Integrable Generalized KdV and MKdV Equations with Spatiotemporally Varying Coefficients
aa r X i v : . [ m a t h - ph ] S e p Integrable Generalized KdV and MKdV Equations withSpatiotemporally Varying Coefficients
Matthew Russo and S. Roy Choudhury
Department of Mathematics, University of Central Florida, Orlando, FL 32816-1364 USA
Corresponding author email: [email protected] 9, 2018
Abstract
A technique based on extended Lax Pairs is first considered to derive variable-coefficient gen-eralizations of various Lax-integrable NLPDE hierarchies recently introduced in the literature. Asillustrative examples, we consider generalizations of KdV equations and three variants of general-ized MKdV equations. It is demonstrated that the techniques yield Lax- or S-integrable NLPDEswith both time- AND space-dependent coefficients which are thus more general than almost allcases considered earlier via other methods such as the Painlev´e Test, Bell Polynomials, and varioussimilarity methods.However, this technique, although operationally effective, has the significant disadvantage that,for any integrable system with spatiotemporally varying coefficients, one must ’guess’ a generaliza-tion of the structure of the known Lax Pair for the corresponding system with constant coefficients.Motivated by the somewhat arbitrary nature of the above procedure, we embark in this paperon an attempt to systematize the derivation of Lax-integrable sytems with variable coefficients.An ideal approach would be a method which does not require knowledge of the Lax pair to anassociated constant coefficient system, and also involves little to no guesswork. Hence we attemptto apply the Estabrook-Wahlquist (EW) prolongation technique, a relatively self-consistent pro-cedure requiring little prior infomation. However, this immediately requires that the techniquebe significantly generalized or broadened in several different ways, including solving matrix par-tial differential equations instead of algebraic ones as the structure of the Lax Pair is deducedsystematically following the standard Lie-algebraic procedure of proceeding downwards from thecoefficient of the highest derivative. The same is true while finding the explicit forms for the various’coefficient’ matrices which occur in the procedure, and which must satisfy the various constraintequations which result at various stages of the calculation.The new and extended EW technique whch results is illustrated by algorithmically derivinggeneralized Lax-integrable versions of the generalized fifth-order KdV, and MKdV equations.Key Words: Generalizing Lax or S-integrable equations, spatially and temporally-dependentcoefficients, generalized Lax Pairs, extended Estabrook-Wahlquist method.1
Introduction
Variable Coefficient Korteweg de Vries (vcKdV) and Modified Korteweg de Vries (vcMKdV)equationshave a long history dating from their derivation in various applications[1]-[10]. However, almostall studies, including those which derived exact solutions by a variety of techniques, as well asthose which considered integrable sub-cases and various integrability properties by methods suchas Painlev´e analysis, Hirota’s method, and Bell Polynomials treat vcKdV equations with coefficientswhich are functions of the time only. For instance, for generalized variable coefficient NLS (vcNLS)equations, a particular coefficient is usually taken to be a function of x [11], as has also beensometimes done for vcMKdV equations[12]. The papers [13]-[14] are somewhat of an exception inthat they treat vcNLS equations having coefficients with general x and t dependences. Variationalprinciples, solutions, and other integrability properties have also been considered for some of theabove variable coefficient NLPDEs in cases with time-dependent coefficients.In applications, the coefficients of vcKdV equations may include spatial dependence, in additionto the temporal variations that have been extensively considered using a variety of techniques. Bothfor this reason, as well as for their general mathematical interest, extending integrable hierarchiesof nonlinear PDEs (NLPDEs) to include both spatial and temporal dependence of the coefficientsis worthwhile.Given the above, we compare two methods for deriving the integrability conditions of botha general form of variable-coefficient MKdV (vcMKdV) equation, as well as a general, variable-coefficient KdV (vcKdV) equation. In both cases,the coefficients are allowed to vary in space ANDtime.The first method employed here is based on directly establishing Lax integrability (or S-integrability to use the