint is sa tisﬁe d b y an y m e r om orphic function such that γ 0 ( ∞ ) = 1 and al l its zer os and p oles c oincide with some zer os of det ˆ D ( λ ) . Note that the matrix C 0 ( λ ) also is not arbitr ary but satisﬁes an analo gic al c onstr aint. W e mak e usual assumptions: γ 0 ( λ ) ≡ 1 and C 0 ( λ ) ≡ Q − 1 (then Ψ and ˆ D are ﬁxed p oin ts of the reduction group [50 ], or , in other w ords, ˆ D and Ψ tak e v alues in the lo op group). Then Ψ( ω k λ ) = Q k Ψ( λ ) Q − k , ˆ D ( ω k λ ) = Q k ˆ D ( λ ) Q − k , (6.5) for k = 1 , . . . , K − 1. Lemma 6.2. L et γ 0 ( λ ) ≡ 1 . If det ˆ D ( λ 1 ) = 0 , then det ˆ D ( ω k λ 1 ) = 0 for k = 1 , . . . , K . Multiplicities of al l these K zer os ar e identic al. Therefore, if ˆ D ( λ ) is a p olynomial of order N (and, as a consequence, det ˆ D ( λ ) has the o r der N n ), then N n has to b e divided by K , i.e., there exists an in teger ˆ N such that N n = ˆ N K (in the t wo-dimensional T o da chain case ˆ N = N ). Moreov er, (6.5) (ev aluated at k = 1) imply t ha t ω N T 0 Q = QT 0 , (6.6) where T 0 is the normalization matrix, compare (3.1). W e hav e to demand that this equation has a solution T 0 6 = 0 (otherwise, the D arb oux matrix cannot b e a p olynomial of order N ). Certainly (6.6) has a solution fo r N suc h that ω N = 1 (in fact, this assumption w as done in [50]). 32 Corollary 6.3. If γ 0 ( λ ) ≡ 1 , then the set of zer os of det ˆ D ( λ ) is given by { ω k λ j | k = 0 , 1 , . . . , K − 1 ; j = 1 , . . . , ˆ N } , wher e λ j ∈ C . Equations (3.11), deﬁning the Darb oux matrix, can b e rewritten as fol- lo ws (taking into accoun t (6.5)): 0 = ˆ D ( ω k λ j )Ψ( ω k λ j ) p j k = Q k ˆ D ( λ j )Ψ( λ j ) Q − k p j k (6.7) F or simplicit y w e a ssume the generic case, i.e., all zeros ω k λ j are pairwise diﬀeren t (and, as a consequence, simple). Then the k ernels of ˆ D ( λ k ) are one-dimensional, whic h means that Ψ( λ j ) Q − k p j k is prop ortional to Ψ( λ j ) p j 0 . Theorem 6.4. Assuming that (6.6) has a solution T 0 6 = 0 we c onstruct the Darb oux matrix (a λ -p olynomial o f or der N ) ac c or ding to Cor ol l a ry 3.7 taking into a c c o unt that its zer os ar e given by ω k λ j (se e Cor ol lary 6.3) and the c orr esp onding eige nve ctors ar e r elate d by p j k = Q k p j 0 ( k = 0 , 1 , . . . , K − 1 ; j = 1 , . . . , ˆ N , wher e ˆ N = N n/K ). This Darb oux matrix pr eserves twiste d lo op gr oup c onstr aints (6.1) and (6.5), i.e., ˜ U ν ( ω λ ) = Q ˜ U ν ( λ ) Q − 1 , etc. Remark 6.5. In the nonisosp e ctr al c ase twiste d r e ductions imp ose c onstr aints on the form of L ν . If (3.4) ar e satisﬁ e d , then also ω λ k , ν = L ν ( x, ω λ k ) . Henc e we get the c ons tr aint: ω L ν ( x, λ ) = L ν ( x, ω λ ) . The particular case K = 2 (i.e., ω = − 1) is very popula r (e.g., this reduction is necessary to deriv e the standard linear problem fo r the famous sine-Gordon equation [56, 60], see also [18]) . This case can b e generalized b y admitting a λ -dep endence of Q (a ctually suc h generalization can b e done for an y K but the results ha ve more complicated form, so w e omit them). One can easily see that Q = Q ( λ ) has to satisfy Q ( − λ ) Q ( λ ) = ϑ 0 ( λ ) I , (6.8) where ϑ 0 is a scalar function suc h tha t ϑ 0 ( − λ ) = ϑ 0 ( λ ) (in particular, w e can t a k e ϑ 0 ( λ ) ≡ 1). Assuming γ 0 = 1 w e ha v e det ˆ D ( − λ ) = det ˆ D ( λ ) whic h means that zeros of det ˆ D ( λ ) app ear in pairs λ k ′ = − λ k . Constan t eigen ve ctors p k ′ and p k satisfy ˜ Ψ( λ k ) p k = 0 and ˜ Ψ( λ k ′ ) p k ′ = 0 which implies ˜ Ψ( λ k ) Q ( − λ k ) p k ′ = 0 . If the zero λ k is simple then p k ′ = Q ( λ k ) p k (the eigen ve ctors a re deﬁned up to a scalar constan t factor, therefore we omitted the facto r ϑ 0 ( λ k )). Moreo ver, the condition (6.6) should be replaced by ω N T 0 Q ∞ = Q ∞ T 0 , where Q ∞ is either Q ( ∞ ) or the co eﬃcien t by the highest p ow er of λ in the asymptotic expansion of Q ( λ ) fo r λ → ∞ . 33 6.2 Realit y condition The condition U ν ( λ ) = U ν ( ¯ λ ) (6.9) (where the bars denote complex conjuga t es) simply means that all co eﬃcien ts of matrices U ν are real. Considering (1.2) w e get Ψ( ¯ λ ) = Ψ( λ ) C ( λ ) (6.10) where C ( λ ) C ( ¯ λ ) = I . Applying (6.1 0) to ˜ Ψ( λ ) = ˆ D ( λ )Ψ( λ ) we o btain the corresp onding constraint on ˆ D ˆ D ( ¯ λ ) = γ ( λ ) ˆ D ( λ ) , (6.11) where γ ( λ ) is a scalar rational f unction whic h satisﬁes γ ( λ ) γ ( ¯ λ ) = 1. Hence ( γ ( λ )) n = det ˆ D ( ¯ λ ) det ˆ D ( λ ) , (6.12) whic h means that γ = w ( ¯ λ ) /w ( λ ), where w ( λ ) is a p olynomial of degree K 6 N (pro vided t ha t ˆ D is a p olynomial of degree N ). Then det D ( λ ) has K zeros of m ultiplicit y n and ( N − K ) n simple zeros. The set of simple zeros is in v arian t with respect to t he complex conjugation, i.e., they a r e either real or form pairs of conjugate n umbers. W e assume the simplest case γ ( λ ) ≡ 1 (and also C ( λ ) = ˜ C ( λ ) ≡ 1). Then all zeros o f det ˆ D ( λ ) are simple, and either λ j ∈ R (then ¯ p j = p j ) or there are pairs λ k ′ = ¯ λ k (then p k ′ = ¯ p k ). 6.3 Unitary reduc tions Unitary reductions (whic h sometimes a re also referred to as reality condi- tions, see for instance [76]) are deﬁned b y U † ¯ ν ( ¯ λ ) = − H U ν ( λ ) H − 1 , (6.13) where the dagger denotes the Hermitean conjugate, H is a constan t Her- mitean matrix ( H † = H ) and ¯ ν means complex conjug a te (necessary if x 1 , x 2 34 are complex , e.g., usually x 1 = z , x 2 = ¯ z in the case of c hiral mo dels, discusse d in Section 6.4). Using ( 6.13) w e obtain f rom (1.2): (Ψ † ( ¯ λ )) , ν ≡ (Ψ( ¯ λ ) , ¯ ν ) † = Ψ † ( ¯ λ ) U † ¯ ν ( ¯ λ ) ≡ − Ψ † ( ¯ λ ) H ( λ ) U ν ( λ ) H − 1 ( λ ) . (6.14) T aking in to account the w ell known formula for diﬀerentiating the inv erse matrix (i.e., (Ψ − 1 ) , ν = − Ψ − 1 Ψ , ν Ψ − 1 ) we transform (6.14) in to H − 1 ((Ψ † ( ¯ λ )) − 1 ) , ν = U ν ( λ ) H − 1 (Ψ † ( ¯ λ )) − 1 , (6.15) and, comparing ( 6.15) with (1.2), we get (Ψ † ( ¯ λ )) − 1 = H Ψ( λ ) C 0 ( λ ) , (6.16) where Ψ( λ ) solv es the system (1 .2) and C 0 ( λ ) is an x - indep enden t matrix. F rom ( 6 .16) we can deriv e C † 0 ( ¯ λ ) = C 0 ( λ ). ˜ Ψ( λ ) satisﬁes the constraint (6.16) with ˜ C 0 in t he place of C 0 . Assuming C 0 ( λ ) = k 0 ( λ ) ˜ C 0 ( λ ), whe re k 0 ( λ ) is an x -indep enden t scalar function, we deriv e the condition ˆ D † ( ¯ λ ) = k 0 ( λ ) H ˆ D − 1 ( λ ) H − 1 , (6.17) whic h is necessary fo r ˆ D to b e t he Darb oux matrix in the case of unita r y reductions. W e po int out that k 0 ( λ ) has to b e a rational f unction. The simplest c hoice k 0 ( λ ) ≡ 1 is not p ossible. Indeed, from (6.17) w e obtain ( k 0 ( λ )) n = det ˆ D ( ¯ λ ) det ˆ D ( λ ) . (6.18) Hence k 0 ( λ ) is a p o lynomial of degree 2 N (provided that det T 0 6 = 0). More- o v er, k 0 ( ¯ λ ) = k 0 ( λ ) whic h means that k 0 ( λ ) is a p o lynomial with real co ef- ﬁcien ts. The set of its zeros is symmetric with resp ect to the real a xis. W e conﬁne ourselv es to the case of k 0 ( λ ) without real ro ots, i.e. k 0 ( λ ) = | det T 0 | 2 ( λ − λ 1 )( λ − ¯ λ 1 ) . . . ( λ − λ N )( λ − ¯ λ N ) where N is the degree of the p olynom ˆ D ( λ ). Then det ˆ D = (det T 0 )( λ − λ 1 ) n − d 1 ( λ − ¯ λ 1 ) d 1 . . . ( λ − λ N ) n − d N ( λ − ¯ λ N ) d N , where d k are some integers, 1 6 d k 6 n − 1 . Note that for n > 2 the zeros of det ˆ D are, as a rule, degenerated. If λ k are pairwise diﬀeren t, then dividing ˆ D by ( λ − λ 1 ) . . . ( λ − λ N ) we o btain t he matrix D (equiv alen t to ˆ D ) b o unded 35 for λ → ∞ (lim λ →∞ D ( λ ) = T 0 ), with singularities at λ = λ k ( k = 1 , . . . , N ). T aking into accoun t det T 0 6 = 0, w e get: D = ˆ D ( λ ) ( λ − λ 1 ) . . . ( λ − λ N ) = N I + N X k =1 A k λ − λ k ! , (6.19) where N = T 0 and A k are some matrices dep enden t on x . The in v erse matrix D − 1 has p oles at λ = ¯ λ k ( k = 1 , . . . , N ). Therefore D is exactly of the form (5.1) with µ k = ¯ λ k . Note that (6.17) can b e rewritten as: D − 1 ( λ ) = H − 1 D † ( ¯ λ ) H . ( 6 .20) In what follo ws w e use natura l notation: | a † i := ( h a | ) † , h b † | := ( | b i ) † . Theorem 6.6. The Darb oux matrix of the form (5. 