On the classification of scalar evolutionary integrable equations in 2+1 dimensions
aa r X i v : . [ n li n . S I] N ov On the classification of scalar evolutionary integrable equations in2 + 1 dimensions
V.S. Novikov and E.V. Ferapontov
Department of Mathematical SciencesLoughborough UniversityLoughborough, Leicestershire LE11 3TUUnited Kingdome-mails:
[email protected]@lboro.ac.uk
Abstract
We consider evolutionary equations of the form u t = F ( u, w ) where w = D − x D y u is thenonlocality, and the right hand side F is polynomial in the derivatives of u and w . The recentpaper [6] provides a complete list of integrable third order equations of this kind. Here weextend the classification to fifth order equations. Besides the known examples of Kadomtsev-Petviashvili (KP), Veselov-Novikov (VN) and Harry Dym (HD) equations, as well as fifthorder analogues and modifications thereof, our list contains a number of equations whichare apparently new. We conjecture that our examples exhaust the list of scalar polynomialintegrable equations with the nonlocality w . The classification procedure consists of twosteps. First, we classify quasilinear systems which may (potentially) occur as dispersionlesslimits of integrable scalar evolutionary equations. After that we reconstruct dispersive termsbased on the requirement of the inheritance of hydrodynamic reductions of the dispersionlesslimit by the full dispersive equation.MSC: 35L40, 35Q51, 35Q58, 37K10, 37K55.Keywords: dispersionless equations, hydrodynamic reductions, dispersive deformations,integrability. Introduction
The classification of integrable 1 + 1 dimensional scalar evolutionary equations, u t = F ( u ) , has been (and still is) a subject of active research within the soliton community. Here u ( x, t )is a scalar potential, and F denotes a differential expression which depends on x -derivativesof u up to some finite order. Although the general classification problem is still out of reach,quite a few important results were obtained under various additional assumptions on F (suchas polynomiality, linearity in the highest derivative, etc). We refer to the review article [12]for a detailed discussion of the classification techniques involved, extensive lists of integrableequations within particularly interesting subclasses, and references.In this paper we apply the novel perturbative approach outlined in [5, 6] to a similar problemin 2 + 1 dimensions, the area where very few classification results are currently available. Themain challenge of higher dimensions is the non-locality of scalar evolutionary integrable equa-tions: the corresponding right hand side F must contain nonlocal variables whose differentialstructure was clarified in [13]. Here we consider equations of the form u t = F ( u, w ) (1)where u ( x, y, t ) is a scalar field and w = D − x D y u is the simplest nonlocality (equivalently, w canbe introduced via the relation w x = u y ). We assume that the right hand side F is polynomial in the x - and y -derivatives of u and w , while the dependence on u and w themselves is allowedto be arbitrary. The paper [6] provides a complete list of integrable third order equations of theform (1), u t = ϕu x + ψu y + ηw y + ǫ ( ... ) + ǫ ( ... ) , (2)where ϕ, ψ and η are functions of u and w , while the terms at ǫ and ǫ are assumed to behomogeneous differential polynomials of the order two and three in the derivatives of u and w (one can show that all terms at ǫ have to vanish). We use the following weighting scheme: u and w are