technical term) as detailed in the following sections. As such, it is fairlygeneral, although subject to the ensuing equations being solvable. We should stress that the com-puter algebra involved is quite challenging, and an order of magnitude beyond that encounteredfor integrable, constant coefficient NLPDEs.However, this first technique, although operationally effective, has the significant disadvantagethat, for any integrable system with spatiotemporally varying coefficients, one must ’guess’ a gen-eralization of the structure of the known Lax Pair for the corresponding system with constantcoefficients. This involves replacing constants in the Lax Pair for the constant coefficient integrablesystem, including powers of the spectral parameter, by functions. Provided that one has guessedcorrectly and generalized the constant coefficient system’s Lax Pair sufficiently, and this is of coursehard to be sure of ’a priori’, one may then proceed to systematically deduce the Lax Pair for thecorresponding variable-coefficient integrable system.Motivated by the somewhat arbitrary nature of the above procedure, we next attempt to sys-tematize the derivation of Lax-integrable sytems with variable coefficients. Of the many techniqueswhich have been employed for constant coefficient integrable systems, the Estabrook-Wahlquist(EW) prolongation technique [15]-[18]is among the most self-contained. The method directly pro-ceeds to attempt construction of the Lax Pair or linear spectral problem, whose compatibilitycondition is the integrable system under discussion. While not at all guaranteed to work, anysuccessful implementation of the technique means that Lax-integrability has already been verifiedduring the procedure, and in addition the Lax Pair is algorithmically obtained. If the techniquefails, that does not necessarily imply non-integrability of the equation contained in the compatibil-ity condition of the assumed Lax Pair. It may merely mean that some of the starting assumptions2ay not be appropriate or general enough.Hence we attempt to apply the Estabrook-Wahlquist (EW) method as a second, more algorith-mic, technique to generate a variety of such integrable systems with such spatiotemporally varyingcoefficients. However, this immediately requires that the technique be significantly generalized orbroadened in several different ways which we then develop and outline, before illustrating this newand extended method with examples.The outline of this paper is as follows. In Section 2, we briefly review the Lax Pair method andits modifications for variable-coefficient NLPDEs, and then apply it to three classes of generalizedvcMKdV equations. In section 3 we consider an analogous treatment of a generalized vcKdVequation. In section 4, we lay out the extensions required to apply the EW procedure to Lax-integrable systems with spatiotemporally varying coefficients. Sections 5 and 6 then illustrate thisnew, extended EW method in detail for Lax-integrable versions of the MKdV and generalizedKorteweg-deVries (KdV) equations respectively, each with spatiotemporally varying coefficients.We also illustrate that this generalized EW procedure algorithmically generates the same resultsas those obtained in a more ad hoc manner in Sections 2 and 3. Some solutions of the generalizedvcKdV equation considered in Sections 3 and 6 are then obtained in Section 7 via the use oftruncated Painlev´e expansions. Section 8 briefly reviews the results and conclusions, and directionsfor possible future work.More involved algebraic details, which are integral to both the procedures employed here, arerelegated to the appendices. In the Lax pair method [20] - [21] for solving and determining the integrability conditions fornonlinear partial differential equations (NLPDEs) a pair of n × n matrices, U and V needs to bederived or constructed. The key component of this construction is that the integrable nonlinearPDE under consideration must be contained in, or result from, the compatibility of the followingtwo linear Lax equations (the Lax Pair) Φ x = U Φ (1)Φ t = V Φ (2)where Φ is an eigenfunction, and U and V are the time-evolution and spatial-evolution (eigenvalueproblem) matrices.From the cross-derivative condition (i.e. Φ xt = Φ tx ) we get U t − V x + [ U, V ] = ˙0 (3)known as the zero-curvature condition where ˙0 is contingent on v ( x, t ) being a solution to thenonlinear PDE. A Darboux transformation can then be applied to the linear system to obtainsolutions from known solutions and other integrability properties of the integrable NLPDE.We first consider the