q-Difference equations of KdV type and "Chazy-type" second-degree difference equations

q -differenc e equations of KdV typ e and “Chazy-t yp e ” secon d-degre e difference equations Chris M Field Korteweg-de V ries Institute for Mathematics, Universiteit v an Amsterda m, Pla ntage Muidergracht 24, 1018 TV Amsterdam, The Nether lands. E-mail: cfield@science.uv a.nl Nalini Joshi School of Mathema tics and Sta tistics F07, Universit y of Sydney , NSW 200 6 Australia. E-mail: nalini@maths.usy d.edu.au F rank W Nijhoff School of Mathema tics, Universit y of Leeds, Leeds LS2 9JT, UK. E-mail: nijhoff@maths.leeds.ac.uk Abstract. By imposing sp ecial compatible similarity constrain ts on a class of int egra ble partial q - differ ence eq uations of KdV-type w e derive a hiera r ch y o f second- degree ordinary q -difference e q uations. The low est (non-trivia l) member of this hierarch y is a seco nd-order second- degree equa tio n which can b e co nsidered as an analogue of equations in the cla s s studied b y Chazy . W e pr e sent co rresp onding isomono dro mic deformation problems and discuss the r elation b etw een this class of difference e q uations and o ther eq uations of Painlev ´ e t yp e. Keywor ds: q -Difference equations, Integrable s ystems, Painlev ´ e eq uations, Lattice equations. AMS c lassification scheme n umbers: 34 M55, 3 7J35, 37 K10, 37K 60, 3 9 A13 1. In tro duction The construction and study of discrete P ainlev ´ e equations has b een a topic of r esearch in terest for almost t w o decades, [28, 1 3 , 35, 21]. Reviews of the sub ject ma y b e found in [14, 16]. The sub ject has culminated in the classification b y H. Sak ai of discrete as w e ll con tin uous P ainlev ´ e equations ba sed o n the a lg ebraic geometry of the corresp onding rational surfa ces asso ciated with t he spaces of initial conditions [37]. As a byproduct of the la t t er treatmen t, a “ mistress” discrete P ainlev ´ e equation with elliptic dep endence on the indep enden t v ariable w as disco v ered. In the history of the P ainlev ´ e program, after the classification results for second- order first-degree equations, Painlev ´ e’s studen ts, Chazy and Ga rnier, [6, 7, 15], in v estigated the classification of second-order second-degree equations a nd third-order Chazy-typ e diff e r enc e e quations 2 first-degree equations. The classification of the second-degree class was completed by Cosgro v e in recen t y ears, [8, 9]. A partial class ification for the third-order case was also obtained b y the aforemen tioned authors. The w ork of Bureau, [3, 4], is also imp o rtan t in this resp ect. No classification results exist for the analogous discrete case and hardly an y examples of second-order second-degree difference equations exist to date, w ith the notable exception of an (additiv e difference) equation giv en b y Est ´ ev ez and Clarkson [10]. A key result of this letter is a second-order second-degree equation, whic h can b e considered as a q -analogue o f an equation in the Chazy-Cosgro v e class, together with its Lax pair (i.e., isomono dromic q - difference problem). This new equation contains four free parameters, whic h suggests that it could b e a q -difference analog ue of the second- order second-degree differen tial equation that is a coun terpart of the sixth P ainlev ´ e equation. There are sev eral forms of a second-order second-degree equation related to the sixth P ainlev ´ e equation that ha v e app eared in the literature, nota bly one deriv ed by F ok as a nd Ablow itz [12] and another app earing in the work of Ok amoto [3 2]. Difference analogues of the F ok as -Ablowitz equation hav e b een prov ided by G r ammaticos and Ramani, [34], but these difference equations w ere all of first-degree. It has b een argued b y these authors that equations that are second-degree in t he highest iterate cannot b e view ed a s “integrable”, how ev er, for the new equation w e establish integrabilit y through a Lax pair in t he form of a n isomono dromic deformation system. F urthermore w e show that the eq uation arises as a similarity reduction fro m a n in tegrable partial q - difference equation. Through t he same pro cedure we a lso construct higher-order second-degree equations, whic h form a hierarc h y asso ciated with the new equation. 