1 6), satisfying the addi- tional c onstr aints µ k = ¯ λ k , | b k 0 i = C 0 ( ¯ λ k ) | a † k 0 i , N † H N = H , rk A k = rk B k , (6.21) ( k = 1 , . . . , N ), pr eserve s the unitary r e duction deﬁne d by (6.13) and (6.16). Pro of: W e will sho w that the constrain ts (6.21), imp osed on D giv en b y (5.16), are suﬃcien t to satisfy the equation (6.20). The condition µ k = ¯ λ k is a lready assumed. Equating normalization matrices in (6.20) we obtain N − 1 = H − 1 N † H . The most con v enien t wa y to pro ceed further is to use symmetric form of the D arb oux matrix (5.16). Equating residua at bo t h sides of (6 .20) w e get B k = A † k (compare (5.1) ) , i.e., N X j =1 | b k i K k j h a j |N − 1 = − N X j =1 H − 1 | a † i K † j k h b † j |N † H . (6.22) In order to satisfy this equation it is suﬃcien t to require | a † k i = H | b k i , (6.23) what implies h b † k | = h a k | H − 1 . Indeed, using (5.13) w e get M † j k = − M k j , and, as a consequence, K † j k = − K k j , compare (5.14). ✷ 36 Remark 6.7. Assuming C 0 ( λ ) = H − 1 we r ew ri te the c ons tr aint (6.16) as Ψ † ( ¯ λ ) H Ψ( λ ) = H , i . e . , Ψ( λ ) takes values in the same lo op gr oup as D ( λ ) , c omp ar e (6.20). This assumption is n ot very r estrictive. It is suﬃci e n t to imp ose it on initial data (at x = x 0 ). Then it holds for any x . Remark 6.8. In the non-isos p e ctr al c ase the unitary r e duction imp os e s c on- str aints L ν ( x, ¯ λ ) = L ν ( x, λ ) on the evolution of λ , c omp ar e R ema rk 6.5. Remark 6.9. By virtue of Pr op osition 5.6 the Darb oux-B¨ acklund tr ansfor- mation for the r e duction S U ( n ) is gener ate d b y the Darb oux matrix f D , wher e D is given by The or em 6.6 and f i s given by (5.25) with rk B k = rk A k and µ k = ¯ λ k , W e p oint o ut tha t the case of k 0 ( λ ) with real ro ots is more diﬃcult. Usually , the assumption is made that λ k are not real, compare [27, 50 , 7 6]. The case of re al λ 1 , µ 1 is solv ed and dis cussed in the case of the binary Darb oux matrix ( N = 1), see [14]. By iterations o ne can obta in more general solutions. Ho w ev er, it w ould b e in teresting to obtain a compact form the Darb oux matrix corresponding to arbitrary set of eigen v alues (symmetric with resp ect to the r eal axis). 6.4 Chiral ﬁelds or harmonic maps As an illustrative example, w e will consider the equation (Φ , 1 Φ − 1 ) , 2 +(Φ , 2 Φ − 1 ) , 1 = 0 , (6.24) whic h des crib es harmonic maps on Lie groups (provide d that Φ ass umes v alues in a Lie group G ) [76, 77] or, in ph ysical con t ext, principal chiral ﬁelds [30, 79]. Adding a nother constrain t, Φ 2 = I , w e get chiral ﬁelds (or sigma mo dels) on symmetric (or Grassmann) spaces [1, 3 1, 61, 79]. The chiral mo del ( 6.24) is integrable and the asso ciated isosp ectral Lax pair Ψ , ν = U ν Ψ is of the fo rm [79]: Ψ , 1 = A 1 1 − λ Ψ , Ψ , 2 = A 2 1 + λ Ψ . (6.25) The Lax pairs considered in [27] and [77] a r e equiv alen t to (6.25) mo dulo simple transformat io ns of the parameter λ . It is con v enien t to denote Φ( x ) = Ψ( x, 0) . (6.26) 37 Then A ν = Φ , ν Φ − 1 or, in o ther w ords, U ν ( λ ) = Φ , ν Φ − 1 1 + ( − 1) ν λ , (6.27) and the compatibility conditions, A 1 , 2 + A 2 , 1 = 0 , A 1 , 2 − A 2 , 1 +[ A 1 , A 2 ] = 0 , (6.28) rewritten in t erms of Φ b ecome iden tical with (6 .2 4). The Darb o ux-B¨ ac klund tra nsformation for Φ (in the case o f the principal GL ( n, C ) c hiral mo del, where there a re no restrictions on Φ except non- degeneracy) is giv en b y ˜ Φ = D (0)Φ (6 .2 9) where D ( λ ) is represen ted, f o r instance, b y the symmetric form ula (5.16). The U ( n ) reduction is deﬁned by the constrain t Φ † Φ = I and adding det Φ = 1 w e get S U ( n ) principal sigma mo del, see fo r instance [3 0 , 77]. These constrain ts are preserv ed b y an appro pria tely mo diﬁed Darb oux ma- trix, see Theorem 6.6 and Remark 6.9. Chiral mo dels on G rassmann spaces can b e c haracterized by the addi- tional constraint: Φ 2 = I . This is a quite non-tr ivial reduction, worth while to b e considered in detail. Prop osition 6.10. Th e L ax p air (6.25) s a tisﬁes the c onstr aints U ν ( λ − 1 ) = Φ , ν Φ − 1 + Φ U ν ( λ )Φ − 1 , ( ν = 1 , 2) (6.30) if and only if Φ 2 = const . Pro of is straigh tforw ard. W e c hec k that Φ U ν Φ − 1 = − U ν iﬀ Φ 2 = const. Then, we compute U ν ( λ − 1 ) = λ Φ , ν Φ − 1 λ + ( − 1) ν = Φ , ν Φ − 1 − U ν ( λ ) = Φ , ν Φ − 1 + Φ U ν ( λ )Φ − 1 , what ends the pro of. ✷ The righ t-hand side of (6.30) has the form of a gauge transformat io n. Hence, w e immediately ha v e the f o llo wing conclusions. 38 Corollary 6.11. I f Φ 2 = I , then (Φ − 1 Ψ( λ − 1 )) , ν = U ν ( λ )Φ − 1 Ψ( λ − 1 ) , (6.31) which me ans that Ψ( λ − 1 ) = ΦΨ( λ ) S 0 ( λ ) , (6.32) wher e S 0 ( λ ) is a c onstant matrix. One c an che ck that S 0 ( λ − 1 ) = S − 1 0 ( λ ) . Prop osition 6.12. Th e D arb oux tr ansformation pr ese rves (6.30) if D ( λ − 1 ) = ˜ Φ D ( λ )Φ − 1 , ˜ Φ = D (0)Φ . (6.33) Pro of. The constrain t (6.32) for ˜ Ψ = D Ψ reads D ( λ − 1 )Ψ( λ − 1 ) = ˜ Φ D ( λ )Ψ( λ ) S 0 ( λ ) . Using (6.32) w e obtain (6.3 3). F inally , we apply (6.26). ✷ Corollary 6.13. The form ula (6.33) i m plies that divisor of p oles of D ( λ − 1 ) has to b e exactly the s a me as divisor o f p o les o f D ( λ ) . Inverting (6.33) w e get that divisors of p oles of D − 1 ( λ − 1 ) and D − 1 ( λ ) also should c oincide. Ther efor e b oth sets of p oles, i.e ., { λ 1 , . . . , λ N } and { µ 1 , . . . , µ N } , ar e invariant with r esp e ct to the inversion λ → λ − 1 . Theorem 6.14. We assume that p oles and zer os ( λ k , µ k ) of the Darb oux matrix (5.16) c an b e c om bine d in the fol lo w ing p a i rs λ k ′ = 1 λ k , µ k ′ = 1 µ k , (6.34) and λ 2 k 6 = 1 , µ 2 k 6 = 1 . We assume also N = I and h a j ′ 0 | = h a j 0 | S 0 ( λ j ) , | b j ′ 0 i = S − 1 0 ( µ j ) | b j 0 i . (6.35) Under these assumptions the Darb oux matrix (5.16) satisﬁes (6.33) and, mor e over, h a j ′ | = h a j | Φ − 1 , | b j ′ i = Φ | b j i . (6.36) 39 Pro of. W e are going to verify that assumptions of the theorem imply (6.3 3). First, we will sho w that assumptions (6.35) imply (6.36). Using (5.1 1), (6.34) and (6.35) we get: h a j ′ | = h a j ′ 0 | Ψ − 1 ( λ j ′ ) = h a j 0 | S − 1 0 ( λ j )Ψ − 1 ( λ j )Φ − 1 = h a j | Φ − 1 , | b j ′ i = Ψ( µ j ′ ) | b j ′ 0 i = ΦΨ( µ j ) S 0 ( µ j ) | b j ′ 0 i = Φ | b j i . (6.37) Then, we compute M j ′ k ′ = h a j ′ | b k ′ i λ j ′ − µ k ′ = h a j | b k i λ j µ k µ k − λ j = − λ j µ k M j k , (6.38) and, vic e versa , M j k = − λ j ′ µ k ′ M j ′ k ′ . Hence, K k ′ j ′ = − 1 µ k λ j K k j , K k j = − 1 µ k ′ λ j ′ K k ′ j ′ . (6.39) Assuming N = I we pro ceed to compute ingredients of the f o rm ula (6.33): D (0) = I − N X j,k =1 | b j i K j k h a k | λ k = I + N X j,k =1 Φ − 1 | b j i K j k h a k | Φ µ j , (6.40) where the second equalit y fo llo ws from: N X j,k =1 | b j i K j k h a k | λ k = − N X j ′ ,k ′ =1 λ k ′ Φ − 1 | b j ′ i  K j ′ k ′ µ j ′ λ k ′  h a k ′ | Φ , (6.41) (primes can b e dropped because w e su m ov er the same set o f indices). Then, w e compute D ( λ − 1 ) and decomp o se it in to the sum of partia l fractions: D ( λ − 1 ) = I − N X j,k =1 | b j i K j k h a k | λ k − N X j,k =1 λ − 2 k | b j i K j k h a k | λ − λ − 1 k . (6.42) Using (6.40), (6.36) and (6 .3 9), w e get (after dropping primes): D ( λ − 1 ) = D (0) + N X j,k =1 λ k Φ − 1 | b j i K j k h a k | Φ µ j ( λ − λ k ) . ( 6 .43) 40 Finally , ˜ Φ D ( λ )Φ − 1 = D (0) + N X j,k =1 Φ | b j i K j k h a k | Φ − 1 λ − λ k + N X i,j,k ,l =1 Φ − 1 | b j i W j k h a k | Φ − 1 µ j ( λ − λ k ) , (6.44) where W j k = N X i,l =1 K j i h a i | Φ 2 | b l i K lk = N X i,l =1 K j i M il ( λ i − µ l ) K lk = ( λ k − µ j ) K j k , (6.45) where w e used Φ 2 = I and (5.24). Substituting W j k in to (6.4 4) and compar- ing the result with (6 .4 3) w e get (6.33). ✷ Usually it is suﬃcien t to a ssume C 0 ( λ ) = H − 1 = const (compare Re- mark 6.7) and S 0 = const (but the assumption S 0 = I can b e to o restrictiv e). Prop osition 6.15. We assume S 0 = const , H = const , S 2 0 = I , H † = H and S † 0 H S 0 = H . We c onsider the Darb oux m atrix (5.1 6) such that N = I , N = 2 K and λ j + K = λ − 1 j , µ j + K = µ − 1 j , µ k = ¯ λ k , | λ j | 2 6 = 0 , ¯ λ j 6 = λ j , h a j 0 | = h b † j 0 | H , h a j + K, 0 | = h b † j + K, 0 | H , h a j + K, 0 | = h a j 0 | S 0 , S 0 | b j + K, 0 i = | b j 0 i , (6.46) wher e j = 1 , . . . , K , k = 1 , . . . , 2 K . Thus al l thes e data c an b e expr esse d by h a 10 | , . . . , h a K 0 | and λ 1 , . . . , λ K . The D a rb oux-B¨ acklund tr ansformation gener ate d by such D arb oux matrix pr eserves r e ductions: Ψ † ( ¯ λ ) H Ψ( λ ) = H and Ψ( λ − 1 ) = Ψ (0)Ψ( λ ) S 0 . Pro of: W e apply Theorems 6.6 and 6.14. It is enough to c hec k whether the equations h a j 0 | = h b † j 0 | H , h a j ′ 0 | = h b † j ′ 0 | H , h a j ′ 0 | = h a j 0 | S 0 , h b † j 0 | = h b † j ′ 0 | S † 0 , are not contradictory . These equations imply h a j 0 | H − 1 = h a j 0 | S 0 H − 1 S † 0 . Hence, using S 2 0 = I , w e obtain the constrain t S † 0 H S 0 = H assuring the compatibilit y of b oth reductions. Finally , w e denote j ′ = j + K . ✷ 41 7 Connec tions with other approac hes In this section we shortly presen t some other metho ds of constructing the Darb oux-B¨ ac klund transformatio n. W e show ho w they are connected with the approach presen ted in this pap er. 7.1 Matrix-v alued sp ectral parameter The name of Darb o ux ﬁrst app eared in the contex t of the dressing transfor- mations in Matve ev’s pap ers (see fo r instance [45 ]) who extended the notion of Dar b oux co v ariance, kno wn in the case of the Sturm-Liouville-Sc hr¨ odinger sp ectral problems, on a rbitrary diﬀeren t ial op erators [46 ]. In o r der to apply Matv eev’s approac h to Zakharo v-Shabat sp ectral prob- lems (1.2) the matrix sp ectral parameter is in t ro duced: Λ := diag( λ 1 , . . . , λ n ) (7.1) (this notat ion should not b e confused with the function Λ describ ed in Re- mark 1.1). W e consider the linear problem o f the form [6 , 46]: Ψ , ν = X j N j X k =1 U ν kj Ψ M k j + N X k =0 V ν k ΨΛ k , (7.2) where U ν kj and V ν k are matrices whic h do not dep end on λ 1 , . . . , λ n and M j := diag  1 λ 1 − a j , . . . , 1 λ n − a j  . The follow ing theorem holds [6 , 46]. Theorem 7.1. Equations (7.2) ar e c ovariant with r esp e ct to the D arb oux tr ansformation ˜ Ψ = Ψ Λ − σ Ψ , σ = Ψ 1 Λ 1 Ψ − 1 1 , (7.3) wher e Ψ 1 is a ﬁxe d solution to (7.2) with Λ r epla c e d b y the d iagonal matrix Λ 1 = diag ( λ 11 , . . . , λ n 1 ) . 42 The linear problem (7.2) is closely related to the follow ing special case o f the standard Zakharo v-Shabat linear problem (1 .2): Φ , ν = X j N j X k =1 U ν kj 1 ( λ − a j ) k Φ + N X k =0 V ν k λ k Φ . (7.4) Namely: Ψ(Λ) = { Φ( λ 1 ) p 1 , . . . , Φ( λ n ) p n } , (7.5) where the notation used o n the righ t-hand side (a matrix as a sequence of columns) is the same as in (2.13) and p 1 , . . . , p n form a constan t basis in C n . The D arb oux matrix generating the transformation (7.3) can b e easily computed (using D = ˜ ΨΨ − 1 ). W e get D (Λ) = ΨΛΨ − 1 − Ψ 1 Λ 1 Ψ − 1 1 . (7.6) If w e put λ 1 = . . . = λ n = λ , (i.e., Λ = λI ) , and p k ≡ e k form the canonical basis in C n (i.e., { p 1 , . . . , p n } = I ) , then Φ( λ ) = Ψ( λI ) ≡ Ψ( λ ). In this case w e obtain D ( λ ) = λI − Ψ 1 Λ 1 Ψ − 1 1 , (7.7) whic h is the starting p oint for the construction of the Da rb oux matrix b y Gu and his collab orat o rs [26, 27, 28, 84]. Sometimes another form is used: D = I − λ Ψ 1 Λ − 1 1 Ψ − 1 1 , (7.8) whic h is equiv alen t to (7.7) a fter c hanging λ → λ − 1 . 7.2 T ransfer matrix form of the Darb oux matrix A rational n × n matr ix function D ( λ ), analytic at inﬁnit y , can b e represen ted in the form [3, 25]: D ( λ ) = N + F ( λI N − A ) − 1 G , (7.9) where A is an N × N matrix, I N is the unit matrix of order N , and N , F, G are matrices of sizes n × n , n × N and N × n , resp ectiv ely . Suc h represen tation is called a “realization” or a “transfer matrix represen tation” of D and the 43 n umber N (i.e., the order of A ) is kno wn as the “state space dimension” of the realization. Realizations a re not unique and can hav e diﬀeren t v alues of the n um b er N . “Minimal realizations” hav e minimal v alue o f N (and the minimal N is called the McMillan degree of D ). Minimal realizatio ns are unique up to a change of the basis in the state space (i.e., F → F T − 1 , A → T AT − 1 and G → T G , for some in v ertible N × N matrix T ) [3, 25]. Prop osition 7.2. I f (7.9) is a r e alization for D , then one of r e a lizations for D − 1 is given by D − 1 ( λ ) = N − 1 − N − 1 F ( λI N − A + G N − 1 F ) − 1 G N − 1 . (7.10) The r e alization (7.10) is minimal iﬀ (7.9) is min i m al, se e [25]. The formu la (7 .1 0) can b e v eriﬁed b y a simple but non- trivial computatio n. The obvious iden tity ( λI N − A + G N − 1 F ) − ( λI N − A ) = G N − 1 F is v ery helpful, compare (7.16). Assuming N = I we consider the so called transfer matrix W A ( x, λ ) = I n − Π ∗ 2 S − 1 ( A − λI N ) − 1 Π 1 , (7.11) where A, S, Π 1 , Π ∗ 2 are some matr ices ( t he star denotes a matrix conjugate, but this is not v ery imp ortan t at this momen t) a nd, moreo v er, the follo wing op erator identit y holds: AS − S B = Π 1 Π ∗ 2 . (7.12) Matrices A, B , Π 1 , Π 2 , S satisfying (7.12) are said to for m an S - colligation [64]. The tra nsfer matrix (7.11 ) can b e used to generate solutions to integrable systems b y the Dar b oux-B¨ ac klund transformation, see [62, 63]. W e can make the following iden tiﬁcation: S = ˆ M , S − 1 = ˆ K , A = diag( λ 1 , λ 2 , . . . , λ N ) , B = diag( µ 1 , µ 2 , . . . , µ N ) (7.13) and, ﬁnally Π ∗ 2 = ( | b 1 i , | b 2 i , . . . , | b N i ) , Π 1 =      h a 1 | h a 2 | . . . h a N |      . (7.14) 44 Corollary 