On a Whitham-Type Equation

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 5 (2009), 101, 7 pages On a Whitham-T yp e Equation Ser gei SAKO VICH Institute of Physics, National A c ademy o f Scienc es, 220 072 Minsk, Belarus E-mail: saks@tut.by Received Septem be r 27, 2009, in f inal form Nov em ber 05, 20 09; P ublished online Nov ember 08 , 20 09 doi:10.38 42/SIGMA.20 09.101 Abstract. The Hunter–Saxton equation a nd the Gurev ich–Zybin system are considered as t wo mutually non-equiv alen t representations of o ne and the same Whitham-t yp e equation, and a ll their common solutions are obtained exa ctly . Key wor ds: nonlinea r PDEs; transfor mations; gener al solutions 2000 Mathematics S ubje ct Classific ation: 35Q58 ; 35C05 1 In tro duction The f ollo w ing Whitham-t y p e equation u t = 2 uu x − ∂ − 1 x u 2 x (1) w as p rop osed recentl y by Pryk arpatsky and Prytula in [1] as a mo del equation describing the short-w a v e p erturbations in an abstract elastic on e-dimensional medium with relaxation and spatial memory ef fects. This equation (1), con taining the ill-def ined term ∂ − 1 x u 2 x , was r epresen ted in [1] by the second- order n onlinear p artial dif feren tial e quation u xt = 2 uu xx + u 2 x , (2) and it was sho wn there that (2) is in tegrable in the sense of p ossessing a bi-Hamilto nian structur e, an inf inite hierarc h y o f conserv ation l a ws, and a Lax-t yp e represen tation. Also, t w o f inite- dimensional reductions of (2) we re obtained in [1], and they turn ed out to b e some integ rable by quadratures dynamical system, w hic h could b e useful for deriving wide classes of exact solutions of (2). A dif f eren t representati on of (1), th e hydro dynamic-t yp e s ystem u t = 2 uu x − v , v t = 2 uv x , (3) w as prop osed recen tly b y Bo goliub o v, Pr yk arpatsky , Gucwa , and Golenia in [2]. In the f irst equation of (3), th e ill-def ined term ∂ − 1 x u 2 x of (1) was replaced by th e new v ariable v , wh ile the time ev olution of v w as determined by an add itional f irst-order equ ation, the s econd equation of (3). It w as announced in [2] that (3) is integ rable in th e s ense of p ossessing a Lax-t yp e represent ation. The system (3 ) wa s called in [2] an in tegrable r egularizatio n of the Whitham- t yp e equation (1). In the presen t pap er, w e consider the nonlinear PDEs ( 2) and (3) in Sectio ns 2 and 3, resp ectiv ely . W e giv e the r eferences whic h show that these equations (2) and (3), esp ecially the f irst one of them, were kn o wn and qu ite well stud ied in the literature prior to [1, 2]. W e put a particular emp hasis on the fac t that the equ ations (2) and (3 ) can b e completely solved by quadratures, and we ref ine the d eriv ations of their general solutions, consistentl y follo wing the 2 S. S ak o v ic h w a y used for the Rab elo equ ations in [3]. W e ha v e the f ollo w ing reasons to re-derive th e general solutions of (2) and (3). Firstly , the dif ferent d eriv ations of the general solutions of (2), giv en in the literature, all o v erlooke d the evident class of x -indep endent solutions. Secondly , the general solution of (3) can b e expr essed n ot only in a parametric form, kno wn in th e literature, but also in an implicit f orm. And thirdly , we need to get th e results in u niform notations, for Section 4, where we mak e a comparison of the general solutions of (2) and (3), taking into account that these t w o representa tions of the Whitham-type equatio n (1) are not equiv alent to eac h other, and ob tain all common solutions of (2) and (3) exactly . 