2. q -Difference Similarity Reduction Lattice equations of KdV-ty p e w ere in tro duced a nd studied o v er the last three decades [19, 29], see [27] for a review. These lattice equations can b e form ulated as partia l difference equations o n a lattice with step sizes that en ter as parameters of the equation. Con v en tionally w e think o f these parameters as fixed constants . How e v er, in agreemen t with the integrabilit y of these equations, t here exists the freedom to take the parameters as functions of the lo cal lattice co ordinate in eac h corr esp o nding direction. In this pap er w e consider the case w hen the parameters dep end expo nentially with base q on the lattice co ordinates. W e w ork in a space F of functions f of arbitrarily man y v ariables a i ( i = 1 , . . . , M for any M ) on whic h we define the q - shift op erations q T i f ( a 1 , . . . , a N ) := f ( a 1 , . . . , q a i , . . . , a M ) . F or u, v , z ∈ F , w e consider the follo wing systems of nonlinear part ia l q - difference equations: ( u − q T i q T j u ) ( q T j u − q T i u ) = ( a 2 i − a 2 j ) q 2 , (2.1) a j ( q T j v ) q T i q T j v + a i ( q T j v ) v = a i ( q T i v ) q T j q T i v + a j ( q T i v ) v (2.2) Chazy-typ e diff e r enc e e quations 3 and a 2 i ( z − q T i z ) ( q T j z − q T i q T j z ) = a 2 j ( z − q T j z ) ( q T i z − q T i q T j z ) , (2.3) where i, j = 1 , . . . , M . Eac h of these systems, (2.1) t o (2 .3), represen ts a m ulti- dimensionally consisten t family of partial difference equations, in the sense of [3 1, 1], whic h implies that they constitute holonomic systems of nonlinear partial q -difference equations. Another w ay to form ulate this prop erty is through an underlying linear system whic h tak es the form q T − 1 i φ = M i ( k ) φ , (2.4) where φ = φ ( k ; { a j } ) is a tw o-comp onen t vec tor- v alued function and b y consistency , q T − 1 i q T − 1 j φ = q T − 1 j q T − 1 i φ , leads to the set of La x equations ( f or eac h pair of indices i, j ) ( q T − 1 i M j ) M i = ( q T − 1 j M i ) M j . (2.5) W e will consider three differen t cases, asso ciated resp ectiv ely with equations (2.1)– (2.3). T o av o id pro liferation o f sym b ols w e use the same sym b ol M i ( k ) for eac h of the resp ectiv e Lax matrices. F or sp ecific choice s of the matrices M i the Lax equations (2.5) lead to the nonlinear equ ations give n ab ov e . In the case of the q -lattice KdV, (2.1), the Lax matrices M i are give n b y M i ( k ; { a j } ) = 1 a i − k a i − q T − 1 i u , 1 k 2 − a 2 i + ( a i + u )( a i − q T − 1 i u ) , a i + u ! . (2.6) In the case of the q -lattice mKdV, (2.2), t he Lax mat r ices M i are give n b y M i ( k ; { a j } ) = 1 a i − k a i ( q T − 1 i v ) /v , k 2 /v q T − 1 i v , a i ! . (2.7) Finally , in the case of the q - lattice SKdV, (2.3), the La x matrices M i are give n by M i ( k ; { a j } ) = a i a i − k 1 , ( k 2 /a 2 i )  z − q T − 1 i z  − 1 z − q T − 1 i z , 1 ! . (2.8) These Lax matrices are straightforw ard generalizations of those with constan t lattice parameters given in e.g. [30]. W e men tion tha t the solutions of the equations (2.1) to (2.3) are related through discrete Miura type relations, namely a i  z − q T − 1 i z  = v  q T − 1 i v  , (2.9 a ) s =  a i − q T − 1 i u  v − a i q T − 1 i v , (2.9 b ) q T − 1 i s = a i v − ( a i + u ) q T − 1 i v , (2.9 c ) where s ∈ F is an auxiliary dep enden t v ariable. F rom these relations, the partial q - difference equations (2.1) to (2 .3) can b e deriv ed by eliminating s . Similarit y reductions of latt ice equations hav e b een considered in [28, 25, 3 1 , 26, 30] where it w as shown that scaling in v ariance of the solution can b e implemen ted through Chazy-typ e diff e r enc e e quations 4 additional compatible constrain ts on the la ttice equations. In the presen t case of (2.1) to (2.3) these constrain ts adopt the f o llo wing form [11] u ( { q − N a i } ) = q − N 1 − λ ( q N − 1)( − 1) P i q log a i 1 + λ ( q N − 1)( − 1) P i q log a i u ( { a j } ) , (2.10 a ) v ( { q − N a i } ) = 1 − λ ( q N − 1)( − 1) P i q log a i 1 + µ ( q N − 1) v ( { a j } ) , (2.10 b ) z ( { q − N a i } ) = q N 1 − µ ( q N − 1) 1 + µ ( q N − 1) z ( { a j } ) , (2.10 c ) where λ and µ are constan t pa r ameters of the reduction a nd N ∈ N represe n ts a “p erio dicity freedom”. The notatio n q log x refers to the logarithm of x with base q . In order to compute the corresponding isomono dromic deformation problems asso ciated with the similarity reductions w e hav e the follow ing constrain ts on the vector function of the Lax pair s. In t he case