7.3. The symmetric r epr esentation of the Darb oux matrix (5.16) c an b e id entiﬁe d with the tr ansfer matrix form (7.11), wher e A is diagonal. The identity (7.12) c oinc i de with the deﬁnition (5.13) of the matrix ˆ M . Constan t matrices A of more general form corresp ond to generalizations of (5.1) ( m ultiple p oles are allow ed). In order to sho w a ﬂa vour of the transfer matrix tec hnique we presen t one of typ ical results. Not e that the pro of of Prop osition 7.4 is similar to some steps in the pro o f of Theorem 5 .5 . Prop osition 7.4. We assume the iden tity (7.12) an d D ≡ w A is given by the formula (7.11). Then D − 1 = I n + Π ∗ 2 ( B − λI N ) − 1 S − 1 Π 1 (7.15) Pro of: W e compute  I n − Π ∗ 2 S − 1 ( A − λI N ) − 1 Π 1   I n + Π ∗ 2 ( B − λI N ) − 1 S − 1 Π 1  = I n + Π ∗ 2 X Π 1 , where X is N × N matrix giv en by X = ( B − λ ) − 1 S − 1 − S − 1 ( A − λ ) − 1 − S − 1 ( A − λ ) − 1 ( AS − S B )( B − λ ) − 1 S − 1 . No w, using the ob vious iden tit y AS − S B = ( A − λ ) S − S ( B − λ ) , (7.16) w e decomp ose the last comp onent of X in to the sum of tw o terms whic h immediately cancel with the ﬁrst t w o comp onen ts of X . Therefore X = 0 whic h ends t he pro of. ✷ V ectorial Da rb oux tr a nsformations constitiue one more appro ac h to Dar- b oux transformations, applied mostly in 2 + 1-dimensional case [43, 44]. Al- though this tec hnique needs no analogue of the Darb oux matrix but the Darb oux transformation is expressed b y a Cauc hy-lik e matrix and imp o r t an t role is play ed by op erator iden tities lik e (7.12). Comparing the results of [44 ] and [62] w e conclude that b oth metho ds are in a v ery close corr esp o ndence (note that the matrix S of [62] corresp onds to the ma t rix Φ o f [44]). 45 7.3 F actorization in lo op group s Giv en a Lie group G w e deﬁne the lo op group of G as the group of smo ot h functions γ : S 1 → G , where S 1 denotes the unit circle on t he complex pla ne ( | λ | = 1) [29, 5 9]. An imp ortant role in the lo op group theory pla ys the Birkhoﬀ factorization theorem. The Birkhoﬀ decomp osition is closely related to the R iemann-Hilb ert problem whic h prov ides a rigorous bac kground for the inv erse scattering metho d [56], see also [29]. In general the Birkhoﬀ f actorization is not explicit. The explicit cases are closely related to the construction of Darb oux matrices [75, 76, 77] (and also to the construction of ﬁnite gap solutions), compare similar ideas in the soliton theory [35, 36]. The a ppro ac h based on t he so called cc-ideals is one more link b etw een the lo op gr o up theory and the theory of solitons [33, 34]. F rom geometrical p oint of view the Lax pair consists of comm uting diﬀer- en tial op erators a nd their compatibility can b e in terpreted as the conditio n that a one-par a meter family of connections is ﬂat: [ ∂ 1 − U 1 ( x, λ ) , ∂ 2 − U 2 ( x, λ )] = 0 (7.17) ( U 1 , U 2 are matrices depending on x through some ﬁelds , say u ). The “trivi- alization” E of a solutio n u is deﬁned as the solution of the system: E , ν = − E U ν , E (0 , λ ) = I . (7.18) Then E ( x, λ ) is holo morphic for λ ∈ C , see [76]. The function E ( x, λ ) is also referred to as an “extended solution”, an “ extended frame” o r simply a “frame”. Comparing (7.18) with (1.2) w e can iden tify E = Ψ − 1 . Actually , (7.18) is the adjoint of (1.2), see also (2.4). Theorem 7.5 (Birkhoﬀ ) . The multiplic ation map µ µ : L + ( GL ( n, C )) × L − ( GL ( n, C )) → L ( GL ( n, C )) is a diﬀe omo rp h ism o n to an op en dense subset of L ( GL ( n, C )) , whe r e • L + ( GL ( n, C )) is the gr oup of holomorp h ic ma p s h + : C → GL ( n, C ) • L − ( GL ( n, C )) is the gr oup of holomorphic ma ps h − : O ∞ → GL ( n, C ) such that h − ( ∞ ) = I , wher e O ∞ is a n e i ghb ourho o d o f λ = ∞ . • L ( GL ( n, C )) is the gr oup o f holomorphic ma p s fr om O ∞ ∩ C to GL ( n, C ) 46 Corollary 7.6. Supp ose that h − h + lies in the image of µ . Th e n, by virtue of the Birkhoﬀ the or em, ther e ex i s ts a unique p air f ± ∈ L ± ( GL ( n, C )) such that h − h + = f + f − . One c an interpr et it a s a “dr essin g action ” of h − on h + and f + is the r esult o f this action, which is d enote d by h − ♯h + = f + . The dressing action seems to “forget” a b out f − . Ho w ev er, it is w ort hwhile to stress that this is f − whic h should b e iden tiﬁed with our D arb oux matrix. On the ot her hand the elemen t h − is deeply hidden (almost non-existing) in other a pproac hes to the construction of Darb oux matrices. In order to explain the dressing action generated b y the Birkhoﬀ decomp osition w e will presen t t he binary D arb oux transformatio n (2.1) in the framew ork of the lo op gro up approac h, following [76]. W e assume that E ( x, λ ) ∈ L + ( GL ( n, C )) is giv en, and w e c ho ose the so called “simple elemen t” h λ 1 ,µ 1 ,π ∈ L − ( GL ( n, C )): h λ 1 ,µ 1 ,π ( λ ) = I + λ 1 − µ 1 λ − λ 1 π , (7.19) where λ 1 , µ 1 are complex parameters, and π is a constan t ( x -indep endent) pro jector in C n (i.e., π 2 = π ). One can easily see that h − 1 λ 1 ,µ 1 ,π = h µ 1 ,λ 1 ,π , compare (2.1) and (2.2). Then, the Birkhoﬀ theorem states that there exists ˜ E ∈ L + ( GL ( n, C )) and D ∈ L − ( GL ( n, C )) suc h that h λ 1 ,µ 1 ,π E ( x, λ ) = ˜ E ( x, λ ) D ( x, λ ) , (7.20) pro vided that the pro duct on the left-hand side b elongs to certain “o p en dense set” of L ( GL ( n, C )). Now, b oth the exact form of D and t his “o p en dense set” can b e f o und b y direct calculation. It is suﬃcien t (similarly as in all other approache s discussed earlier) to compare the residua at b oth sides of the equation (7.20). Hence D = I + λ 1 − µ 1 λ − λ 1 P , (7.21) where P is deﬁned b y (2 .11), where V k er = k er π and V im = im π . The op en dense set from the Birkhoﬀ theorem is deﬁned by : ker P ∩ im P = { 0 } . W e remark, by the wa y , t ha t the Birkhoﬀ theorem assumes the isosp ectral case and the canonical normalization ( N = I ). Note that (7.20) implies ˜ E = hE D − 1 = h Ψ − 1 D − 1 = ( D Ψ h − 1 ) − 1 (where h denotes the simple elemen t). Therefore, ˜ Ψ ≡ E − 1 = D Ψ h − 1 , whic h is equiv alen t (b ecause h do es not dep end o n x ) to the usual formula ˜ Ψ = D Ψ (compare Remark 1.4). 47 8 In v arian ts of the Darb oux transformation The Da rb oux transformation changes the matrices U ν in to new matrices ˜ U k of the same fo rm. By in v aria nts of the D arb oux tra nsformation w e mean constrain ts on co eﬃcien ts of U ν whic h are preserv ed by t he transformation, see [14] (compare also [69], where one may ﬁnd man y examples). The in- v arian ts are ve ry useful in the construction of D arb oux matrices in purely algebraic w ay , without referring to the sp ecial boudary conditions a nd to the scattering theory (whic h is a usual practice, compare [27, 76, 84]). Here w e simplify the a pproac h of [14] and extend it o n the non-isosp ectral p olynomial case. Moreo v er, w e show t ha t our a pproac h w orks also in muc h more general case: when the Lax pair is singular a t some ﬁxed v alues o f the sp ectral para meter. In this section w e denote U 1 = U , U 2 = V . 