assumed to have order zero, their derivatives u x , u y , w x , w y are of order one, theexpressions u xx , u xy , u yy , w yy , u x , u x u y , u y , u x w y , u y w y , w y are of order two, and so on. Assumingthat the dispersionless limit of the equation (2), u t = ϕu x + ψu y + ηw y , w x = u y , (3)is linearly nondegenerate (the property to be clarified in Sect. 2.2), and satisfies the condition η = 0 (which is equivalent to the requirement that the dispersion relation of the system (3)defines an irreducible conic), we have the following result: Theorem 1 [6] Up to invertible transformations, the examples below provide a complete list ofintegrable third order equations (2) with η = 0 whose dispersionless limit is linearly nondegen- rate: KP equation u t = uu x + w y + ǫ u xxx ,modif ied KP equation u t = ( w − u / u x + w y + ǫ u xxx ,Gardner equation u t = ( βw − β u + δu ) u x + w y + ǫ u xxx ,V N equation u t = ( uw ) y + ǫ u yyy ,modif ied V N equation u t = ( uw ) y + ǫ u yy − u y u ! y ,HD equation u t = − wu y + uw y − ǫ u (cid:18) u (cid:19) xxx ,def ormed HD equation u t = δu u x − wu y + uw y − ǫ u (cid:18) u (cid:19) xxx ,Equation E u t = ( βw + β u ) u x − βuu y + w y + ǫ [ B ( u ) − βu x B ( u )] ,Equation E u t = 43 β u u x + ( w − βu ) u y + uw y + ǫ [ B ( u ) − βu x B ( u )] , here B = βuD x − D y , β =const. The main result of this paper is a generalisation of the above classification to fifth orderequations, u t = ϕu x + ψu y + ηw y + ǫ ( ... ) + ǫ ( ... ) + ǫ ( ... ) + ǫ ( ... ) , (4)where the terms at ǫ k are assumed to be homogeneous differential polynomials of the order k + 1in the derivatives of u and w , respectively. We also assume that the ǫ term depends on at leastone of the possible fifth derivatives u xxxxx , u xxxxy , . . . , i.e. the equation (4) is of order 5. Theorem 2
Up to invertible transformations, the examples below provide a complete list of in-tegrable fifth order equations (4) with η = 0 whose dispersionless limit is linearly nondegenerate: BKP equation u t = 5( u + w ) u x + 5 uu y − w y + 5 ǫ ( uu xxx + u xxy + u x u xx ) + ǫ u xxxxx ,CKP equation u t = 5( u + w ) u x + 5 uu y − w y + 5 ǫ ( uu xxx + w xxx + 52 u x u xx ) + ǫ u xxxxx ,HD equation u t = 15 wu y − uw y + 5 ǫ (cid:20) u xxy u − u (cid:16) u x u y u (cid:17) x (cid:21) − ǫ u (cid:18) u (cid:19) xxxxx ,Equation E u t = 4 γ u x u + 5(3 w − γu ) u y − uw y + 5 ǫ (cid:20) γ u (cid:18) u (cid:19) xxx + u xxy u − u (cid:16) u x u y u (cid:17) x (cid:21) − ǫ u (cid:18) u (cid:19) xxxxx ,Equation E u t = 4 γ u x u + 5(3 w − γu ) u y − uw y + 5 ǫ " γ u (cid:18) u (cid:19) xxx − γ u x u − (cid:18) u (cid:19) xxy + (cid:16) u x u y u (cid:17) x − u y u (cid:0) uu xx − u x (cid:1) − ǫ (cid:20) u (cid:18) u (cid:19) xxxxx − (cid:18) (2 uu xx − u x ) u (cid:19) x (cid:21) .
3e point out that the last two examples from Theorem 2 are apparently new. The equation E can be viewed as a deformation of the fifth order Harry Dym equation HD : it reducesto HD when γ = 0. Although each equation appearing in Theorems 1-2 gives rise to anintegrable hierarchy, the corresponding higher flows will not belong to the class (1): they willnecessarily have a more complicated nonlocality. Preliminary calculations suggest that thereexist no seventh order equations of the form (1). This leads to the following Conjecture
Up to invertible transformations, Theorems 1-2 provide a complete list of integrableevolutionary equations of the form (1) which are polynomial in the derivatives of u and w . Remark.