of (2 .1) w e ha v e φ ( q N k ; { a j } ) = (2.11)  1 + λ ( q N − 1)( − 1) P i q log a i  , 0 − 2 λq q N − 1 q − 1 ( P i a i ) ( − 1) P i q log a i , q N  1 − λ ( q N − 1)( − 1) P i q log a i  ! φ ( k ; { q − N a j } ) . In the case of (2.2) φ ( q N k ; { a j } ) = (2.12)  1 + λ ( q N − 1)( − 1) P i q log a i  0 0 q − N (1 + µ ( q N − 1)) ! φ ( k ; { q − N a j } ) . In the case of (2.3) φ ( q N k ; { a j } ) = (2.13)  1 − µ ( q N − 1)  0 0 q − N (1 + µ ( q N − 1)) ! φ ( k ; { q − N a j } ) . The similarity constrain ts, (2.11) t o (2.13), in conjunction with the discrete linear equations (2.6) to (2.8 ) can b e used t o deriv e corresp onding q - isomono dromic deformation problems. That is, (2.11) to (2.13) lead to q -difference equations in the sp ectral v ariable k , hence together with the la ttice equation Lax pairs we obta in q - isomono dromic deforma t ion problems for the corresp onding reductions. R emarks: (i) The similarity constrain ts ab ov e w ere obtained through a n a pproac h based o n Jac kson-t yp e integrals, the details of whic h will b e presen ted elsewhere [11]. By construction, these constrain ts are compatible with the underlying la ttice equations, whic h can b e chec ked a p osteriori by an explicit calculation, presen ted in the app endix. (ii) In this approac h, the dynamics in terms of the v ariables a i app ear through appropriately c hosen q -a nalogues of exp onen tial functions, whereas the relev ant Jac kson integrals exhibit an in v ariance through scaling by factors q N . Chazy-typ e diff e r enc e e quations 5 (iii) The par a meters λ and µ arise in this setting through b oundary contributions in a manner analogo us to the deriv atio n in [30]. In the remainder of this letter our aim is to implemen t the similarit y constrain t to obtain exp licit reductions to ordinary q - difference equations. F or simplicity we consider only the reduction of the q - mKdV equation (2 .2), leaving considerations of the q -KdV and q -SKdV to a f uture publication [11]. There are t w o p ossible scenarios to deriv e similarit y reductions of the lattice equations using the constrain t (2.10 b ). “Perio dic” similarity r e duction: By fixing M = 2 and allowing N t o v a r y , w e select t w o lattice directions, sa y the v ariables a 1 and a 2 , and consider similarity reductions with differen t v alues of N . This is a q -v arian t of the p erio dic staircase t yp e reduction of partial difference equations on the t w o-dimensional lattice. F or instance, with N = 2 the reduction is a second-order first- degree q - P ainlev ´ e equation. Increasing N leads to q -difference Painlev ´ e t yp e equations of increasing order. How eve r, we will not pursue this route here but leav e it to a subsequen t publication [11]. The se reductions are reminiscen t of the w ork [18, 36, 17]. Multi-variable similarity r e duction: The similarity constraints prov ide the mec hanism to couple together tw o or more lattice directions. By considering the case N = 1 we implemen t t he similarit y cons traints on an extended lattice of three or more dime nsions in order to o btain coupled o rdinary q - difference equations, in a w a y that is r eminiscen t of the appro a c h of [31 ]. This is considered in the next section. W e hav e not considered the more general case of arbitra r y M , N ∈ N , whic h w e will p ostp one to a future publication [11]. 3. Multi-v ariable similarity reduction In this section w e consider explicitly the M = 1, 2 a nd 3 cases for N = 1. F or simplicit y w e shall in what follows denote the co efficien t in (2.10 b ) as γ , i.e. γ = 1 − λ ( q − 1)( − 1) P i q log a i 1 + µ ( q − 1) ⇒ v ( { q − 1 a i } ) = γ v ( { a i } ) , (3.1) where γ alt ernat es b etw ee n tw o v alues, i.e., q T 2 i γ = γ . In con trast to the usual difference case whic h was explored in [31] where in the case of t w o v a r ia bles we obtain a nontrivial O∆E as a reduction, in the q -diff erence case w e ha v e to consider at least three indep enden t v ariables to o btain a nontrivial system of O∆Es as a reduction. In [31] the compatibility b et w een the similarity constrain t and the la ttice system w as established and led to a system o f higher order difference equations in the reduction, namely equations whic h w ere on the leve l of the first Gar nier system. In con trast to the q = 1 w ork, the 3D similarity constrain t here is somewhat simpler and leads to a second-order equation (whic