8.1 Linear inv arian ts for p olynomial Lax pairs W e consider Lax pairs with the f o llo wing λ -dep endence: U = ∞ X k =0 u k λ N − k ≡ λ N u , V = ∞ X k =0 v k λ M − k ≡ λ M v , (8.1) where N , M are ﬁxed p ositive in tegers (not to b e confused with the notation of previous sections) and u k = u k ( x ), v k = v k ( x ) ( k = 0 , 1 , 2 , . . . ). Usually the sums are ﬁnite (i.e., u k = v k = 0 for suﬃcien tly large k ), and this typical case (p olynomial in λ and λ − 1 ) corresp onds to man y classical soliton equations. In particular b oth U and V can b e p olynomials in λ (in this case, for N = 1 , w e get famous AKNS hierarc h y). W e also assume a similar λ -dep endence of the deriv ativ es of λ : λ, 1 = ∞ X k =0 a k λ N ′ − k , λ, 2 = ∞ X k =0 b k λ M ′ − k , (8.2) where N ′ , M ′ are g iven in tegers ﬁxed b y the assumption a 0 6 = 0, b 0 6 = 0 (in the nonisosp ectral case). The co eﬃcien ts a k = a k ( x ), b k = b k ( x ) hav e t o satisfy compatibilit y conditions resulting from λ, 12 = λ, 21 (some examples can b e f o und in [11, 14, 69 ]). W e consider the D arb oux transformat io n of U + H V , where H = H ( x, λ ) is a ﬁxed function H ( x, λ ) = λ N − M h ( x, λ ) ≡ λ N − M ( h 0 + h 1 λ − 1 + h 2 λ − 2 + . . . ) , (8.3) 48 where h 0 , h 1 , h 2 , . . . are give n functions of x . W e assume that H is unc hanged b y the Da rb oux tr a nsformation (and U, V are transformed, as usual, accord- ing to (1.4)). The Darb o ux transformation yields ( ˜ U + H ˜ V ) D = D , 1 + hD , 2 + D ( U + hV ) , (8.4) whic h reduces to ( ˜ u + h ˜ v ) D − D ( u + hv ) = λ − N D , 1 + hλ − M D , 2 . (8.5) W e assume that D is analytic at λ = ∞ : D = T 0 + T 1 λ − 1 + T 2 λ − 2 + . . . , det T 0 6 = 0 , (8.6) i.e., D = λ − N ˆ D , where ˆ D is given b y (3.1). The idea of linear in v ariants is quite ob vious. Supp o se that for λ ≈ ∞ the rig h t- hand side of (8.5) b eha v es as λ − K , where K > 1. Then the ﬁrst K terms of the T a ylor expansion (in λ − 1 ) of the left-hand side are equal to zero. The ﬁrst tw o of these equations read: ( ˜ u 0 + h 0 ˜ v 0 ) T 0 = T 0 ( u 0 + h 0 v 0 ) , ( ˜ u 1 + h 0 ˜ v 1 + h 1 ˜ v 0 ) T 0 + ( ˜ u 0 + h 0 ˜ v 0 ) T 1 = T 0 ( u 1 + h 0 v 1 + h 1 v 0 ) + T 1 ( u 0 + h 0 v 0 ) . The a sumption u 0 + h 0 v 0 = 0 implies ˜ u 0 + h 0 ˜ v 0 = 0 (provide d that det T 0 6 = 0). Then, adding the second assumption: u 1 + h 0 v 1 + h 1 v 0 = 0, w e obtain as a consequenc e ˜ u 1 + h 0 ˜ v 1 + h 1 ˜ v 0 = 0. Thus we ha v e tw o expressions in v aria nt with r esp ect to the Da rb oux t r ansformation. Considering the ﬁrst k (where k 6 K ) equations w e g et an inv ar ia n t system of k equations. W e pro ceed to estimate K . The leading terms of the rig ht-hand side of (8.5) are given b y: λ − N ( T 0 , 1 − a 0 T 1 λ N ′ − 2 + . . . ) + h 0 λ − M ( T 0 , 2 − b 0 T 1 λ M ′ − 2 + . . . ) (8 .7 ) Therefore K > k max 1 , where k max 1 = − 1 + min { N , M , N + 2 − N ′ , M + 2 − M ′ } , (8.8) what can b e summarized as follows. 49 Prop osition 8.1. Supp ose that 0 6 k 6 k max 1 and h 0 , h 1 , . . . , h k ar e given functions of x . Then the system of k + 1 line ar c onstr aints u j + j X i =0 h i v j − i = 0 , ( j = 0 , 1 , . . . , k ) , (8 .9) is invariant with r esp e ct to Darb oux tr ansformations such that det T 0 6 = 0 . In some sp ecial cases, w e can form ula t e stronger prop ositions (i.e., w e ha ve more inv ariants). In t he isosp ectral case w e can replace k max 1 b y k ′ max 1 = − 1 + min { N , M } , (8.10) (the same result is v alid when N ′ 6 2 and M ′ 6 2). In the case of the canonical normalizatio n ( T 0 = I ) we can replace k max 1 b y k ′′ max 1 = min { N , M , N + 1 − N ′ , M + 1 − M ′ } . (8.11) Belo w w e presen t one more example. Prop osition 8.2. Supp os e that min { M , N + 2 − N ′ , M + 2 − M ′ } > N (it implies , in p articular, k max 1 = N − 1 ), functions h 0 , h 1 , . . . , h k ( k 6 N ) ar e given, and T 0 assume values in some matrix Lie gr oup G . Then, the fol lowing system of k + 1 line ar c onstr aints is invaria nt w ith r esp e ct to the Darb oux tr ansformation: u j + j X i =0 h i v j − i = 0 , ( j = 0 , 1 , . . . , k − 1) , u k + k X i =0 h i v k − i ∈ g , (8.12) wher e g is the Lie algebr a of the Lie gr oup G . The pro of of t his prop osition is analogical to the pro of of Prop osition 8.1: w e consider co eﬃcien ts by p o we rs of λ − 1 in (8.5). Only the last step has to b e treated in a diﬀeren t w ay . Assuming that the ﬁrst k constraints hold, the co eﬃcien ts b y λ − k yield ˜ u k + k X i =0 h i ˜ v k − i = T 0 u k + k X i =0 h i v k − i ! T − 1 0 + δ k N T 0 , 1 T − 1 0 . (8.13) No w the pro of follows immediatelly from w ell kno wn prop erties of matrix Lie groups ( T g T − 1 ⊂ g and T , 1 T − 1 ∈ g , pro vided that T = T ( x ) ∈ G ). 50 8.2 Bilinear in v arian ts for p olynomial L ax pairs Assuming the p olynomial form ( 8 .1) of U, V w e consider the Dar b oux trans- forms of bilinear f o rms T r( U 2 ), T r( V 2 ) and T r( U V ). W e presen t computa- tions for the last case (the o ther t w o cases are analogical). In this setion w e use notation: A · B ≡ T r( AB ). F rom (1 .4) w e get T r( ˜ U ˜ V ) − T r ( U V ) = T r( D , 1 D − 1 D , 2 D − 1 + D , 1 V D − 1 + D , 2 U D − 1 ) . (8.14) The leading terms of the righ t-hand side of (8.14) r ead λ − ( N + M ) T r  ( T 0 , 1 − a 0 T 1 λ N ′ − 2 + . . . ) T − 1 0 ( T 0 , 2 − b 0 T 1 λ M ′ − 2 + . . . ) T − 1 0  , λ − N T r  ( T 0 , 1 − a 0 T 1 λ N ′ − 2 + . . . ) v 0 T − 1 0  , λ − M T r  ( T 0 , 2 − b 0 T 1 λ M ′ − 2 + . . . ) u 0 T − 1 0  . (8.15) Th us the right-hand side o f (8.1 4 ) b ehav es as λ − K , where K will b e estimated b elo w. W e assume that D and D − 1 are analytical at λ = ∞ (i.e., det T 0 6 = 0). Considering co eﬃcien ts b y λ − j ( j = 0 , 1 , 2 , . . . ) in the form ula (8.14), w e obtain the f o llo wing in v arian ts: f 0 := u 0 · v 0 , f 1 := u 0 · v 1 + u 1 · v 0 , f 2 := u 0 · v 2 + u 1 · v 1 + u 2 · v 0 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . f k := u 0 · v k + u 1 · v k − 1 + . . . + u k · v 0 , (8.16) where k < K . In order to form ulat e a mo r e precise statement, w e deﬁne k max 2 = min { k max 1 , k mn } , (8.17) where k mn = M + N − 1 + min { 0 , 2 − N ′ , 2 − M ′ , 4 − M ′ − N ′ } and k max 1 is giv en b y (8.8). Prop osition 8.3. B iline ar expr essions f k ( k = 0 , . . . , k max 2 ), gi v en by (8.16), ar e pr eserve d by the Darb oux tr ansf o rmation (i.e., ˜ f k = f k ) pr ovide d that det T 0 6 = 0 . 51 Remark 8.4. If M > N > 0 , N ′ 6 N + 2 , M ′ 6 M + 2 , then k max 2 = k max 1 . In some cases w e can formu late stronger prop ositions. F or N ′ 6 2, M ′ 6 2 (including the isospectral case) k max 2 in Prop o sition 8.3 can b e replaced b y k ′ max 2 = − 1 + min { N , M , N + M } . (8.18) If the normalizatio n is canonical ( T 0 = I ) w e can replace k max 2 b y k ′′ max 2 = min { k ′′ max 1 , k ′′ mn } , (8.19) where k ′′ mn = M + N + min { 1 , 2 − N ′ , 2 − M ′ , 3 − M ′ − N ′ } . Analogical conside rations can be done for T r U 2 and T r V 2 . T o obtain the ﬁnal results (see b elow ) it is enough to substitue M → N , M ′ → N ′ in the ﬁrst case, and N → M , N ′ → M ′ in the second case. Prop osition 8.5. Supp os e that 0 6 k 6 k max 3 , wher e k max 3 = min { N − 1 , N + 1 − N ′ , 2 N − 1 , 2 N + 1 − N ′ , 2 N + 3 − 2 N ′ } and g 0 , g 1 , . . . , g k ar e given functions of x . Then the bilin e a r c onstr aints g 0 := u 0 · u 0 , g 1 := u 0 · u 1 + u 1 · u 0 , g 2 := u 0 · u 2 + u 1 · u 1 + u 2 · u 0 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . g k := u 0 · u k + u 1 · u k − 1 + . . . + u k · u 0 , (8.20) ar e pr eserve d by the D arb oux tr ansformation such that det T 0 6 = 0 . Prop osition 8.6. Supp os e that 0 6 k 6 k max 4 , wher e k max 4 = min { M − 1 , M + 1 − M ′ , 2 M − 1 , 2 M + 1 − M ′ , 2 M + 3 − 2 M ′ } and h 0 , h 1 , . . . , h k ar e given functions of x . Then the biline ar c onstr aints h 0 := v 0 · v 0 , h 1 := v 0 · v 1 + v 1 · v 0 , h 2 := v 0 · v 2 + v 1 · v 1 + v 2 · v 0 , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . h k := v 0 · v k + v 1 · v k − 1 + . . . + v k · v 0 , (8.21) ar e pr eserve d by the D arb oux tr ansformation such that det T 0 6 = 0 . 