The assumption of polynomiality is essential: there exist examples of integrableequations of the form (1) where the right hand side F is an infinite series in ǫ . As an illustration,let us consider integrable differential-difference equations of the Toda lattice, v t = v △ − ( w ) , w x = △ + ( v ) , where △ − ( w ) = w ( y ) − w ( y − ǫ ) ǫ , △ + ( v ) = v ( y + ǫ ) − v ( y ) ǫ . Introducing the variable u by the formula △ + ( v ) = u y , one can rewrite the equations of theToda lattice in such a way that the nonlocality w will be of the required form, u t = D − y △ + (cid:0) △ − ( u y ) △ − ( w ) (cid:1) , w x = u y . Expanding the first equation in powers of ǫ one obtains an infinite series, u t = uw y + ǫ
12 ( uw yy ) y + ..., w x = u y . Examples of this type will be outside the scope of this paper.The structure of the paper is as follows. Following [6], in Sect. 2.1 we review the classificationof integrable quasilinear systems of the form (3). In Sect 2.2 we outline the general procedurewhich, starting with an integrable dispersionless system, allows one to systematically reconstructdispersive corrections. This procedure is applied in Sect. 2.3 to the case of fifth order equations(4). For the reader’s convenience, in Sect. 3 we present Lax pairs for all equations appearing inTheorems 1-2.
The proof consists of two steps. In Sect. 2.1 we review the classification of integrable quasilinearsystems (3) which may (potentially) occur as dispersionless limits of integrable soliton equations.In Sect. 2.2 we discuss the general procedure of the reconstruction of dispersive corrections basedon the requirement of the inheritance of hydrodynamic reductions. This procedure is applied tofifth order equations in Sect. 2.3, leading to the proof of Theorem 2.
For a system of the form (3), u t = ϕu x + ψu y + ηw y , w x = u y , ϕ, ψ and η , ϕ uu = − ϕ w + ψ u ϕ w − ψ w ϕ u η , ϕ uw = η w ϕ u η , ϕ ww = η w ϕ w η ,ψ uu = − ϕ w ψ w + ψ u ψ w − ϕ w η u + 2 η w ϕ u η , ψ uw = η w ψ u η , ψ ww = η w ψ w η ,η uu = − η w ( ϕ w − ψ u ) η , η uw = η w η u η , η ww = η w η ;we assume η = 0: this is equivalent to the requirement that the dispersion relation of the system(3) defines an irreducible conic. The integrability conditions are straightforward to solve. Firstof all, the equations for η imply that, modulo translations and rescalings, one can set η = 1, η = u or η = e w h ( u ). We will consider all three possibilities case-by-case below. Notice that ϕ and ψ are defined up to additive constants which can always be set equal to zero via theGalilean transformations of the initial system (3). Moreover, the integrability conditions areform-invariant under transformations of the form˜ ϕ = ϕ − sψ + s η, ˜ ψ = ψ − sη, ˜ η = η, ˜ u = u, ˜ w = w + su, s = const, which correspond to the following transformations preserving the structure of system (3):˜ x = x − sy, ˜ y = y, ˜ u = u, ˜ w = w + su. All our classification results are formulated modulo this equivalence.
Case 1: η = 1. Then the remaining equations imply ψ = αw + f ( u ) , ϕ = βw + g ( u ), where f and g satisfy the linear ODEs f ′′ = α ( f ′ − β ) , g ′′ = 2 αg ′ − βf ′ − β . The subcase α = 0 leads to polynomial solutions of the form ψ = γu, ϕ = βw − β ( β + γ ) u + δu. (5)Up to equivalence transformations, the case α = 0 leads to exponential solutions, ψ = w + βe u , ϕ = αe u , (6)where α, β, γ are arbitrary constants. Case 2: η = u . Then the remaining equations imply ψ = αw + f ( u ) , ϕ = βw + g ( u ), where f and g satisfy the linear ODEs uf ′′ = α ( f ′ − β ) − β, ug ′′ = 2 αg ′ − βf ′ − β . The case α / ∈ { , − , − / } leads to power-like solutions of the form ψ = αw + γu α +1 , ϕ = δu α +1 . (7)5he subcase α = 0 leads to logarithmic solutions, ψ = − βu ln u − βu, ϕ = βw + β u ln u + δu. (8)The subcase α = − ψ = − w + γ ln u, ϕ = δ/u. (9)Finally, the subcase α = − / ψ = − w + γ √ u, ϕ = δ ln u. (10) Case 3: η = e w h ( u ). Then the remaining equations imply ψ = e w f ( u ) , ϕ = e w g ( u ) where f , g and h satisfy the nonlinear system of ODEs, h ′′ = f ′ − g, g ′′ h = 2 f g ′ − gf ′ − g , f ′′ h = 2 hg ′ − gh ′ + f f ′ − f g. Although the structure of the general solution is this system is quite complicated, one can showthat Case 3 cannot occur as the dispersionless limit of an integrable soliton equation.