h is of second-degree, and is a principal result of this letter). Chazy-typ e diff e r enc e e quations 6 Two-variable c ase Let us no w select among the collection of v ariables { a j } tw o sp ecific ones whic h for simplicit y w e will call a and b . D enote the q -shifts in these v ar iables b y an o v er-tilde e and an ov er-hat b resp ectiv ely . Equation (2.2) ma y now be written b b v b e v + a b v v = a e v b e v + bv e v , (3.2) where the ov er-tilde e refers to the q -translation a 7→ q a and the ov er-hat b refers t o the q -t ranslation b 7→ q b (so if v ≡ v ( a, b ), e v ≡ v ( q a, b ), v e ≡ v ( q − 1 a, b ), b v ≡ v ( a, bq ), . . . ). Equation (3.1) gives the constrain t v = γ b e v to imp ose on (3.2) (where b e γ = e e γ = γ ) . This leads to the linear first- order (in that it is a tw o point) ordinary difference equation v e = C ˜ v , (3.3) where C = e γ ( aγ + b ) / ( a + bγ ) . In t he app endix the consiste ncy b et w een the lattice equation (3.2) a nd the constraint (3.3) is sho wn b y direct computation. Thr e e-variabl e c ase T ak e three copies of the la ttice mKdV equation with a 1 = a , a 2 = b , a 3 = c , b b v b e v + a b v v = a e v b e v + bv e v , (3.4 a ) c v e v + av v = a e v e v + cv e v , (3.4 b ) c v b v + bv v = b b v b v + cv b v , (3.4 c ) where c = q c , together with the constraint v ( q − 1 a, q − 1 b, q − 1 c ) = γ v ( a, b, c ) . (3.5) (The similarity constraint is sho wn b y a direct computatio n to b e compatible with the m ultidimensionally consisten t system of mKdV lattice equations in the app endix.) W e no w pro ceed to deriv e the reduced system whic h leads to a (higher-degree) or dina r y q -difference equation in terms of one sele cted independent v ariable, sa y the v ariable a . The remaining v ariables b and c will play t he role of parameters in the reduced equation. Thus , w e can deriv e the follow ing system of tw o coupled O∆Es for v ( a, b, c ) and w ( a, b, c ) ≡ v ( a, b, q − 1 c ): γ v = w a e γ e v − b w e a w e − b e γ e v , (3.6 a ) w e = v aw − c v e a v e − cw , (3.6 b ) where e e γ = γ . W e consider the system ( 3.6 a ) a nd ( 3 .6 b ) to constitute a q -Painlev ´ e system with four free parameters. Chazy-typ e diff e r enc e e quations 7 The system (3.6 a ) and ( 3 .6 b ) can b e reduced to a second-order second-degree ordinary difference equation as follo ws. In tro duce t he v ariables X = v w , V = e v v , W = e w w , (3.7) then from (3.6 a ) w e o btain e γ e v w e = e γ V X W e = aγ X + b bγ X + a , (3.8) whereas from (3.6 b ) w e get W = V X e X = a + q − 1 cX V q − 1 c/X + aV , (3.9) using also the definitions (3.7). Thus, w e obtain a quadratic equation for V in terms of X and e X and hence also we ha v e W in terms of X and e X . Inserting these in to (3.8) w e obtain a second-order algebraic equation for X . Alternative ly , av o iding the emergence of square ro ots, the follow ing second-order second-degree equation for X may b e deriv ed " e γ 2 e X X e −  aγ X + b bγ X + a  2 # 2 = e γ c 2 a 2 1 X  aγ X + b bγ X + a   e γ e X (1 − X X e ) + q − 1 (1 − X e X ) aγ X + b bγ X + a  ×  q − 1 e γ X e (1 − X e X ) + (1 − X X e ) aγ X + b bγ X + a  . (3.10) W e consider this second-degree equation to b e one of the main results of this letter. W e no w pro ceed to presen t the Lax pair f o r the q -P ainlev ´ e system (3.6 a ) and (3.6 b ) a nd the second-order second-degree equation (3.10). The L a x pair is fo r med b y considering the compatibilit y of t w o paths on the lattice: along a ‘perio d’ then in the a direction a nd ev olving in the a direction then a long a ‘p erio d’. Using (2.12) the ev olution along a p erio d is conv e rted in to a dilat io n of the sp ectral parameter, k , by q . The res ult is the following isomono dromic q -difference system for the vec tor φ ( k ; a ) whic h using the results o f section 2 yields φ ( k ; q − 1 a ) = M ( k ; a ) φ ( k ; a ) , (3.11 a ) φ ( q k ; a ) = L ( k ; a ) φ ( k ; a ) , (3.11 b ) where M ( k ; a ) = 1 a − k   a v e /v k 2 /v v e a   , (3.12 a ) and L ( k ; a ) = 1 a − k aγ v / e v k 2 / e v q − 1 γ v q − 1 a ! b e γ e v /w k 2 /w e γ e v b ! cw /v k 2 /v w c ! (3.12 b ) Chazy-typ e diff e r enc e e quations 8 where we hav e suppressed the dependence on the v ariables b and c (whic h no w play the role of pa r ameters) and omitted the unnecessary prefactors ( b − k ) − 1 and ( c − k ) − 1 , as w ell as an ov er fa ctor q − 1 (1 + µ ( q − 1)). The