52 8.3 In v arian ts for general Lax pairs Let us consider matrices U and V in the neigh b ourho o d of λ = λ 0 , where U, V hav e p oles of N -th and M -th order, resp ectiv ely , i.e., U = ∞ X k =0 u k ( λ − λ 0 ) k − N , V = ∞ X k =0 v k ( λ − λ 0 ) k − M , (8.22) where u k = u k ( x ), v k = v k ( x ) ( k = 0 , 1 , 2 , . . . ). W e are going to sho w tha t the general case reduces to the p olynomial case discussed ab ov e. Indeed, it is suﬃcien t to use the fo llo wing parameter z in the neighbourho o d of λ 0 : z = ( λ − λ 0 ) − 1 , (8.23 ) and then the Lax pair (8.22) b ecomes identical with (8.1) . Note that z → ∞ for λ → λ 0 . W e assume that the Darb o ux matrix D is analytic at λ 0 : D = T 0 + T 1 ( λ − λ 0 ) + T 2 ( λ − λ 0 ) 2 + . . . = T 0 + z − 1 T 1 + z − 2 T 2 + . . . where ma t r ices T k dep end on x . In the nonisosp ectral case w e transform the equations (1 .3) to the f orm (8.2): z , ν = − z 2 L ν ( x, λ 0 + z − 1 ) , (8.24) where L ν ha ve to b e expanded in the Laurent (or T ay lor) series at z = ∞ . In order to obtain linear in v ariants we consider the linear com bination of matrices U, V , giv en by : U + ( λ − λ 0 ) M − N hV (8.25) where h = h ( x, y ; λ ) ≡ ∞ X k =0 ( λ − λ 0 ) k h k ( x, y ) = ∞ X k =0 h k z − k (8.26) is a give n scalar function, holomorphic at λ = λ 0 . Finally , w e arrive a t an exact analogue of Prop osition 8.1. Bilinear in v ariants can b e treated in the same wa y . W e obtain exact analogues of Prop ositions 8.3 , 8.5 and 8.6. Corollary 8.7. The p olynomial c ase c an b e tr e ate d as a sp e cial sub c ase, deﬁne d by λ 0 = ∞ . It is enough to c h ange variables in formulas (8.22): λ → λ − 1 (and λ 0 → λ − 1 0 ). Then , making the limi t λ 0 → 0 , we get (8.1) . 53 8.4 Application to the KdV equation W e will sho w adv an tages of Da rb oux in v ariants considering the case o f the KdV equation. Our appro a c h consists in characteriz ing t he Lax pair in terms of some algebraic constrain ts (see [1 3, 14]) and then sho wing that these con- strain ts are preserv ed b y the Darb oux-B¨ ac klund transformation. Prop osition 8.8. The L ax p air (4.11) c an b e uniquely char acterize d by the fol lowing set of a l e b r aic c onstr aints: 1. U is line ar in λ ( U = u 0 λ + u 1 ), T r U = 0 , 2. V is quadr atic in λ ( V = v 0 λ 2 + v 1 λ + v 2 ), T r V = 0 , 3. u 0 =  0 0 − 1 0  , u 1 is oﬀ-diagonal. 4. u 0 − 1 4 v 0 = 0 , u 1 − 1 4 v 1 ∈ g , wher e g is the 1-dimensio nal Lie algebr a gener ate d by  0 0 1 0  , 5. u 0 · v 1 + u 1 · v 0 = − 8 , v 1 · v 1 + 2 v 0 · v 2 = 0 , 6. U ( λ ) = U ( ¯ λ ) , V ( λ ) = V ( ¯ λ ) . Pro of: The ﬁrst four prop erties imply t he follow ing form of U, V : u 0 =  0 0 − 1 0  , u 1 =  0 p u 0  , v 0 =  0 0 − 4 0  , v 1 =  0 4 p q 0  , v 2 =  − a b c a  , (8.27) where u, p, q , a, b, c are some complex ﬁelds. Bilinear constraints 5 yield − 8 p = − 8 , 8 pq − 8 b = 0 , (8.28) i.e., p = 1, q = b . Now compatibility conditions yield the KdV equation (4.10) and expressions (4.1 4) fo r a, b, c . The last pro p ert y implies u ∈ R . ✷ The ﬁrst tw o constrain ts a r e preserv ed b y an y Darb oux transformation constructed in the standa r d w ay , e.g., using Corollary 3.7 (and the traceless- ness is pres erv ed b y virue of R emark 3.3, pro vided that det N = const). The 54 constrain ts 6 imp ose r estrictions on λ k and p k , see Section 6.2. In order to preserv e the third constraint w e ha v e to use freedom in the choice of the nor- malization matrix T 0 ≡ N . F rom t he ﬁrst equation o f (3.19) w e get (ta king in to accoun t the form of u 0 giv en by the third constraint): N = f  1 0 α 1  , (8.29) where f , α are some functions. In the sequel w e put f = 1 (th us det N = 1) . Then, denoting T 1 =  c 1 c 2 c 3 c 4  , w e rewrite the second equation of (3.1 9) as  0 1 ˜ u 0  =  − α 1 u − α 2 α  +  − c 2 0 c 1 − αc 2 − c 4 c 2  +  0 0 α, 1 0  . (8.30) Hence: α = − c 2 , ˜ u = u − α 2 − α c 2 + α , 1 + c 1 − c 4 . (8.31) The constraint 4 is preserv ed b y virtue of Pro p osition 8.2 . Other prop o- sitions fr o m Section 8 are to o weak for our presen t purp oses. Indeed, in the KdV case we hav e k max 2 = 0 and k max 4 = 1. Therefore the preserv ation of the constraints 5 do es not follow from Prop ositions 8.3 and 8.6. F ortunatelly , the sp ecial f o rm o f matrices u 0 , v 0 and T 0 (giv en b y (8.2 9) with f = 1) for KdV equation allows us to reconsider the b ehaviour of leading terms (8.15). W e easily see that any matrix pro duct con ta ining only matrices from the set { T 0 , T − 1 0 , T 0 , 1 , T 0 , 2 , u 0 , v 0 } (and among them a t least one matrix from t he set { T 0 , 1 , T 0 , 2 , u 0 , v 0 } ) is propo rtional to u 0 , and, as a consequence , it has v anishing trace. Hence, in the case of the KdV equation the D ar- b oux tra nsfor ma t io n preserv es constrain ts (8.16) for k = 0 , 1 and constrain ts (8.21) fo r k = 0 , 1 , 2. In particular, the constraints 5 of Prop osition 8.8 are preserv ed. Corollary 8.9. T h e Darb oux tr ansformation (deﬁne d as in Cor ol la ry 3.7) pr eserves al l c onstr aints deﬁning the KdV L ax p air (se e Pr op os i tion 8.8) pr ovide d that we imp ose r e ality r estrictions o n λ k , p k (se e Se ction 6.2) and ﬁx the normalization matrix ac c o r ding to the formula (8.29) wher e f = 1 and α is exp r esse d by the matrix T 1 , namely α = − c 2 . In the case of the elemen tary Darb oux matrix det T 0 = 0 and considera- tions presen ted in t his section are not applicable. It would b e in teresting to extend the theory of D a rb oux inv ariants on the case det T 0 = 0. 55 9 Conclud ing remarks In this pap er w e gav e a uniﬁed view on the Darb oux-B¨ ac klund transforma- tions for 1 + 1-dimensional in tegrable systems of nonlinear partial diﬀeren tial equations. In particular, w e discussed in detail relationships b etw een v arious approac hes to t he construction of the D a rb oux matrix. Darb oux-B¨ ac klund tra nsfor mat ions ha v e b een extended in man y dir ec- tions. First of all, they are applicable to 2 + 1-dimensional in tegrable systems [9, 24, 27, 46, 54], including self-dual Y ang-Mills equations [27, 55, 78]. Then, w e hav e 0+ 1-dimensional systems , e.g., ordinary diﬀeren tia l equations of non- linear quan tum mec hanics [21, 2 4, 37]. Darb o ux tra nsformations w ere also constructed in the sup ersymmetric case [42, 48 ] and in the non-comm utat ive case [65]. Matrix represen tations of sp ectral problems and Darb oux transformatio ns are not alwa ys conv enien t. Impressiv e examples are asso ciated with Cliﬀord algebras. It is enough to compare the pap er [15], where mainly the matrix approac h w as used, with subsequen t pa p ers [5, 