following three variants of generalized variable-coefficient MKdV(vcMKdV) equations: v t + a v xxx + a v v x = 0 (4)3 t + b v xxxxx + b v v xxx + b vv x v xx + b v x + b v v x = 0 (5) v t + c v xxxxxxx + c v v xxxxx + c vv xx v xxx + c v x v xxx + c v x v xx + c v v xxx + c v v x v xx + c v v x + c v v x = 0 (6)These equations, which we shall always call the physical (or field) NLPDEs to distinguish themfrom the many other NLPDEs we encounter, will be Lax-integrable or S-integrable if we can finda Lax pair whose compatibility condition (3) contains the appropriate equation ((4), (5) or (6)).One expands the Lax pair U and V in powers of v and its derivatives with unknown functions ascoefficients. This results in a VERY LARGE system of coupled NLPDEs for the variable coefficientfunctions in (4)-(6). Upon solving these (and a solution is not guaranteed, and may prove to beimpossible to obtain in general for some physical NLPDEs), we simultaneously obtain the Lax pairand integrability conditions on the a i , b i , and c i for which (4)-(6) are Lax-integrable.The results, for which the details are given in Appendix A, are given in the following threesubsections. a i a a x − a a a x a xx + a a a xxx − K t K a + a a t − a a xxx +3 a xx a a x − a x a a x + 3 a x a a xx = 0 (7)where K ( t ) is an arbitrary function of t . b i b = 2( b + b ) (8) b = H ( t )(2 b − b ) (9)12 b x b xx b + 12 b b xx b x − b xx b x b + 4 b b xxx b − b b xxx b − b b x b x +24 b b x b x + 12 b b x b xx − b b x b xx + 4 b b b xxx − b b b xxx − b x b b x +6 b x b b x − b x b b xx + b xxx b − b b x − b b xxx + 24 a x b − b b x b xx b + 6 b b b x b xx = 0 (10)4 b b x b x + 9600 b b x b x − b b x b x + 1200 b b x b x − b x b b x − b x b b x +1920 b x b x b + 120 b x b b x + 3840 b b x − b b x + 7680 b x b b x b x − b x b b x b x +1920 b x b b x b x − b x b x b b x + 2880 b x b x b b x − b x b x b b x − b xxxxx b + b xxxxx b + 10 b xxxx b x b − b xxxx b b x − b xxxx b b x + 80 b xxxxx b b − b xxxxx b b +40 b xxxxx b b − b xxxxx b b + 32 b b b xxxxx + 2 b b xxxxx b − b b b xxxxx − b b b xxxxx − b x b b xx − b x b b xx + 30 b x b b xx + 160 b x b b xxxx − b x b b xxxx + 10 b x b b xxxx − b x b b xxxx + 1920 b xx b b x − b xx b b x + 480 b xx b x b − b xx b b x + 320 b xx b b xxx − b xx b b xxx + 20 b xx b xxx b − b xx b b xxx − b xxx b b x − b xxx b b x +80 b xxx b x b + 20 b xxx b b x + 320 b xxx b b xx − b xxx b b xx + 20 b xxx b xx b − b xxx b b xx + 960 b b b xx b xxx b − b b b xx b xxx b − b b b xx b xxx b +240 b b b xx b + 240 b b b b xx b xxx − b b b b xx b xxx + 480 b b b x b xxxx b − b b b x b xxxx b − b b b x b b xxxx + 120 b b b x b b xxxx − b b b xxxx b b x +120 b b b xxxx b b x + 120 b b b b x b xxxx − b b b b x b xxxx − b x b b x b xx b x +2880 b x b b x b x b xx − b x b b x b xx b + 2880 b x b b x b b xx − b x b b xx b b x +720 b x b b b x b xx − b x b x b xx b b x + 720 b x b b b x b xx − b x b x b xx b b x +720 b x b x b b x b xx + 1920 b x b b x b xxx b − b x b b x b b xxx − b x b b xxx b b x +480 b x b b b x b xxx − b x b b x b xxx b + 480 b x b b x b b xxx + 480 b x b b xxx b b x − b x b b b x b xxx + 2880 b xx b b x b xx b − b xx b b x b b xx − b xx b b xx b b x +720 b xx b b b x b xx − b xx b b x b xx b + 720 b xx b b x b b xx + 720 b xx b b xx b b x − b xx b b b x b xx + 11520 b b b x b xx b x − b b x b xx b b x − b b b x b x b xx +2880 b b x b b x b xx − b b b x b xx b x + 2880 b b x b xx b b x + 2880 b b b x b x b xx − b b x b b x b xx − b b b x b xx b − b b b x b xx b xx − b b x b xx b b xx − b b b x b b xx + 1440 b b b xx b b x + 1440 b b b xx b x b xx + 360 b b xx b b x b xx +360 b b b b x b xx − b b b x b xxx b + 960 b b b x b b xxx − b b b x b xxx b x − b b x b xxx b b x + 960 b b b x b x b xxx + 240 b b x b b x b xxx − b b b xxx b b x +240 b b b b x b xxx − b x b b xx b b xx + 720 b x b b xx b b xx + 2880 b xx b b x b b x − b xx b b x b b x − b xxx b b x b b x + 480 b xxx b b x b b x + 5760 b x b b x b xx b b x − b x b b x b b x b xx + 2880 b b b x b xx b b xx − b b b xx b b x b xx + 80 b b xx b xxx b +1920 b b b x b xxx b b x − b b b x b b x b xxx − b b b xx b xxx + 320 b b b xx b xxx − b b xx b b xxx + 320 b b b xxx b xx − b b xxx b b xx − b b b xx b xxx + 20 b b b xx b xxx − b b b x b xxxx + 40 b b x b xxxx b + 160 b b b x b xxxx − b b x b b xxxx + 160 b b b xxxx b x − b b xxxx b b x − b b b x b xxxx + 10 b b b x b xxxx + 5760 b x b b x b xx − b x b b x b xx b x b xx b b x − b x b b x b xxx − b xx b b x b xx − b xx b x b b xx +1440 b x b b xx b x − b x b b x b xx + 1440 b x b x b xx b − b x b x b b xx − b x b b x b xx − b x b b x b xxx + 640 b x b b x b xxx + 640 b x b b xxx b x +160 b x b x b xxx b − b x b x b b xxx − b x b xxx b b x + 40 b x b b x b xxx − b xxxx b b x b +960 b xx b b x b xx + 960 b xx b b xx b x − b xx b b x b xx + 240 b xx b x b xx b − b xx b xx b b x + 60 b xx b b x b xx − b xxxx b b x b + 240 b xxxx b b x b +160 b xxxx b b b x − b xxxx b b b x + 40 b xxxx b b b x − b b b x b xx + 3840 b b x b xx b +3840 b b b x b xx − b b x b b xx + 960 b b b xx b x − b b xx b b x − b b b x b xx +240 b b b x b xx + 2880 b b b x b xx + 720 b b x b xx b + 720 b b b x b xx + 180 b b x b b xx − b b b xx b x − b b xx b b x − b b b x b xx − b b b x b xx + 1920 b b b x b xxx +480 b b x b xxx b − b b b x b xxx − b b x b b xxx + 480 b b b xxx b x + 120 b b xxx b b x − b b b x b xxx − b b b x b xxx − b b b xxxxx b + 48 b b b xxxxx b − b b b xxxxx b +32 b b b b xxxxx − b b b b xxxxx + 8 b b b b xxxxx + 960 b x b b xx b xx + 1440 b x b b xx b +360 b x b b b xx − b x b b xx b − b x b b b xx − b x b xx b b xx − b x b b xxxx b +160 b x b b b xxxx + 240 b x b b b xxxx − b x b b b xxxx − b x b b xxxx b + 40 b x b b b xxxx − b xx b b x b x + 1440 b xx b b x b x − b xx b b x b + 240 b xx b b b x − b xx b x b b x +360 b xx b x b b x − b xx b b xxx b + 320 b xx b b b xxx + 480 b xx b b xxx b − b xx b b b xxx − b xx b b xxx b + 80 b xx b b b xxx + 640 b xxx b b x b x + 960 b xxx b b x b + 240 b xxx b b b x − b xxx b b x b − b xxx b b b x − b xxx b x b b x − b xxx b b xx b + 320 b xxx b b b xx +480 b xxx b b xx b − b xxx b b b xx − b xxx b b xx b + 80 b xxx b b b xx + 160 b xxxx b b x +120 b x b xx b = 0 (11)where H ( t ) is an arbitrary function of t and b or b are arbitrary. The latter equation may looka bit daunting but is quite easily managed with the aid of a CAS (in this case MAPLE) once b and b are given. c i c = − c + 5 c − c + 2 c (12) c = 23 c + 4 c (13) c x c − c c x − c c x + 2 c x c + c x c − c x c = 0 (14) − c x c + 12 c x c + c x c − c x c = 0 (15)5 Gc − Hc x c x c + Hc c xx + 2 Hc x c − Hc c c xx = 0 (16)14 c c x − c c c xx − c c x + 30 c c c xx + 60 c x c c x − c xx c +4 c x c c x − c x c c x + 7 c xx c − c c x + 2 c c c xx − c xx c = 0 (17)68 c c x − c c c x c xx + 3 a c c xxx − c x c c x + 9 c x c c xx − c c x +60 c c c x c xx − c c c xxx + 60 c x c c x − c x c c xx − c xx c c x +10 c xxx c + 6 c x c c x − c x c c xx − c xx c c x + 9 c xx c c x − c xxx c − c c x + c c c x c xx − c c c xxx + c xxx c = 0 (18) c xxxxxxx c + 5040 c x c c x − c xxxxxx c c x − c c c xxxxxxx − c xxx c c x c xx +140 c x c c xxx − c x c c xxxxxx − c xx c c x − c xx c c xxxxx + 840 c xxx c c x +210 c xxx c c xx − c xxx c c xxxx − c xxxx c c x − c xxxx c c xxx − c xxxxx c c xx + 5040 c c c x c xx c xxx − c c c x c xx c xxxx − c x c c xx + H t H c − c c t − c c x + 15210 c c c x c xx − c c c x c xxx + 42 c xxxxx c c x − c c c xx c xxx − c c c x c xxxxx + 70 c c c xxx c xxxx + 42 c c c xx c xxxxx +14 c c c x c xxxxxx − c c c x c xx + 2520 c c c x c xx − c c c x c xxx − c x c c x c xx + 7560 c x c c x c xx + 3360 c x c c x c xxx − c x c c x c xxxx +210 c x c c xx c xxxx + 84 c x c c x c xxxxx + 5040 c xx c c x c xx − c xx c c x c xx − c xx c c x c xxx + 420 c xx c c xx c xxx + 210 c xx c c x c xxxx +280 c xxx c c x c xxx + 210 c xxxx c c x c xx − c x c c x c xx c xxx +840 c c c x c xxxx = 0 (19) − c c x + 720 c c x + 120 c c x + 40 c xxx c c x − c xxx c c xx + 6 c c c xxxxx − c x c c xx + 30 c x c c xxxx + 360 c xx c c x + 60 c xx c c xxx − c xxx c c x +60 c xxx c c xx − c x c c xx + 5 c x c c xxxx + 60 c xx c c x + 10 c xx c c xxx +10 c xxx c c xx + c c c xxxxx − c xx c c x − c xx c c xxx − c c c xxxxx − c x c c xxxx + 30 c xxxx c c x + 5 c xxxx c c x − c xxxx c c x − c xxxxx c − c xxxxx c − c x c c x − c x c c x + 240 c x c c x + 480 c c c x c xx − c c c x c xxx + 40 c c c xx c xxx + 20 c c c x c xxxx − c x c c x c xx +120 c xx c c x c xx − c c c x c xx + 540 c c c x c xx + 360 c c c x c xxx − c xxx c c x + 60 c x c c xx + 2 c xxxxx c − c c c x c xx − c c c xx c xxx +80 c x c c x c xxx + 180 c x c c x c xx + 60 c c c x c xxx − c c c xx c xxx − c c c x c xxxx + 1080 c x c c x c xx − c x c c x c xxx − c xx c c x c xx − c x c c x c xxx − c xx c c x c xx − c c c x c xx + 90 c c c x c xx − c c c x c xxxx = 0 (20)where G ( t ) and H ( t ) are arbitrary functions of t .7 The Generalized Variable Coefficient Fifth-order KdV (vcKdV)Equation