Given an integrable dispersionless system of the form (3), one has to reconstruct dispersiveterms. This can be done by requiring that all hydrodynamic reductions of the dipersionlesssystem are inherited by its dispersive counterpart [5, 6]. Following [6], we will illustrate thisprocedure with the example of the KP equation, u t = uu x + w y + ǫ u xxx , w x = u y . The dispersionless KP (dKP) equation, u t = uu x + w y , w x = u y , possesses one-phase solutions of the form u = R , w = w ( R ) where the phase R ( x, y, t ) satisfiesa pair of Hopf-type equations R y = µR x , R t = ( µ + R ) R x ; (11)here µ ( R ) is an arbitrary function, and w ′ = µ . Equivalently, one can say that Eqs. (11)constitute a one-component hydrodynamic reduction of the dKP equation. Although the dKPequation is known to possess infinitely many N -component reductions for arbitrary N [7, 8, 9, 10],one-component reductions will be sufficient for our purposes. The main observation of [5] is that all one-component reductions (11) can be deformed into reductions of the full KP equation byadding appropriate dispersive terms which are polynomial in the x -derivatives of R . Explicitly,one has the following formulae for the deformed one-phase solutions, u = R, w = w ( R ) + ǫ (cid:18) µ ′ R xx + 12 ( µ ′′ − ( µ ′ ) ) R x (cid:19) + O ( ǫ ) , (12)6otice that one can always assume that u remains undeformed modulo the Miura group [2]. Thedeformed equations (11) take the form R y = µR x + ǫ (cid:18) µ ′ R xx + 12 ( µ ′′ − ( µ ′ ) ) R x (cid:19) x + O ( ǫ ) ,R t =( µ + R ) R x + ǫ (cid:0) (2 µµ ′ + 1) R xx + ( µµ ′′ − µ ( µ ′ ) + ( µ ′ ) / R x (cid:1) x + O ( ǫ ) , (13)see [5]. In other words, the KP equation can be ‘decoupled’ into a pair of (1 + 1)-dimensionalequations (13) in infinitely many ways, indeed, µ ( R ) is an arbitrary function. The series in (12)and (13) contain even powers of ǫ only, and do not terminate in general.Conversely, the requirement of the inheritance of all one-component reductions allows oneto reconstruct dispersive terms: given the dKP equation, let us look for a third order dispersiveextension in the form u t = uu x + w y + ǫ ( ... ) + ǫ ( ... ) , w x = u y , (14)where the terms at ǫ and ǫ are homogeneous differential polynomials in the derivatives of u and w of the order two and three, respectively. We require that all one-component reductions (11)can be deformed accordingly, so that we have the following analogues of Eqs. (12) and (13), u = R, w = w ( R ) + ǫ ( ... ) + ǫ ( ... ) + O ( ǫ ) , (15)and R y = µR x + ǫ ( ... ) + ǫ ( ... ) + O ( ǫ ) , R t = ( µ + R ) R x + ǫ ( ... ) + ǫ ( ... ) + O ( ǫ ) , (16)respectively. In Eqs. (15) and (16), dots denote terms which are polynomial in the derivativesof R . Substituting Eqs. (15) into (14), and using (16) along with the consistency conditions R ty = R yt , one arrives at a complicated set of relations allowing one to uniquely reconstructdispersive terms in (14): not surprisingly, we obtain that all terms at ǫ vanish, while the termsat ǫ result in the familiar KP equation. Moreover, one only needs to perform calculations upto the order ǫ to arrive at this result. It is important to emphasise that the above procedureis required to work for arbitrary µ : whenever one obtains a differential polynomial in µ whichhas to vanish due to the consistency conditions, all its coefficients have to be set equal to zeroindependently. Another observation is that the reconstruction procedure