17], whic h are m uch shorter, more g eneral a nd mo r e elegan t. All these pap ers consider binary Darb oux transformation. An extension on multipole case is not so ob vious, compare [20], where some progress in this direction is describ ed. There exist other generalizations of the Da rb oux transformation on sp ectral problems with v alues in abstract asso ciativ e alg ebras [10, 2 4]. The discrete case is (to some exten t) very similar to the con tin uous case. Man y asp ects (e.g., those concerning the rational dep endence on λ and the lo op group structure) are just rep etitions from the con tinuous case, compare [24, 39, 4 5]. It is tempting to apply the ideas of time scales [8], all the more so that in the “classic al” case of the pseudospherical surfaces we succede d to construct t he Darb oux-B¨ ac klund transformation on arbitrary time scale [19], th us treating the discrete and con t inuous case in a unifo rm w a y . How ev er, some p oin ts see m to b e more diﬃcult in the discrete case, e.g., Darbo ux in v ariants a re not form ulated yet. Actually , it is not so easy ev en to ﬁnd an appropria te discretization o f a giv en in tegrable system, esp ecially if t he asso ciated linear problem is non-isosp ectral. A cknow le dgements : I am grateful to Maciej Nieszporski for man y fruiful discussions . 56 References [1] J.-P .Antoine, B.Piette: “Cla ssical nonlinear σ models on Grassma nn manifolds of compact or noncompact type”, J. Math. Phys. 28 (1 987) 2753 -2762 . [2] W.I.Arnold: O r dinary diﬀer ential e quations , Nauk a, Moscow 1971 [in Russian]. [3] H.Bar t, I .Gohberg, M.A.Kaas ho ek: Minimal factorization of matrix and op er ator functions , Birkh¨ auser, Base l 1 979. [4] V.A.Belinsky , V.E.Zakha rov: “ In tegration of Einstein’s equations by means o f the inv erse scattering technique a nd construction of exac t solutions ”, Sov. Phys. JETP 48 (1978) 985-9 94. [5] W.Bierna cki, J.L.Cie ´ sli ´ nski: “A compact form o f the Da rb oux-B¨ acklund transfor- mation for some sp ectr al problems in Cliﬀor d a lgebras ”, Phys. L ett. A 288 (2001) 167-1 72. [6] A. I. Bob enko: “Da rb oux transformations and canonical tra nsformations” , V estnik LGU 22 (1982) 1 4-21 [in Russian]. [7] A.I.Bob enko: “ Surfaces in T e rms of 2 b y 2 Matrices. Old and New In tegra ble Cases”, [in:] Harmonic maps and inte gr able syst ems (Asp ects of Ma thematics, v o l. 23 ), pp. 83-12 8; eds. A.P .F ordy , J.C.W oo d; Vieweg, Brunswick 1 994. [8] M.Bohner , A.Peterson: Dynamic e quations on time sc ales: an intr o duction with applic ations , Birkh¨ auser, Bos ton 2 001. [9] M.Boiti, B.Kono p elchenk o, F.P empinelli: “ B¨ acklund transformations via gauge transformatio ns in 2 + 1 dimensions”, Inv. Pr ob. 1 (198 5) 33 -56. [10] A.Bo utet de Monv el, V.Marchenk o: “ Generalizatio n of the Da rb oux transfor m”, Matem. ﬁz. anal. ge om. 1 (19 94) 4 79-50 4. [11] S.P .Burtsev, V.E .Zakharov, A.V.Mikhailov: “Inv er se scatter ing metho d with v a ri- able sp ectral para meter”, T e or. Ma t. Fiz. 70 (1987 ) 32 3-34 1 [in Russia n]. [12] J .Cie ´ sli´ nski: “An eﬀective metho d to compute N -solito n Darb oux matrix and N - soliton surfaces”, J. Ma th. Phys. 32 (199 1) 2 395-2 399. [13] J .Cie ´ sli´ nski: “Algebraic repr esentation of the linear pr oblem as a method to construct the Darb oux-B¨ acklund t ransforma tion”, Chaos, Solitons and F r actals 5 (1995) 2303- 2313. [14] J .Cie ´ sli´ nski: “An algebra ic metho d to construct the Darb o ux matrix”, J. Math. Phys. 36 (19 95) 5670- 5706. [15] J .Cie ´ sli´ nski: “The Darb oux-Bianchi t ransforma tion for isothermic surfaces. Classical results versus the soliton appro ach”, Diﬀ. Ge om. App l. 7 (1997) 1-28. [16] J .Cie ´ sli´ nski: “The Darb oux-Bia nchi-B¨ a cklund transformation and soliton surfaces”, [in:] Nonline arity and Ge ometry , pp. 81-1 07; edited b y D.W´ ojcik, J.Cie ´ sli ´ nski; Polish Scient iﬁc Publishers PWN, W arsaw 19 98. [17] J .L.Cie ´ sli ´ nski: “The Dar b oux-B¨ acklund tr ansformation without using a matrix r ep- resentation”, J. Phys. A: Math. Gen. 33 (2000) L363 -L368 . [18] J .L.Cie ´ sli ´ nski: “ The s tructure o f sp ectra l problems and geo metry: hyperb olic sur- faces in E 3 ”, J. Phys. A: Math. Gen. 3 6 (2 003) 6 423- 6440. [19] J .L.Cie ´ sli ´ nski: “P seudospherica l surfaces on time scales: a geometric deﬁnition a nd the sp ectral approach”, J. Ph ys. A: Math. The or. 42 (2007) 1 2525 -1253 8. 57 [20] J .L.Cie ´ sli ´ nski, W.Biernacki: “ A new approa ch to the Darb oux-B¨ a cklund transfor- mation versus the standar d dressing metho d”, J . Phys. A: Math. Gen. 38 (2005) 9491- 9501. [21] J .L.Cie ´ sli ´ nski, M.Cza chor, N.V.Ustinov: “Darb o ux cov ariant equatio ns of von Neu- mann type and their generaliz ations”, J. Ma th. Phy s. 4 4 (2 003) 1 763-1 780. [22] A.Cole y , D.Levi, R.Milson, C.Ro gers, P .Win ternitz (editor s): B¨ ackl und and Darb oux tr ansformations . The ge ometry of solitons , CRM P ro c. Le ct. No tes, v o l. 29 , AMS, Providence RI, 200 1. [23] G.Dar b oux: “Sur une prop osition relative aux ´ equa tions lin´ ea ires”, Compt. R end. 94 (1882) 1456 -1459 . [24] E .V.Doktorov, S.B.Leble: A dr essing metho d in mathematic al physics , Springer, Dor- drech t 2007 . [25] I.Go hberg, M.A.Kaasho ek, A.L.Sakhnovich: “Scatter ing problems for a canonical system with a pseudo-exp onential p otential”, Asymptotic Anal. 29 (2002) 1-38. [26] G.C.Gu: “B¨ acklund tr ansformations and and Darb oux transformations”, [in:] Soli- ton t he ory and its applic ations , pp. 122 -151; edited by G.C.Gu, Springer, Berlin 1995. [27] G.C.Gu, H.S.Hu, Z .X.Zhou: Darb oux t r ansformations in inte gr able systems. The ory and their applic ations to ge ometry , Spring er, Dordrech t 2005. [28] G.C.Gu, Z.X.Zho u: “O n Darbo ux transfo rmations for solito n equations in hig h- dimensional spacetime”, L ett. Math. Phys. 32 (1994 ) 1-10 . [29] M.A.Guest: Harmonic maps, lo op gr oups, and inte gr able systems , Ca mbridge Univ. Press, Cambridge 199 7. [30] J .Harnad, Y.Sain t-Aubin, S.Shnider: “Supe rp osition of s olutions to B¨ acklund trans- formations for SU( n ) principal σ -mo del”, J. Math. Phys. 25 (1984 ) 368 -375. [31] J .Harnad, Y.Saint-Aub in, S.Shnider: “B¨ acklund transforma tions for nonlinear sigma mo dels with v alues in Riemannian sy mmetric spa ces”, Commun. Math. Phys. 92 (1984) 329- 367. [32] J .Harnad, Y.Sain t-Aubin, S.Shnider: “The soliton correla tion matrix and the reduc- tion problem for in tegrable systems”, Commun. Ma th. Phys. 93 (19 84) 33-56 . [33] B.K .Harriso n: “ Uniﬁcation o f Erns t-equation B¨ acklund transforma tions us ing a mo diﬁed W ahlquis t-Estabro o k technique”, J. Math. Phys. 24 (1983 ) 217 8-218 7. [34] C.Ho ens elaers, W.K.Schief: “Pr olonga tion structures for Ha rry Dym type equations and B¨ acklund tra nsformations of cc-idea ls”, J. Phys. A: Math. Gen. 25 (1992) 601-6 22. [35] A.R.Its: “ Liouville’s theorem and the inv ers e scatter ing metho d”, [in:] Zapiski Nau. Sem. LO MI 133 (1984) 1 13-12 5 [in Russian]. [36] I.M.K richev er : “Metho ds o f algebra ic geometr y in the theory of non-linear equa - tions”, Russian Math. S u rveys 32 (6 ) (1977) 185-21 3. [37] S.B.Le ble, M.Cz achor: “ Darb oux-integrable nonlinear Lio uville-von Neumann equa - tion”, Phys. R ev. E 58 (199 8) 7091-7 100. [38] S.B.Le ble, N.V.Ustinov: “Deep r eductions for matrix L ax s ystem, in v ariant forms and element ary Darb oux transfor ms”, J . Phys. A: Math. Gen. 26 (19 93) 5 007-5 016. 