2 The Hun ter–Saxton equation The nonlinear e quation (2) is f ar not no vel . Up to a scale transformation of its v ariables, this is the celebrated Hunter–Saxton equation [4], sometimes r eferred to as the Hunter–Zheng equation [5]. The Hunter–Saxto n equation has b een studied in almost all resp ects, includ ing its complete solv ability by quadr atures [4, 6, 7, 8], relationship with the Camassa–Holm equation and the L iouville equation [6 , 7 ], bi-Hamiltonian formulat ion [5, 9], in tegrable f inite-dimensional reductions [5 , 10 ], global solution pr op erties [11, 12], and geometric int erpretations [13, 14], to men tion on ly a few of numerous publications on this equ ation. In our o pinion, the most imp ortan t feature o f the Hun ter–Saxton equatio n is the p ossi- bilit y to obtain its general solution in a closed f orm. This equation is linearizable [8], or C-in tegrable in the Calogero’s terminology , but it also b elongs to a subset of C-in tegrable equations whose general solutions ca n b e exp ressed in a clo sed form. Suc h co mpletely sol- v able equ ations of the Liouville equation’s t yp e dif f er from other C-in tegrable equations of the Burgers equation’s t yp e, and from th e so-called S-int egrable (completely integrable, or Lax in- tegrable) equ ations of the sine-Gordon equation’s t yp e, in many resp ects. F or example, the Liouville equation p ossesses a con tin uum of v ariat ional s ymmetries (hence, a con tin uum of non trivial conserv ation la ws ) a nd sev eral Lax- t yp e represen tations whic h all turn out to be equiv alent to conserv ation la ws [15]. Leaving a s tudy of such prop erties of the Hu n ter–Saxton equation for a separate publication, here w e only co ncen trate on its general solution. The general solutio n of (2) can b e obtained in a parametric form, in at least thr ee dif feren t wa ys [4, 7, 8 ]. The deriv atio n we give b elo w is s imilar to th e original one of [4], but dif fers from it b y a more p recise treatmen t of the arbitrariness of the trans formation in v olv ed , in the spirit of [3]. Making the trans formation x = x ( y , t ) , x y 6 = 0 , u ( x, t ) = a ( y , t ) , (4) where th e function x ( y , t ) is initially not f ixed, and using the id en tities u x = a y x y , u t = a t − a y x t x y , u xx = a y y x 2 y − a y x y y x 3 y , u xt = a y t x y − a y y x t + a y x y t x 2 y + a y x y y x t x 3 y , (5) w e b ring th e nonlinear equation (2) int o the form a y t + a 2 y x y = ∂ y  ( x t + 2 a ) a y x y  . (6) No w we see from (6) that it is exp edient to f ix the function x ( y , t ) of the tran sformation (4 ) b y the condition x t + 2 a = 0, wh ic h brings the equation in to a constant -c haracteristic form and On a Whitham-Type Equ ation 3 considerably simplif ies it. Doing this, we f ind that the transformation (4) w ith a = − 1 2 x t (7) relates the second-order equation (2 ) with the third-ord er equation x y tt − x 2 y t 2 x y = 0 (8) whic h f ollo w s fr om (6) and (7). Through th e transformation (4) w ith (7), the general solution of the third-order equation (8) represent s the general solution of the second-order equation (2) parametrically , with y b eing the parameter. Note, h o w ev er , that, according to the Cauc h y–Ko v alevsk a ya theorem [16], the gene- ral s olution of (8) must contai n three arbitrary fun ctions of one v ariable, w hereas the general solution of (2) must con tain only t w o arbitrary functions of one v ariable. Th is r edundant arbi- trariness in x ( y , t ), caused by the inv ariance of (8) w ith resp ec t to an arbitrary transformation y 7→ Y ( y ) whic h has n o ef fect on u ( x, t ) of (2), can b e eliminated by the follo w ing normalizatio n of the parameter y . W e rewrite (8) in the form ∂ t  x − 1 / 2 y x y t  = 0 , in tegrate o v er t , and get x − 1 / 2 y x y t = f ( y ) , (9) where f ( y ) is an arb itrary function. F or any non zero fu nction f ( y ), we can set, without loss of generalit y , f = 2 in (9 ) b y an appropr iate transform ation y 7→ Y ( y ) whic h do es not c hange the corresp ondin g solutions of (2), where the v al ue 2 is c hosen for co n v enience only . The case of f = 0 must b e considered separately . C onsequen tly , all solutions of the second-order equation (2) are repr esen ted parametrically by all solutions of th e second-order equation (9) w ith f = 0 and f = 2 through the transf ormation (4) with (7). The case of f = 0 in (9) is x y t = 0, wh ic h immediately leads us thr ough (7) and (5) to a y = 0 and u x = 0, that is, to the evident class of solutions u = τ ( t ) (10) of (2), with any function τ ( t ). In the case of f = 2, w e in tegrate (9) ov er t and get x y =  t + φ ( y )  2 , (11) with an y function φ ( y ). Then, in tegrating (11) o v er y and usin g (7) and (4), we obtain the follo wing class of solutions of (2), determined parametrically: x = y t 2 + 2 t Z φ ( y ) dy + Z φ ( y ) 2 dy + ψ ( t ) , u ( x, t ) = − y t − Z φ ( y ) dy − 1 2 ψ ′ ( t ) , (12) where y is the p arameter, φ ( y ) and ψ ( t ) are arbitrary functions, and the p rime d enotes the deriv ative . The expressions (10) and (12) toget her constitute the general solution of the s econd- order n onlinear p artial dif feren tial equation (2 ). Some w ords are du e on the obtained general s olution of (2). It f ollo ws from (12) that u x = − 1 t + φ ( y ) . (13) 4 S. S ak o v ic h According to this relation (13), the condition u x 6 = 0 is satisf ied for an y fu nction φ ( y ), which pro v es that the class of solutions (12) do es n ot co v er solutions of t he class (1 0). F or some unknown reasons, only the parametric expressions (12) were called the general solution of (2) in [4, 7, 8], wh ereas the solutions (10) w ere omitted there. Also, the relation (13) make s clear that all solutions of (2), except for those of the class (10), inevitably p ossess singularities of the t yp e u x = ±∞ , w hen consid ered on the in terv al −∞ < t < ∞ . The trans formation (4), u sed for obtaining the general solution of (2 ), is applicable everywhere outsid e those singularities u x = ±∞ , b eca use the condition x y 6 = 0 is satisf ied due to (11). This inevitable presence of singularities in the solutions (12) w as noticed in [4]. In the next section, we sho w that nontrivial solutions of the repr esen tation (3) of the Whitham-t yp e equation (1) not necessarily con tain blo w-ups of deriv ativ es. 3 The Gurevic h–Zybin system Pro ceeding to the h yd ro dyn amic-t yp e system (3 ), w e note that this i s the one-dimensional reduction of the Gurevic h–Zybin sys tem [17, 18] w hic h can b e completely solv ed by quadra- tures [18, 19]. F or an earlie r app earance of (3) in plasma physic s, one can co nsult Section 3 of [20]. In [21], a bi-Hamiltonian structur e and a zero-curv ature represent ation were found an d studied for the s ystem (3). Bel o w we sho w ho w to obtain the general solution of (3) in an implicit f orm, follo wing the wa y u sed in [3]. Applying the transformation x = x ( y , t ) , x y 6 = 0 , u ( x, t ) = a ( y , t ) , v ( x, t ) = b ( y , t ) (14) to the sys tem (3), with x ( y , t ) b eing not f ixed initially , we obtain a t − ( x t + 2 a ) a y x y + b = 0 , b t − ( x t + 2 a ) b y x y = 0 . (15) Then we f ix the function x ( y , t ) in (14) and (15) by the condition x t + 2 a = 0, and thus get a = − 1 2 x t , b = 1 2 x tt , x ttt = 0 , that is, x = α ( y ) t 2 + β ( y ) t + γ ( y ) , a = − α ( y ) t − 1 2 β ( y ) , b = α ( y ) , (16) where α ( y ) , β ( y ) , γ ( y ) are three arbitrary fu nctions, of w hic h at least one is non-constant due to x y 6 = 0. The expr essions (14) a nd (16) represen t the general solution of the sys tem (3) parametrically , with y b eing the parameter. An app ropriate tr ansformation y 7→ Y ( y ), whic h has no ef fect on solutions u ( x, t ), v ( x, t ) of (3), may b e used to f ix an y on e of th e three arbitrary functions in (16) . This parametric general solution of (3) can b e expressed in an implicit form, as f ollo ws. When the f unction α ( y ) is non-constan t, w e r eplace a ( y , t ) and b ( y , t ) in (16) b y u ( x, t ) and v ( x, t ), resp ectiv ely , then eliminate y from th e resulting expr essions, and thus obtain x + v t 2 + 2 ut + µ ( v ) = 0 , u + v t + ν ( v ) = 0 , (17) where µ ( v ) and ν ( v ) are arb itrary fun ctions (expressible in terms of the arbitrary fun ctions α , β , γ ). When α ( y ) is constant bu t β ( y ) is not, w e do the same and get x + ξ t 2 + 2 ut + ρ ( u + ξ t ) = 0 , v = ξ , (18) On a Whitham-Type