does not necessarilylead to a unique dispersive extension like in the dKP case: one and the same dispersionlesssystem may possess essentially non-equivalent dispersive extensions. In particular, VN andmodified VN equations from Theorem 1, as well as BKP and CKP equations from Theorem 2have coinciding dispersionless limits.Let us now turn to the general case of dispersionless equations of the form (3), u t = ϕu x + ψu y + ηw y , w x = u y . The corresponding one-component reductions are of the form u = R , w = w ( R ) where R ( x, y, t )satisfies a pair of Hopf-type equations R y = µR x , R t = ( ϕ + ψµ + ηµ ) R x ;7ere µ ( R ) is an arbitrary function, and w ′ = µ . Let us seek a third order dispersive deformationof system (3) in the form u t = ϕu x + ψu y + ηw y + ǫ ( ... ) + ǫ ( ... ) , w x = u y , and postulate that one-phase solutions can be deformed accordingly, u = R, w = w ( R ) + ǫ ( ... ) + ǫ ( ... ) + O ( ǫ ) , where R y = µR x + ǫ ( ... ) + ǫ ( ... ) + O ( ǫ ) , R t = ( ϕ + ψµ + ηµ ) R x + ǫ ( ... ) + ǫ ( ... ) + O ( ǫ ) . Proceeding as outlined above we reconstruct dispersive terms.
Remark.
We point out that the formulae for dispersive deformations contain the expression η w µ + ( ψ w + η u ) µ + ( ϕ w + ψ u ) µ + ϕ u in the denominator. Since µ is assumed to be arbitrary, this expression is nonzero unless ϕ, ψ, η satisfy the relations η w = 0 , ψ w + η u = 0 , ϕ w + ψ u = 0 , ϕ u = 0 . (17)These relations characterise the so-called totally linearly degenerate systems . Dispersive deforma-tions of such systems do not inherit hydrodynamic reductions, and require a different approachwhich is beyond the scope of this paper. In this Section we summarize the classification results for integrable fifth order equations (4), u t = ϕu x + ψu y + ηw y + ǫ ( ... ) + ǫ ( ... ) + ǫ ( ... ) + ǫ ( ... ) , which are obtained by adding dispersive terms to integrable dispersionless candidates from Sect.2.1 (one can show that all terms at ǫ and ǫ have to vanish). Thus, we follow the classificationof Sect. 2.1. We concentrate on the case when the ǫ -terms contain at least one fifth orderderivative of u or w , and skip all cases leading to third order equations which were alreadyclassified in [6]. Case 1:
We verified that the exponential solutions (6) do not survive, so that all non-trivialexamples come from the polynomial case (5), η = 1 , ψ = γu, ϕ = βw − β ( β + γ ) u + δu. A detailed analysis of dispersive deformations leads to the constraints γ = β, δ = 0. Modulorescalings, this gives BKP/CKP equations. Case 2:
One can prove that none of the logarithmic cases (8), (9) and (10) survive, so that allnon-trivial examples come from the power case (7), η = u, ψ = αw + γu α +1 , ϕ = δu α +1 . The further analysis leads to the only possibility α = −
3. Modulo rescalings, the case δ = γ = 0gives the HD equation. The case of nonzero δ and γ leads to the new equations E and E . Case 3:
One can show that no examples from this class possess fifth order dispersive extensions.8
Lax pairs
For the reader’s convenience, in this section we bring together Lax pairs for all equations ap-pearing in Theorems 1-2. We emphasise that our classification scheme does not assume theexistence of a Lax pair: these come as the result of direct calculations once the classificationis completed. We refer to [11, 17] for an alternative approach to the classification of integrablesystems in 2 + 1 dimensions based on postulating the structure of a Lax pair.