58 [39] D.Levi, O.Ragnisco, M.Br uschi: “ Extension of the Zakha rov-Shabat Gener alized Inv erse Metho d to Solv e Diﬀeren tial-Diﬀerence and Diﬀerence-Diﬀerence Equations” , Nuovo Cim. A 58 (19 80) 56-66 . [40] D.Levi, O.Ragnisco, A.Sym: “Dressing Metho d vs. Cla ssical Darb oux T r ansforma- tion”, Nuovo Cim. B 83 (1984 ) 34 -41. [41] D.Levi, A.Sym, S.W o jciecho wsk i: “A hierarch y of c oupled Korteweg-de V ries equa- tions a nd the nor malisation conditions of the Hilb ert-Riemann problem”, J. Phys. A: Math. Gen. 16 (1983 ) 2423 -2432 . [42] Q .P .L iu, M.Ma ˜ nas: “ Darb oux transfor mations for the Manin-Radul sup ers ymmetric KdV equation”, Phys. L ett. B 394 (19 97) 337-34 2. [43] Q .P .L iu, M.Ma˜ nas: “V ec torial Dar b oux tra nsformations for the Ka domtsev- Petviash vili hierar ch y”, J. Nonl. Sci. 9 (199 9) 213-23 2. [44] M.Ma ˜ nas: “Darb oux transfo rmations for the nonlinear Schr¨ odinger eq uations”, J. Phys. A: Ma th. Gen. 29 (1996 ) 772 1-773 7. [45] V.B.Ma tveev: “ Darb oux trans formation and the explicit solutions of diﬀerential- diﬀerence and diﬀerence -diﬀerence evolution eq uations. I” , L ett. Math. Phys. 3 (1979) 217-2 22. [46] V.B.Ma tveev, M.A.Salle: Darb oux T r ansformations and S olitons , Springer-V erlag, Berlin-Heidelb erg 199 1. [47] R.Meinel, G.Neugebauer, H.Steudel: Soli tonen. N ichtline ar e Struktu r en , Academie V erla g, Berlin 1 991 [in German]. [48] A.V.Mikhailov: “Integrabilit y o f super symmetric genera lizations of class ical chiral mo dels in tw o-dimensional s pace-time”, Pis’ma ZhETF 28 (19 78) 554-558 [in Rus- sian]. [49] A.V.Mikhailov: “Reductions in integrable systems. The reduction g roup”, Pis’ma ZhETF 32 (1980) 187-1 92 [in Russian]. [50] A.V.Mikhailov: “ The reduction problem and the in verse sca ttering method”, Physic a D 3 (1981 ) 73-1 17. [51] G.Neuge bauer, D.Kra mer: “Einstein-Maxwell solitons” , J . Phys. A: Math. Gen. 16 (1983) 1927 -193 6. [52] G.Neuge bauer, R.Meinel: “General N -solito n solution of the AK NS c lass on ar bi- trary background”, Phys. L ett. A 1 00 (1 984) 467- 470. [53] F.Nijhoﬀ, J.Atkinson, J.Hietarinta: “ Soliton solutions for ABS lattice equa tions: I Cauch y Matrix appro ach”, J. P hys. A: Math. Th e or. 4 1 (2 009), this is sue. [54] J .J.C.Nimmo: “Darb oux tr ansformations for a t w o-dimensional Zakharov- Shabat/AKNS sp ectral problem”, Inv. Pr ob. 8 (199 2) 219 -243. [55] J .J.C.Nimmo, C.R.Gilson, Y.Oh ta : “Applica tions of Darboux transformations to the self-dual Y a ng-Mills eq uations”, The or. Math. Phys. 122 (2000 ) 23 9-24 6. [56] S.P .Novik ov, S.V.Manakov, L.P .Pitaievsky , V.E.Zakhar ov: The ory of solitons , Con- sultants Burea u, New Y o rk 1984. [57] W.Oe vel, W.Sc hief: “ Darb oux theorems a nd the KP hiera rch y” , [in:] Applic ations of analytic and ge ometric metho ds to nonline ar diﬀer ential e quations , pp. 193-2 06; edited b y P .A.Clarkson; Kluw er, Dordrech t 1993. 59 [58] Q .H.Park, H.J.Shin: “Darb o ux transformation and Cr um’s formula for multi- comp onent integrable equatio ns”, Physic a D 157 (200 1) 1-1 5. [59] A..N.Pr essley , G.B.Segal: L o op gr oups , Oxford Univ. P ress, Oxfor d 1986 . [60] C.Ro gers, W.K.Schief: B¨ acklund and Darb oux tr ansformations. Ge ometry and mo d- ern ap plic ations in soliton the ory , Cambridge Univ. Pres s, Cambridge 200 2. [61] Y.Saint-Aubin: “B¨ acklund transformations and soliton- t yp e solutio ns for σ mo dels with v alues in real Grassmannia n spaces” , L ett . Math. Phys. 6 (19 82) 441-4 47. [62] A.L.Sak hnovic h: “Dressing pro cedure for solutions of nonlinear equations and the metho d of op erator identities”, Inv. Pr ob. 10 (199 4) 69 9-710 . [63] A.Sak hnovic h: “Iterated B¨ acklund-Darb oux transfor mation and tr ansfer matrix - function (nonisosp ectral case)” , Chaos, Solitons & F r actals 7 (199 6) 1 251-1 259. [64] L.A.Sak hnovic h: “ F acto rization problems and operato r identities”, Usp. Mat. N auk 41 (1986) 3-55 ( R ussian Ma th. Surv. 41 (198 6) 1-4 1). [65] U.Sale em, M.Ha ssan: “Lax pa ir a nd Darb oux tr ansformatio n of a noncomm utative U ( N ) principa l chiral mode l”, J . Phys. A: Math. Gen. 39 (2 006) 1168 3-116 96. [66] D.H.Sattinger , V.D.Zurkowski: “Ga uge theory of B¨ acklund transformations . II”, Physic a D 26 (198 7) 225 -250. [67] W.K.Schief, C.Rogers : “Lo ewner tra nsformations: a djoint and binary Darb oux c on- nections”, Stud. Appl. Math. 100 (1998) 391- 422. [68] A.B.Sha bat: “Inv ers e scattering problem fo r a s ystem o f diﬀerential equa tions”, F unk. Anal. Prilozh. 9 (1975) 7 5-78 [ F unc. Anal. Appl. 9 (19 75) 24 4-247 ]. [69] H.J.Shin: “Darb o ux inv ariants of in tegrable equations with v ariable sp ectra l para m- eters”, J. Phys. A : Ma th. The or. 41 (200 8) 28 5201. [70] H.Steudel, R.Meinel, G.Neugebauer: “V andermonde-like deter minants a nd N -fold Darb oux/B¨ a cklund trans formations”, J. Math. Phys. 38 (199 7) 4692-4 695. [71] A.Sym: “Soliton surfaces” , L ett . Nuovo Cim. 33 (1982) 394-4 00. [72] A.Sym: “Soliton Surfaces I I. Geo metric Uniﬁcation of Solv a ble Nonlinearities”, L et t . Nuovo Cim. 36 (19 83) 30 7-31 2. [73] A.Sym: “Soliton surfac es VI. Gauge Inv ariance a nd Final F ormulation of the Ap- proach”, L ett . Nuovo Cim. 41 (1984 ) 35 3-360 . [74] A.Sym: “Soliton surfaces and their application. Soliton geometry from spectra l p rob- lems.”, [in:] Ge ometric A sp e cts of the Einst ein Equations and Inte gr able Systems (Lecture Notes in Ph ysics No.239), pp. 154-2 31; edited b y R.Martini; Springer, Berlin 1985. [75] C.L.T erng: “ Soliton equa tions a nd diﬀere nt ial geometry” , J. Diﬀ. Ge om. 45 (1 997) 407-4 45. [76] C.L.T erng, K.Uhlenbeck: “ B¨ acklund transformations and loop group actions”, Com- mun. Pur e Appl. Math. 53 (200 0) 1-75. [77] K .Uhlen be ck: “Harmonic maps into Lie g roup: classic al s olutions of the chiral mo del”, J. D iﬀ. Ge om. 30 (198 9) 1-5 0. [78] N.V.Ustinov: “The reduced self-dual Y ang-Mills equation, binary and inﬁnitesimal Darb oux transforma tions”, J. Math. Phys. 39 97 6-985 . [79] V.E.Z akharov, A.V.Mikhailov: “Relativistically inv ariant systems integrable by the inv erse scattering metho d”, ZhETF 74 (197 8) 195 3-197 3 [in Russian]. 60 [80] V.E.Z akharov, A.V.Mikhailo v: “On the In tegrability of Classic al Spinor Models in Two-dimensional Space-Time”, Comm. Math. Phys. 74 (198 0) 21- 40. [81] V.E.Z akharov, A.B.Shabat: “Integration of nonlinear equations of mathematical ph ysics by the inv er se scattering metho d. I”, F unk. Anal. Pril. 8 (1 974) 43- 53 [in Russian]. [82] V.E.Z akharov, A.B.Shabat: “Integration of nonlinear equations of mathematical ph ysics b y the inv er se scattering method. I I”, F unk. Anal. Pri l. 13 (1979 ) 13-22 [in Russian]. [83] Y.B.Ze ng, W.X.Ma, Y.J.Shao: “Two binary Dar bo ux tra nsformations for the K dV hierarch y with self-consis tent so urces”, J. Math. Phys. 42 (2001) 2113 -212 8. [84] Z.X.Zho u: “Darbo ux tr ansformations and exact solutions of tw o-dimensio nal C (1) l and D (2) l +1 T o da equatio ns”, J. Ph ys. A : Math. Gen. 39 (200 6) 5727-5 737. 61

Algebraic construction of the Darboux matrix revisited

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