consistenc y condition obtained from the t w o w a ys of expres sing φ ( q k ; q − 1 a ) in terms of φ ( k ; a ) is formed by the Lax equation L ( k ; q − 1 a ) M ( k ; a ) = M ( q k ; a ) L ( k ; a ) . (3.13) A gauge transformatio n can b e obtained expressing the Lax ma t r ices in terms o f the v a r iables in tro duced in ( 3.7). Setting M ( k ; a ) = 1 a − k   a/ V e k 2 1 a V e   , (3.14 a ) L ( k ; a ) = 1 a − k e γ ( abγ X + k 2 ) k 2 ( aγ X + b ) /V q − 1 e γ V ( a + bγ X ) q − 1 ( ab + k 2 γ X ) ! c/X k 2 1 /X c ! , (3 .14 b ) the Lax equations (3.13) (replacing L and M b y L and M resp ectiv ely) yield a set of relations equiv alen t to t he followin g t w o equations: e γ V V e X e = aγ X + b a + b γ X , (3.15) aV 2 + q − 1 c  1 X − e X  V − a e X X = 0 , (3.16) using also e γ = γ e . This set follows directly from (3.8) and (3.9). Th us (3.14 a ) and (3.14 b ) form a q -isomono dromic Lax pair f or the second-degree equation (3.10). F our-variable c ase: Supp ose we hav e 4 v ariables a i , i = 1 , . . . , 4. Select a = a 1 to b e the indep enden t v a r iable aft er reduction. In tro duce the dep enden t v ariables w j − 2 = q T − 1 j v , j = 3 , 4. Then directly from the q -lattice mKdV equation (2.2) w e hav e the set of equations e w j = v aw j − a j +2 v e a v e − a j +2 w j , j = 1 , 2 , (3.17) where as b efore the tilde denotes a q -shift in the v a riable a . A t the same time the m ultiply shifted ob ject q T − 1 3 q T − 1 4 v e can b e expressed in a unique wa y (due to the CA C prop erty ) in terms of v e and q T − 1 j v = w j − 2 , j = 3 , 4, b y iterating the relev an t copies of the q -lattice mKdV equation in the v ariables a j , j 6 = 2, leading to an expression of the form q T − 1 3 q T − 1 4 v e =: F ( v e , w 1 , w 2 ), where F is easily obtained explicitly . Imp osing the similarit y constraint (3.1) w e obtain e γ q T 2 v = F ( v e , w 1 , w 2 ) and inserting this expression in to the q - lattice mKdV (2.2) with i = 1 , j = 2 w e obtain  a + a 2 γ a 3 w 2 − a 4 w 1 a 3 w 1 − a 4 w 2   a 2 e γ − 1 F ( v e , w 1 , w 2 ) − a e v  = ( a 2 2 − a 2 ) v e v . (3.18) Chazy-typ e diff e r enc e e quations 9 With the explicit form of F ( v e , w 1 , w 2 ) equation (3.18) reads ( a 2 2 − a 2 ) γ e γ e v ( a 3 w 1 − a 4 w 2 )  a ( a 2 3 − a 2 4 ) v e + a 3 ( a 2 4 − a 2 ) w 1 + a 4 ( a 2 − a 2 3 ) w 2  = [( a 2 a 3 − γ aa 4 ) w 2 + ( γ aa 3 − a 2 a 4 ) w 1 ]  a ( a 2 3 − a 2 4 )( a 2 w 1 w 2 − e γ a e v v e ) +  a 2 a 4 ( a 2 − a 2 3 ) v e − aa 3 ( a 2 4 − a 2 ) e γ e v  w 1 +  a 2 a 3 ( a 2 4 − a 2 ) v e − e γ aa 4 ( a 2 − a 2 3 ) e v  w 2  , (3.19 a ) and this is supplemen ted b y the t w o equations av w 1 + a 3 w 1 e w 1 = a 3 v v e + a v e e w 1 , (3.19 b ) av w 2 + a 4 w 2 e w 2 = a 4 v v e + a v e e w 2 , (3.19 c ) whic h is equiv a lent to a fiv e-p oint (fourth-order) q -difference equation in t erms of v alone, con taining five free parameters: a 2 , a 3 , a 4 , λ and µ (inside γ and e γ ). This w ould b e an algebraic equation, so w e pro ceed as fo llo ws in order to deriv e a higher-degree q -difference system. In tro duce the v ariables X i = v w i , W i = e w i w i , i = 1 , 2 , (3.20) while retaining the v ariable V = e v /v as b efore. By definition we ha v e V W i = e X i X i , i = 1 , 2 , (3.21) and from (3 .19 b ), (3.19 c ) we obtain W i = q a + a i +2 V X i q aV + a i +2 /X i = V X i e X i , i = 1 , 2 , (3.22) whilst from ( 3 .19 a ) w e get ( a 2 2 − a 2 ) γ e γ V  a 3 X 1 − a 4 X 2    a a 2 3 − a 2 4 V e + a 3 a 2 4 − a 2 X 1 + a 4 a 2 − a 2 3 X 2   =  a 2 a 3 − γ aa 4 X 2 + γ aa 3 − a 2 a 4 X 1    a ( a 2 3 − a 2 4 )( a 2 X 1 X 2 − e γ a V V e ) +   a 2 a 4 a 2 − a 2 3 V e − aa 3 ( a 2 4 − a 2 ) e γ V   1 X 1 +   a 2 a 3 a 2 4 − a 2 V e − e γ aa 4 ( a 2 − a 2 3 ) V   1 X 2   . (3.23) F ro m (3.22) w e o bta in the set of quadratic equations fo r V q a X i e X i V 2 + a i +2  1 e X i − X i  V − q a = 0 , i = 1 , 2 , (3.24) Chazy-typ e diff e r enc e e quations 10 from whic h by eliminating V w e o bta in h a 3 (1 − X 1 e X 1 ) X 2 − a 4 (1 − X 2 e X 2 ) X 1 i h a 3 (1 − X 1 e X 1 ) e X 2 − a 4 (1 − X 2 e X 2 ) e X 1 i = q 2 a 2 ( X 1 e X 2 − X 2 e X 1 ) 2 . (3.25) F urthermore, solving V from the quadratic system as V = q a X 2 e X 1 − X 1 e X 2 a 3 (1 − X 1 e X 1 ) X 2 − a 4 (1 − X 2 e X 2 ) X 1 , (3.26) and inserting this into (3.23) we obtain a second-order equation in both X 1 , X 2 coupled to the equation (3 .2 5) whic h is first o r der in b oth X 1 , X 2 . It is this coupled system of tw o equations in X 1 , X 2 whic h forms our higher or der generalisation of (3.10). The system of (3.23) a nd (3.25) with (3.26) constitutes a third- order system with fiv e pa rameters. The deriv ation o f the Lax pair follows the same approac h as that f o r the three- v a r iable case (with an extra factor in L due to the additional la ttice direction). W e omit details here, whic h w e inten d to publish in the future [11 ]. Beyond the fo ur-va riable c ase: It is straigh t-forward to g ive the form of the full hierarc h y , ho w ev er due to lac k o f space w e p ostp one this un til a lat er publication [1 1]. 