Since both KP and modified KP equations are particular cases of the Gardner equation, we willskip the first two examples.The
Gardner equation , u t = ( βw − β u + δu ) u x + w y + ǫ u xxx , possesses the Lax pair [11] ǫ ψ xx + ǫ √ ψ y − βuψ x ) + δ uψ = 0 ,ǫψ t = 4 ǫ ψ yyy − √ βǫ (2 ψ xx + u x ψ x ) + ǫ ( βw + β u + δu ) ψ x + ǫ δ u x − βδ √ u + δ √ w. The
VN equation , u t = ( uw ) y + ǫ u yyy , possesses the Lax pair [16, 15] ǫ ψ xy + 13 uψ = 0 ,ψ t = ǫ ψ yyy + wψ y . The modified VN equation , u t = ( uw ) y + ǫ u yy − u y u ! y , possesses the Lax pair [1] ǫ ψ xy − ǫ u y u ψ x + 13 uψ = 0 ,ψ t = ǫ ψ yyy + wψ y + 12 w y ψ. The
HD equation , u t = − wu y + uw y − ǫ u (cid:18) u (cid:19) xxx , possesses the Lax pair [11] ǫu ψ xx + 1 √ ψ y = 0 ,ψ t = 4 ǫ u ψ xxx + √ ǫwu − ǫ u x u ! ψ xx . deformed HD equation , u t = δu u x − wu y + uw y − ǫ u (cid:18) u (cid:19) xxx , possesses the Lax pair [6] ǫ u ψ xx + ǫ √ ψ y + δ u ψ = 0 ,ψ t = 4 ǫ u ψ xxx + √ ǫwu − ǫ u x u ! ψ xx + δu ψ x + − δu x u + √ δw ǫu ! . The
Equation E , u t = ( βw + β u ) u x − βuu y + w y + ǫ [ B ( u ) − βu x B ( u )] , possesses the Lax pair [6] ǫ ψ xy = ǫ βuψ xx + 13 ψ,ψ t = ǫ β u ψ xxx − ǫ ψ yyy + 3 ǫ β uu y ψ xx + βwψ x . The
Equation E , u t = 43 β u u x + ( w − βu ) u y + uw y + ǫ [ B ( u ) − βu x B ( u )] , possesses the Lax pair [6] ǫ ψ xy = ǫ βuψ xx + 13 uψ,ψ t = ǫ β u ψ xxx − ǫ ψ yyy + 3 ǫ β uu y ψ xx + β u ψ x + wψ y + βuu y ψ. The
BKP equation , u t = 5( u + w ) u x + 5 uu y − w y + 5 ǫ ( uu xxx + u xxy + u x u xx ) + ǫ u xxxxx , possesses the Lax pair [11] ψ y + uψ x + ǫ ψ xxx = 0 ,ψ t + 5( u − w ) ψ x + ǫ (15 uψ xxx + 15 u x ψ xx + 10 u xx ψ x ) + 9 ǫ ψ xxxxx = 0 . The
CKP equation , u t = 5( u + w ) u x + 5 uu y − w y + 5 ǫ ( uu xxx + u xxy + 52 u x u xx ) + ǫ u xxxxx , ψ y + uψ x + 12 u x ψ + ǫ ψ xxx = 0 ,ψ t + 5( u − w ) ψ x + 5( uu x − u y ) ψ + ǫ (15 uψ xxx + 452 u x ψ xx + 352 u xx ψ x + 5 u xxx ψ ) + 9 ǫ ψ xxxxx = 0 . The
HD equation , u t = 15 wu y − uw y + 5 ǫ (cid:20) u xxy u − u (cid:16) u x u y u (cid:17) x (cid:21) − ǫ u (cid:18) u (cid:19) xxxxx , possesses the Lax pair [11] ψ y + ǫ u ψ xxx = 0 ,ψ t + 15 ǫ wu ψ xxx + ǫ (cid:20) u ψ xxxxx − u x u ψ xxxx + 15 u (cid:18) u (cid:19) xx ψ xxx (cid:21) = 0 . The