Equ ation 5 where ρ is an arbitrary f unction of its argument, and ξ is an arbitrary co nstan t. When α ( y ) and β ( y ) are constant but γ ( y ) is not, w e get u = η t + ζ , v = − η , (19) where η and ζ are a rbitrary constan ts. These expressions (17) –(19) together constitute the general solution of the n onlinear system (3). Unlik e all non trivial solutions of the Hunter–Saxton equation, some solutions of the Gurevic h– Zybin system (3), of the class (17), do not co n tain b lo w-ups of deriv ativ es. Indeed, it follo ws from (17) that the expression for an y deriv ativ e of u or v conta ins o nly some degree of the expression t 2 + 2 tν ′ ( v ) − µ ′ ( v ) in its denominator, for example, u x = − t − ν ′ ( v ) t 2 + 2 tν ′ ( v ) − µ ′ ( v ) , v x = 1 t 2 + 2 tν ′ ( v ) − µ ′ ( v ) , where the prime denotes the deriv at iv e. Clearly , it is p ossible to c ho ose the functions µ and ν so that t 2 + 2 tν ′ ( v ) − µ ′ ( v ) 6 = 0 holds on the whole in terv al −∞ < t < ∞ . 4 Discussion The general solutions of the Hu n ter–Saxton equatio n (2) and the Gurevic h–Zybin system (3) are quite d if ferent in their structure. The general solution of (2) is giv en in the p arametric form (12), exce pt for the explicit solutions (10). The general solution of (3) is giv en in th e implicit form (17 ) and (18), except for the explicit solutions (19). F rom this p oin t of view, the Hun ter–Saxton equation (2) and the Gurevic h–Zybin system (3) are v ery simila r to the exp- Rab elo equation u xt = exp u − (exp u ) xx and th e quadratic Rab elo equation u xt = 1 + 1 2 ( u 2 ) xx , resp ectiv ely [3]. The nonlinear PDEs (2) and (3) w ere considered in [1, 2] as t w o w ell-def ined r epresent ations of the Whitham-t yp e equation (1) wh ic h itself con tains the ill-def ined term ∂ − 1 x u 2 x . Eviden tly , these t w o representa tions are not equiv alen t to eac h other. The Gur evic h–Zybin system (3) can b e re-written as the second-order equation u tt − 4 uu xt + 4 u 2 u xx − 2 u x u t + 4 uu 2 x = 0 (20) for u ( x, t ) w ith the def inition v = 2 uu x − u t for v ( x, t ), and this s econd-order equation (20) dif f ers from the Hunt er–Saxton equation (2). F or th is reason, one ma y wonder whether the PDEs (2) and (3) ha v e an y common n on trivial solutions at all. It can b e found easily that the compatibilit y cond ition f or the equations (2) and (20) is u tt = 4 u 2 u xx + 2 u x u t . (21) Alternativ ely , in the v ariables u and v , the compatibilit y condition for the PDEs (2 ) and (3) is v x = u 2 x . (22) One can f ind all common solutions of (2 ) and (3) b y applying the condition (21 ) to the ge- neral solution (10) and (12) of the Hunter–Saxton equation, or, alternativ ely , b y app lying the condition (22) to th e general solution (17)–(19) of th e Gurevic h–Zybin system. Using the con- dition (21), we get τ ′′ = 0 (23) 6 S. S ak o v ic h from (10) , and ψ ′′′ = 0 (24) from (12) . Using the condition (22), we get µ ′ + ν ′ 2 = 0 (25) from (17 ), while (18) do es not satisfy (22), and (19 ) s atisf ies (22) id en tically . It is quite ob vious that (10) w ith (23) is equiv al en t to (19), and that (12) with (24) is equiv alen t to (17) with (25). Th us, summarizing the result in a nonrigourous wa y , we can sa y that the degree of arbitrari- ness of common n on trivial solutions of th e Hunter–Saxto n equation (2) and the Gurevich–Zybin system (3) is one arbitrary function of one v ariable. Ac kno wledgemen t The author is d eeply grateful to Professor E.V. F erap onto v and Professor M .V. P a vlo v for p ointi ng out th e origin of the sys tem (3), to the referees for their useful su ggestions, and to th e Max-Planc k -Institut f ¨ ur Mathematik for hospitalit y and supp ort. References [1] Pryk arpatsky A .K., Prytula M.M., The gradient-holonomic in tegrabilit y analysis of a Whitham-typ e n on- linear dynamical mo del for a relaxing medium with spatial memory , Nonli ne arity 19 (2006), 2115–2122. [2] Bogoliubov N.N. 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