Here, we will apply the technique of the last section in exactly the same fashion to generalizedvcKdV equations, but will omit the details for the sake of brevity. Please note that the coefficients a i in this section are totally distinct or different from those given the same symbols in the previoussection. All equations in this section are thus to be read independently of those in the previous one. Consider the generalized fifth-order vcKdV equation in the form u t + a uu xxx + a u x u xx + a u u x + a uu x + a u xxx + a u xxxxx + a u + a u x = 0 (21)As before, we consider the generalized variable-coefficient KDV equation to be integrable if wecan find a Lax pair which satisfies (3). In the method given in (cite Khwaja) one expands theLax pair U and V in powers of u its derivatives with unknown function coefficients and require(3) to be equivalent to the nonlinear system. This results in a system of coupled PDEs for theunknown coefficients for which upon solving we simultaneously obtain the Lax pair and integrabilityconditions on the a i . The results, for which the details are given in Appendix B, are as follows a − = H − a (22) a = a H (cid:18)(cid:18) H a (cid:19) t + (cid:18) H a a (cid:19) xxx + (cid:18) H a a (cid:19) xxxxx + (cid:18) H a a (cid:19) x (cid:19) (23)where H − ( t ) are arbitrary functions of t and a , a , a and a are taken to be arbitrary functionsof x and t . This form helps to give integrability conditions to sub-equations of (21). For examplean integrable variable-coefficient KDV equation would require that H − ( t ) = a − ( x, t ) = 0. For a ( x, t ) = 0 we would need to further require (through a little algebraic manipulation) that thefollowing be satisfied (cid:18) H ( t ) a (cid:19) t + H ( t ) (cid:18) a a (cid:19) xxx = 0 (24)Having considered these two examples of various vcMKdV equations, as well as the vcKdV equa-tion, we shall now proceed to consider whether these results may be recovered in a more algorithmicmanner. As discussed in Section 1, it would be advantageous if they could be obtained without: a.requiring to know the form of the Lax Pair for the corresponding constant-coefficient Lax-integrableequation, and b. requiring to generalize this constant-coefficient Lax Pair by guesswork. Towardsthat end, we now proceed to consider how this may be accomplished by generalizing and extendingthe Estabrook-Wahlquist technique to Lax-integrable systems with variable coefficients.
In the standard Estabrook-Wahlquist method one begins with a constant coefficient NLPDE and as-sumes an implicit dependence on u ( x, t ) and its partial derivatives of the spatial and time evolutionmatrices ( F , G ) involved in the linear scattering problem ψ x = F ψ, ψ t = G ψ F and G are connected via a zero-curvature condition (independence ofpath in spatial and time evolution) derived by mandating ψ xt = ψ tx . That is, it requires F t − G x + [ F , G ] = 0provided u ( x, t ) satisfies the NLPDE.Considering the forms F = F ( u, u x , u t , . . . , u mx,nt ) and G = G ( u, u x , u t , . . . , u kx,jt ) for the spaceand time evolution matrices where u px,qt = ∂ p + q u∂x p ∂t q we see that this condition is equivalent to X m,n F u mx,nt u mx, ( n +1) t − X j,k G u jx,kt u ( j +1) x,kt + [ F , G ] = 0From here there is often a systematic approach[15]-[18] to determining the form for F and G whichis outlined in [17] and will be utilized in the examples to follow.Typically a valid choice for dependence on u ( x, t ) and its partial derivatives is to take F todepend on all terms in the NLPDE for which there is a partial time derivative present. Similarlywe may take G to depend on all terms for which there is a partial space derivative present. Forexample, given the Camassa-Holm equation, u t + 2 ku x − u xxt + 3 uu x − u x u xx − uu xxx = 0 , one would consider F = F ( u, u xx ) and G = G ( u, u x , u xx ). Imposing compatibility allows one todetermine the explicit form of F and G in a very algorithmic way. Additionally the compatibilitycondition induces a set of constraints on the coefficient matrices in F and G . These coefficientmatrices subject to the constraints generate a finite dimensional matrix Lie algebra.In the extended Estabrook-Wahlquist method we allow for F and G to be functions of t , x , u and the partial derivatives of u . Although the details change, the general procedure will remainessentially the same. We will begin by equating the coefficient of the highest partial derivativeof the unknown function(s) to zero and work our way down until we have eliminated all partialderivatives of the unknown function(s). This typically results in a large partial differential equation (in the standard Estabrook-Wahlquistmethod, this is an algebraic equation) which can be solved by equating the coefficients of the differentpowers of the unknown function(s) to zero.