4. Conclusion and discussion In this letter w e hav e pr esen ted the results of a sc heme to derive par t ia l q -difference equations of KdV type and consisten t symmetries o f the equations and demonstrated ho w it can b e imple men ted. Lax matrices follow fr om this approac h. A notable result is the deriv ation of the higher-degree equation (3.10), show ing that the sc heme presen ted here allow s for the deriv ation of new results within the field of discrete integrable systems . The first-, second- and third-order mem b ers of the N = 1 hierarch y ha v e b een presen ted. The sc heme con tin ues to giv e successiv ely higher-order equations b y considering success iv ely higher dimensions o f the orig inal latt ice equation. One may ask the nat ural question a s to whether this giv es an ‘interpo la ting’ hierarc h y whic h, contrary to the usual cases, increases the order and num ber o f parameters of the equations by one in each step, r a ther t ha n a t w o step increase. A further natural qu estion connected with this hierarch y is it s relation to the q -Garnier systems of Sak ai [38]. W e will presen t full details of the sc heme f r om whic h the lattice equations (2 .1) to (2.3) a nd their asso ciated constraints fo llo w in a future publication [11]. There we will consider the most general case of symmetry reductions (arbitra ry N ∈ N ) of all three lattice equations. W e also in tend to return in a future publication to the question of limits and degeneracies o f the equations presen ted in this pap er. These include the q → 1 con tin uum limit, the q → 1 discrete limit and the q → 0 crystal or ultradiscrete limit. Chazy-typ e diff e r enc e e quations 11 Ac kno wledgemen ts During the writing of this letter C.M. Field has b een supp o rted by the Australian Researc h Council Disco v ery Pro ject G ran t #DP0664624 and b y the Netherlands Organization for Scien tific Researc h (NW O) in the VIDI-pro ject “Symmetry and mo dularity in exactly solv able mo dels”. N. Joshi is also supp orted by the Australian Researc h Council Disco v ery Pro ject Gran t #DP0664 6 24. P art o f the w ork presen ted here w as p erformed whilst N. Joshi w as visiting the Unive rsit y of Leeds. 5. App endix In this app endix we presen t the result of explicit calculations showing the consistency of the latt ice equations a nd constrain ts. Two-variable c onsistency W e shall chec k the consistency b etw ee n the lattice equation ( 3 .2) and the constraint v = γ b e v b y direct computation. This computation is illustrated in the followin g diagram: . . . . . . . . ② ② ✐ × ✐ ⊗ ✐ v 0 v 1 v 12 v 2 v − 1 v − 1 , − 2 v − 2 Fig 1. Consistency on t he 2D lat t ice. Assuming t he v a lues v 0 , v 1 as indicated in F ig 1 a re giv en, we compute successiv ely v 12 , v 2 etc., where the subscripts refer to the shifts in the lattice v a r ia bles a , b resp ectiv ely , as is eviden t from Fig 1. Poin ts ot her than v 0 and v 1 are compu ted using either the lattice equation (indicated by × ) or by using the similarity constrain t (indicated by  ) . The v a lue v − 1 , − 2 is t he first po int whic h can be calculated in t w o differen t w a ys (hence indicated in the diagram b y ⊗ ). Witho ut making any particular a ssumptions on ho w γ dep ends on a and b , a straightforw ard calculation sho ws that the t w o wa ys of computing v − 1 , − 2 are indeed the same, f o r any ch oice of initial data v 0 and v 1 , provided t ha t γ obeys the relation  a + b γ b + aγ  b e  a + b γ b + aγ  − 1 = e γ b γ . (5.1) Chazy-typ e diff e r enc e e quations 12 A particular solution of this relation is b e γ = γ ⇔ b γ = e γ , (5.2) and hence e e γ = γ implying that γ is an alternating “constan t” whic h is in a ccordance with the v alue give n in (2 .10 b ). The reduced