Equation E , u t =4 γ u x u + 5(3 w − γu ) u y − uw y + 5 ǫ (cid:20) γ u (cid:18) u (cid:19) xxx + u xxy u − u (cid:16) u x u y u (cid:17) x (cid:21) − ǫ u (cid:18) u (cid:19) xxxxx , possesses the Lax pair ψ y − γu ψ x + ǫ u ψ xxx = 0 ,ψ t + (cid:18) γ u − γwu (cid:19) ψ x + 15 ǫ (cid:20)(cid:16) wu − γu (cid:17) ψ xxx + 3 γu x u ψ xx + 2 γu (cid:16) u x u (cid:17) x ψ x (cid:21) + ǫ (cid:20) u ψ xxxxx − u x u ψ xxxx + 15 u (cid:18) u (cid:19) xx ψ xxx (cid:21) . The
Equation E , u t =4 γ u x u + 5(3 w − γu ) u y − uw y + 5 ǫ " γ u (cid:18) u (cid:19) xxx − γ u x u − (cid:18) u (cid:19) xxy + (cid:16) u x u y u (cid:17) x − u y u (cid:0) uu xx − u x (cid:1) − ǫ (cid:20) u (cid:18) u (cid:19) xxxxx − (cid:18) (2 uu xx − u x ) u (cid:19) x (cid:21) , possesses the Lax pair ψ y + (cid:16) γu x u + u y u (cid:17) ψ − γu ψ x + ǫ (cid:20) u ψ xxx + (cid:18) u xx u − u x u (cid:19) ψ x + (cid:18) u (cid:16) u xx u (cid:17) x + 34 u x u (cid:19) ψ (cid:21) = 0 , t + (cid:18) u u y + 2 γu x ) w u − γu y u − γ u x u − w y (cid:19) ψ + (cid:18) γ u − γwu (cid:19) ψ x ++ ǫ (cid:20) wu − γ ) u ψ xxx + (cid:18) γu x u + 15 u y u (cid:19) ψ xx ++ (cid:18) γu (cid:16) u x u (cid:17) x + 15 u x u y u + (90 uu xx − u x ) w u (cid:19) ψ x ++ (cid:18) γ (cid:16) u xx u (cid:17) x − γu (cid:18) u x u (cid:19) x + 52 (cid:16) u xy u (cid:17) x + 152 u w (cid:16) u xx u (cid:17) x + 45 u x u w + 15 u x u y u (cid:19) ψ (cid:21) ++ ǫ (cid:20) u ψ xxxxx − u x u ψ xxxx + (cid:18) − u xx u + 225 u x u (cid:19) ψ xxx + (cid:18) (cid:16) u xx u (cid:17) x + 180 u x u (cid:19) ψ xx ++ (cid:18) (cid:16) u xxx u (cid:17) x − (cid:16) u x u xx u (cid:17) x − u xx u + 8256 u (cid:18) u x u (cid:19) x − u x u (cid:19) ψ x + (cid:18) − (cid:18) u (cid:19) xxxxx + 1952 (cid:16) u x u xxx u (cid:17) x + 1352 (cid:18) u xx u (cid:19) x − (cid:18) u x u xx u (cid:19) x − u x u xx u ++ 33752 (cid:18) u x u (cid:19) x + 3645 u x u (cid:19) ψ (cid:21) . We do not exclude a possibility that simpler Lax pair can be found in this case.
Acknowledgements
The research of EVF was partially supported by the European Research Council AdvancedGrant FroM-PDE.
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