This final step induces a set of constraints on thecoefficient matrices in F and G . Another big difference which we will see in the examples comes inthe final and, arguably, the hardest step. In the standard Estabrook-Wahlquist method the final stepinvolves finding explicit forms for the set of coefficient matrices such that they satisfy the contraintsderived in the procedure. Note these constraints are nothing more than a system of algebraic matrixequations. In the extended Estabrook-Wahlquist method these constraints will be in the form ofmatrix partial differential equations which can be used to derive an integrability condition on thecoefficients in the NLPDE.
As we are now letting F and G have explicit dependence on x and t and for notational clarity,it will be more convenient to consider the following version of the zero-curvature conditionD t F − D x G + [ F , G ] = 0 (25)where D t and D x are the total derivative operators on time and space, respectively. Recall thedefinition of the total derivative 9 y f ( y, z, u ( y, z ) , u ( y, z ) , . . . , u n ( y, z )) = ∂f∂y + ∂f∂u ∂u ∂y + ∂f∂u ∂u ∂y + · · · + ∂f∂u n ∂u n ∂y Thus we can write the compatibility condition as F t + X m,n F u mx,nt u mx, ( n +1) t − G x − X j,k G u jx,kt u ( j +1) x,kt + [ F , G ] = 0It is important to note that the subscripted x and t denotes the partial derivative with respect toonly the x and t elements, respectively. That is, although u and it’s derivatives depend on x and t this will not invoke use of the chain rule as they are treated as independent variables. This willbecome more clear in the examples of the next section.Note that compatibility of the time and space evolution matrices will yield a set of constraintswhich contain the constant coefficient constraints as a subset. In fact, taking the variable coefficientsto be the appropriate constants will yield exactly the Estabrook-Wahlquist results for the constantcoefficient version of the NLPDE. That is, the constraints given by the Estabrook-Wahlquist methodfor a constant coefficient NLPDE are always a proper subset of the constraints given by a variable-coefficient version of the NLPDE. This can easily be shown. Letting F and G not depend explicitlyon x and t and taking the coefficients in the NLPDE to be constant the zero-curvature conditionas it is written above becomes X m,n F u mx,nt u mx, ( n +1) t − X j,k G u jx,kt u ( j +1) x,kt + [ F , G ] = 0which is exactly the standard Estabrook-Wahlquist method.The conditions derived via mandating (25) be satisfied upon solutions of the vc-NLPDE maybe used to determine conditions on the coefficient matrices and variable-coefficients (present in theNLPDE). Successful closure of these conditions is equivalent to the system being S-integrable. Amajor advantage to using the Estabrook-Wahlquist method that carries forward with the extensionis the fact that it requires little guesswork and yields quite general results.In Khawaja’s method[14]-[22] an educated guess is made for the structure of the variable-coefficient pde Lax pair based on the associated constant coefficient Lax pair. That is, Khawajaconsidered the matrices F = U = (cid:20) f + f q f + f qf + f r f + f r (cid:21) and G = V = (cid:20) g + g q + g q x + g rq g + g q + g q x + g rqg + g r + g r x + g rq g + g r + g r x + g rq (cid:21) where f i and g i unknown functions of x and t which satisfy conditions derived by mandating thezero-curvature condition be satisfied on solutions of the variable-coefficient NLPDE. In fact, in aprevious paper Khawaja derives the associated Lax pair via a similar means where he begins withan even weaker assumption on the structure of the Lax pair. This Lax pair is omitted from thepaper as it becomes clear the zero-curvature condition mandates many of the coefficients be zero.10 n ideal approach would be a method which does not require knowledge of the Lax pair to anassociated constant coefficient system and involves little to no guesswork. The extended Estabrook-Wahlquist does exactly this. It will be shown that the results obtained from Khawaja’s method arein fact a special case of the extended Estabrook-Wahlquist method. We now proceed with the variable coefficient MKdV equation as our first example of thisextended Estabrook Wahlquist method.