equation in this case is (3.3), whic h can b e readily in tegrated. More generally , equation (5.1) can b e resolv ed by setting a + b γ b + aγ = e ν b ν , γ = b e ν ν , (5.3) leading to the consequence that ν has to solve the q - lattice mKdV (3.2). In princip le we could take for ν any solution of the reduced equation (3.3) and use this to parametrise the reduced equation for v via the relations (5.3). In an y ev en t, we see that the tw o - v a r iable case do es not lead to inte resting nonlinear equations. Thr e e-variabl e c onsistency In this case the consistency diagram is as follows: ❣ ❣ ⊗ ✇ ❣ × ✇ ✇ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ v 0 v 1 v 2 v − 3 v 1 , 2 v − 2 , − 3 v − 1 , − 3 v − 1 , − 2 , − 3 Fig 2. Consistency on t he 3D lat t ice. A similar nota tion as the pr evious case is use d as is eviden t from F ig 2. The initial dat a v 0 , v 1 and v 2 are giv en, and the indicated v alues on the v ertices ar e computed either b y using one of the lattice equations (3.4 a ) to (3 .4 c ) or the similarit y constraint (3.5) o v er the diagonal. Th us, v 1 , 2 is obtained f rom (3.4 a ) yielding v 1 , 2 = v 0 av 2 − bv 1 av 1 − bv 2 , whilst from t he similarit y constrain t w e obtain v − 1 , − 3 = γ 2 v 2 , v − 3 = γ 1 , 2 v 1 , 2 , v − 2 , − 3 = γ 1 v 1 , assuming that γ shifts along the lattice, indicated b y the indices, and finally the v alue of v − 1 , − 2 , − 3 can b e computed in tw o differen t w a ys, leading to v − 1 , − 2 , − 3 = γ 0 v 0 = av − 2 , − 3 − bv − 1 , − 3 av − 1 , − 3 − bv − 2 , − 3 v − 3 = aγ 1 v 1 − bγ 2 v 2 aγ 2 v 2 − bγ 1 v 1 γ 1 , 2 v 0 av 2 − bv 1 av 1 − bv 2 , Chazy-typ e diff e r enc e e quations 13 leading quadratic iden tit y in v 1 and v 2 . Assuming tha t the latter m ust hold iden tically , and thus setting all co efficien ts equal to zero, we obtain the follow ing conditions on γ : γ 1 , 2 , 3 = γ 1 = γ 2 = γ 3 , from whic h we conclude that γ is an alternat ing “constan t”, f or instance γ = α β ( − 1) n + m + ... ( α, β constan ts) (5.4) and this leads to the conditions fro m which it is easily de duced that the form (3.1) of γ satisfies these conditio ns. References [1] Bob enko A. and Suris Y.: I ntegrable systems on quad-graphs, Int. Ma th. R es. Not. 11 573– 611 (2002) [2] Birkhoff G. D.: The generalized Riemann problem for linear differential eq uations and the allied problems for linear difference and q -differ ence equa tions, Pr o c. Am. A c ad. Sci. 49 5 2 1–56 8 (19 1 3) [3] Bureau F. J.: ´ Equations diff´ erentielles du second ordr e en Y et du sec ond degr´ e en ¨ Y dont l’int ´ egra le g´ en´ erale est ´ a p oints critiques fixes , Ann. Mat. Pur a Appl. 91 (4) 16 3 –281 (1972 ) [4] Bureau F. J.: Sur des syst` emes diff´ erentiels non lin´ eaires du troisi` eme ordre et les ´ equa tions diff ´ erentielles non lin ´ eaires asso c i´ ees, A c ad. Ro y. Belg. Bul l. Cl. Sci. 73 (5) no . 6-9 , 335–35 3 (1987) [5] Cap el H. W., Nijhoff F. W. and Papageorgio u V. G.: Co mplete integrability of Lag rangia n mappings and la ttices of K dV t yp e Phys. L ett. A 155 377 –387 (1991 ) [6] Chazy J.: Sur les ´ equations diff´ erentielles du second or dre ` a p oints critiques fix es Comptes R endus de l’A c ad ´ emie des Scienc es, Paris 148 138 1-138 4 (190 9 ) [7] Chazy J.: Sur les ´ eq uations diff´ erentielles du troisi` eme o rdre et ordre sup´ erieur dont l’in t´ egrale g´ en´ erale a s es p oints critiques fixes, A cta Math. 34 317– 385 (1911) [8] Cosgrove C.: Chaz y’s seco nd-degree Painlev´ e equations, J. Ph ys. A: Math. Gen. 39 1 1955- 11971 (2006) [9] Cosgrove C.M., Scoufis G.: Painlev ´ e Classification o f a Class of Differential E q uations o f the Second O rder a nd Second Deg ree, Stud. Appl. Math. 8 8:25–8 7 (1993) [10] Est´ evez P . G. and Clarkson P . A.: Discrete equations and the singular manifold metho d, In: D. Levi a nd O. Ragnisco (eds), SI D E III—symmetries and inte gr ab ility of differ enc e e qu ations CRM Pro ceeding s and Lecture Notes, 25. pp 1 3 9–14 6, American Mathematical So ciety , Providence, RI, (2000) [11] Field C.M., Joshi N. a nd Nijhoff F.W.: in prepar a tion. [12] F ok as A.S. and Ablowitz M.J.: A unified appr oach to tra nsformations and e lement ary solutions of Painlev ´ e equations, J. Math. Phys. 2 3 2 033–2 042 (1 9 82) [13] F ok as A.S., Its A.R. and Kita ev A.V., Disc r ete Painlev ´ e Equa tions and their Appeara nce in Quantum Gravit y , Comm. Math. Phys. 142 313 – 344 (1991) [14] Grammatico s B., Nijhoff F. W. and Ramani A.: Discrete P ainlev´ e equa tions, In: R. Con te (ed), The Painlev ´ e Pr op e rty: One Ce ntu ry L ater , pp 413 –516, Springer- V erlag, New Y ork (19 9 9) [15] Garnier R.: Sur des ´ equations diff ´ erentielles du troisi` eme ordr e dont l’int ´ egr ale g´ e n´ er ale est uniforme et sur une classe d’ ´ equations nouvelles d’or dre sup ´ erieur, Ann. ´ Ec ol. Norm. Sup. 29 1–126 (1 912) [16] Grammatico s B. and Ramani A.: Discrete Painlev´ e equa tio ns: A rev ie w, In: B. Grammatico s, Y. Kossmann-Sch warzbac h and T. T a mizhmani (eds) Discr et e Inte gr able Systems , Lect. Notes Phys. 644 , pp. 24 5–32 1, Springer V er lag, 2004 ), (2004 ) [17] Grammatico s B., Rama ni A., Sa tsuma J., Willox R. and Carstea A.: Reductio ns of integrable lattices, J. Nonl. Math. Phys. 12 Suppl. 36 3–37 1 (200 5) Chazy-typ e diff e r enc e e quations 14 [18] Hay M., Hietarin ta J., Jos hi N. and Nijhoff F. W.: A Lax pair for a lattice mo dified KdV equation, reductions to q -Painlev´ e equations a nd asso ciated Lax pairs, J. Phys. A: Math. The or. 40 F61– F73 (2007) [19] Hirota R.: Nonlinear Partial Difference Equatio ns I. A difference analo g ue of the Korteweg-de V ries equation J. Phys. So c. Jap an 43 (4) 14 24-14 33 (19 7 7) [20] Hydon P . and Rasin O.: Symmetries of integrable differe nc e equatio ns on the quad-gr aph, Stud. Appl. Math. 11 9 (3) 253 -269 (20 0 7) [21] Jimbo M. and Sak ai H.: A q -a nalog of the sixth Painlev ´ e equation, L et t Ma th. Phys. 38 145–15 4 (1996) [22] Ka jiwara K., Noumi M. and Y amada Y.: A study o n the fourth q - Painlev ´ e equation, J. Phys. A 34 8 563– 8581 (2 001) [23] Ka jiwara K., Noumi M. and Y amada Y.: Discrete dynamica l systems with W ( A (1) m − 1 × A (1) n − 1 ) symmetry , L ett. Math. Phy s. 60 21 1-219 (20 02) [24] Ka jiwara K., Noumi M. and Y a ma da Y.: q - Painlev ´ e systems arising fr o m q -KP hierarch y , L ett. Math. Phys. 62 25 9 –268 (2002 ) [25] Nijhoff F. W.: On some “Sch w arzia n” E quations and their discrete analogues , In: A.S. F o k as and I.M. Gel’fand (eds), Algebr aic A s p e cts of Inte gr able Systems: In memory of Ir ene D orfman , pp 237–2 60. Birkh¨ auser V erla g (199 6) [26] Nijhoff F. W.: Discrete Painlev´ e equations and sy mmetr y reductio n on the lattice, In: A.I. bob enko and R. Seiler (eds) Discr ete Inte gr able ge ometry and Physics , pp. 209– 243, Oxford Univ. Press , (1999) [27] Nijhoff F. W. and Capel H. W.: The discrete Korteweg-de V ries equatio n, A cta Appl. Math. 39 133–1 58 (1995) [28] Nijhoff F. W. and Papageorgio u V.G.: Similar ity r eductions of integrable lattices and dis crete analogues o f the Painlev´ e I I equation, Phys. L ett . A 153 337 –344 (1991 ) [29] Nijhoff F. W., Q uisp el G.R.W. and Cap el H.W.: Direct lineariza tion of nonlinear difference- difference e q uations, Phys. L ett. A 97 125 – 128 (1983) [30] Nijhoff F. W., Ramani A., Grammatico s B. and O hta Y.: On dis c r ete Painlev´ e equations asso ciated with the la ttice K dV systems a nd the Painlev ´ e VI equation, Stu d. Appl. Math. 106 2 61–31 4 (2001) [31] Nijhoff F. W. and W a lker A. J.: The disc rete a nd contin uous Painlev´ e VI hier arch y and the Garnier systems, Glasgow Math. J. 4 3A 1 0 9–12 3 (2001) [32] Ok amoto K .: Studies on the Painlev´ e equa tions. 1. 6th Painlev´ e equatio n P VI. A nn. Math. 146 337–3 81 (1987) [33] Papageorg iou V. G., Nijhoff F. W., Grammaticos B. and Ramani A.: Is omono dromic de fo rmation problems for discrete analogues of Painlev´ e equations, Phys. L et t. A 164 57 –64. (199 2) [34] Ramani A. and Grammaticos B.: O n the discre te fo r m of the F ok as-Ablowitz equation, Chaos, Solitons and F r actals 24 13 31-13 35 (2005) [35] Ramani A., Gr ammaticos B. a nd Hietarinta J.: Discrete versions of the Painlev´ e eq uations, Phys. R ev. L ett. 67 18 29–18 32 (19 9 1) [36] Sahadev an R. and Cap el H. W.: Complete in tegrability a nd singular ity co nfinemen t of nonautonomous modified Ko rteweg-de V ries and sine Gordon mappings, Physic a A 330 373– 390 (2003) [37] Sak ai H.: Rational s urfaces asso cia ted with affine r o ot s y stems and geometry of the Painlev ´ e equations, Comm. Math. Phys. 22 0 1 65–22 9 (2001) [38] Sak ai H.: A q - a nalog of the Gar nier s ystem, F unkcial. Ekvac. 48 273–2 97 (2005 ) [39] Sak ai H.: Lax form of the q -Painlev ´ e equation asso ciated with the A (1) 2 surface, J. Phys. A: Math. Gen. 3 9 1 2203– 1221 0 (2006)