For this example we consider the mKdV equation given by v t + b v xxx + b v v x = 0 (26)where b and b are arbitrary functions of x and t . Following the procedure we let F = F ( x, t, u ) , G = G ( x, t, u, u x , u xx )Plugging this into (25) we obtain F t + F v v t − G x − G v v x − G v x v xx − G v xx v xxx + [ F , G ] = 0 (27)Using (26) to substitute for v t we have F t − G x − ( G v + b F v v ) v x − G v x v xx − ( G v xx + b F v ) v xxx + [ F , G ] = 0 (28)Since F and G do not depend on v xxx we can set the coefficient of the v xxx term to zero from whichwe have G v xx + b F v = 0 ⇒ G = − b F v v xx + K ( x, t, v, v x )Substituting this into (28) we have F t + ( b F v ) x v xx − K x + b F vv v x v xx − K v v x − b F v v v x − K v x v xx − b [ F , F v ] v xx + [ F , K ] = 0 (29)Since F and K do not depend on v xx we can equate the coefficient of the v xx term to zero fromwhich we require ( b F v ) x + b F vv v x − K v x − b [ F , F v ] = 0 (30)Solving for K we have K = ( b F v ) x v x + 12 b F vv v x − b [ F , F v ] v x + K ( x, t, v )Substituting this expression into (29) we have 11 t − ( b F v ) xx v x −
12 ( b F vv ) x v x + ( b [ F , F v ]) x v x − K x − ( b F vv ) x v x − b F vvv v x − K v v x + b [ F , F vv ] v x − b F v v v x + [ F , ( b F v ) x ] v x + 12 b [ F , F vv ] v x − b [ F , [ F , F v ]] v x +[ F , K ] = 0 (31)Since F and K do not depend on v x we can equate the coefficients of the v x , v x and v x termsto zero from which we obtain the system O ( v x ) : F vvv = 0 (32) O ( v x ) : 12 ( b F vv ) x + ( b F vv ) x − b [ F , F vv ] − b [ F , F vv ] = 0 (33) O ( v x ) : ( b F v ) xx − ( b [ F , F v ]) x + K v + b F v v − [ F , ( b F v ) x ] + b [ F , [ F , F v ]] = 0 (34)Since the MKdV equation does not contain a vv t term and for ease of computation we takerequire F vv = 0 from which we have F = X ( x, t ) + X ( x, t ) v . For the O ( v x ) equation we solve for K and thus have K = − ( b X ) xx v + ( b [ X , X ]) x v + [ X , ( b X ) x ] v + 12 [ X , ( b X ) x ] v − b [ X , [ X , X ]] v − b X v − b [ X , [ X , X ]] v + X ( x, t ) (35)Substituting this expression for K into (31) we obtain X ,t + ( b X ) xxx v − ( b [ X , X ]) xx v + 13 ( b X ) x v − ([ X , ( b X ) x ]) x v −
12 ([ X , ( b X ) x ]) x v +( b [ X , [ X , X ]]) x v + 12 ( b [ X , [ X , X ]]) x v − X ,x − [ X , ( b X ) xx ] v + [ X , ( b [ X , X ]) x ] v + X ,t v − b [ X , X ] v + [ X , [ X , ( b X ) x ]] v + 12 [ X , [ X , ( b X ) x ]] v − b [ X , [ X , [ X , X ]]] v − b [ X , [ X , [ X , X ]]] v − [ X , ( b X ) xx ] v + [ X , ( b [ X , X ]) x ] v + [ X , [ X , ( b X ) x ]] v +[ X , X ] + 12 [ X , [ X , ( b X ) x ]] v − b [ X , [ X , [ X , X ]]] v − b [ X , [ X , [ X , X ]]] v +[ X , X ] v = 0 (36)Since the X i do not depend on v we can equate the coefficients of the different powers of v tozero. We thus obtain the constraints 12 (1) : X ,t − X ,x + [ X , X ] (37) O ( v ) : X ,t − ([ X , ( b X ) x ]) x + ( b [ X , [ X , X ]]) x − [ X , ( b X ) xx ] + [ X , ( b [ X , X ]) x ] − ( b [ X , X ]) xx + ( b X ) xxx + [ X , [ X , ( b X ) x ]] − b [ X , [ X , [ X , X ]]]+[ X , X ] = 0 (38) O ( v ) : −
12 ([ X , ( b X ) x ]) x + 12 ( b [ X , [ X , X ]]) x + 12 [ X , [ X , ( b X ) x ]] − [ X , ( b X ) xx ] − b [ X , [ X , [ X , X ]]] + [ X , ( b [ X , X ]) x ] − b [ X , [ X , [ X , X ]]]+[ X , [ X , ( b X ) x ]] = 0 (39) O ( v ) : 13 ( b X ) x − b [ X , X ] + 12 [ X , [ X , ( b X ) x ]] − b [ X , [ X , [ X , X ]]] = 0 (40)Note that if we decouple the O ( v ) equation into the following equations b [ X , X ] − ( b X ) x = 0 (41) b [ X , X ] − ( b X ) x = 0 (42)we find that the O ( v ) equation is immediately satisfied and the O ( v ) equation reduces to[ X , X ] + X ,t = 0 (43) Again we should note that had we opted instead for the forms X = (cid:20) g ( x, t ) g ( x, t ) g ( x, t ) g ( x, t ) (cid:21) , X = (cid:20) f ( x, t ) f ( x, t ) f ( x, t ) f ( x, t ) (cid:21) , X = (cid:20) f ( x, t ) f ( x, t ) f ( x, t ) f ( x, t ) (cid:21) we would obtain an equivalent system to that obtained in [ ] for the mKdV. The additional un-known functions which appear in Khawaja’s method ( [ ] ) can again be introduced with the propersubstitutions via their functional dependence on the twelve unknown functions given above. Therefore utilizing the same generators as in the generalized KdV equation we obtain the systemof equations f f − f f = 0 (44)( b f j ) x = ( b f j ) x = 0 , j = 3 , f g − f g = 0 (46) f jt + ( − j f j ( g − g ) = 0 , j = 3 , g jx + ( − j ( f g − f g ) = 0 , j = 1 , f jt − g ( j +1) x + ( − j f j ( g − g ) = 0 , j = 2 , j ( x, t ) = F j ( t ) b ( x, t ) , j = 3 , g ( x, t ) = F ( t ) g ( x, t ) F ( t ) (51) f ( x, t ) = F ( t ) f ( x, t ) F ( t ) (52) g ( x, t ) = G ( t ) (53) g ( x, t ) = G ( t ) (54)Subject to the constraints (cid:18) F j b (cid:19) t + ( − j ( G − G ) = 0 , j = 3 , (cid:18) b F j b (cid:19) x = 0 j = 3 , f jt − g ( j +1) x + ( − j f j ( G − G ) = 0 , j = 2 , F , b , b and G we obtain F ( t ) = c F ( t ) F ( t ) (58) b ( x, t ) = F ( x ) F ( t ) (59) b ( x, t ) = F ( x ) F ( t ) (60) G ( t ) = F ( t ) F ′ ( t ) − F ( t ) F ′ ( t ) + G ( t ) F ( t ) F ( t ) F ( t ) F ( t ) (61) g ( x, t ) = Z [ f ( x, t )] t F ( t ) F ( t ) − f ( x, t ) F ′ ( t ) F ( t ) + f ( x, t ) F ( t ) F ′ ( t ) F ( t ) F ( t ) dx + F ( t ) (62)where F and F are arbitrary functions in their respective variables and c is an arbitrary constant. The Lax pair for the variable-coefficient MKdV equation with the previous integrability condi-tions is thus given by F = X + X v (63) G = − b X v xx − b X v + X (64)Next, we illustrate our generalized Estabrook-Wahlquist technique by applying it to the gener-alized fifth-order vcKdV equation. 14 The Generalized Fifth-Order KdV (vcKdV) Equation Recon-sidered
As a second example of the extended EW technique, let us consider the generalized KDV equation u t + a uu xxx + a u x u xx + a u u x + a uu x + a u xxx + a u xxxxx + a u + a u x = 0 (65)where a − are arbitrary functions of x and t . As with the last example, we will go through theprocedure outlined earlier in the paper and show how one can obtain the results previously obtainedfor the constant coefficient cases. Running through the standard procedure we let F = F ( x, t, u )and G = G ( x, t, u, u x , u xx , u xxx , u xxxx ). Plugging this into (25) we obtain F t + F u u t − G x − G u u x − G u x u xx − G u xx u xxx − G u xxx u xxxx − G u xxxx u xxxxx + [ F , G ] = 0 (66)Next, substituting (65) into this expression in order to eliminate the u t yields F t − F u (cid:0) a uu xxx + a u x u xx + a u u x + a uu x + a u xxx + a u xxxxx + a u + a u x (cid:1) − G x − G u u x − G u x u xx − G u xx u xxx − G u xxx u xxxx − G u xxxx u xxxxx + [ F , G ] = 0 (67)Since F and G do not depend on u xxxxx we can equate the coefficient of the u xxxxx term to zero.This requires that we must have G u xxxx + a F u = 0 ⇒ G = − a F u u xxxx + K ( x, t, u, u x , u xx , u xxx )Now updating (67) we obtain F t − F u (cid:0) a uu xxx + a u x u xx + a u u x + a uu x + a u xxx + a u + a u x (cid:1) + a x F u u xxxx + a F xu u xxxx − K x − K u u x − K u x u xx − K u xx u xxx − K u xxx u xxxx + a F uu u x u xxxx − [ F , F u ] a u xxxx + [ F , K ] = 0 (68)Since F and K do not depend on u xxxx we can equate the coefficient of the u xxxx term to zero.This requires that we have a x F u + a F xu + a F uu u x − K u xxx − [ F , F u ] a = 0 (69)Thus, integrating with respect to u xxx and solving for K we have K = a x F u u xxx + a F xu u xxx + a F uu u x u xxx − [ F , F u ] a u xxx + K ( x, t, u, u x , u xx ) (70)Now we update (68) by plugging in our expression for K to obtain F t − F u (cid:0) a uu xxx + a u x u xx + a u u x + a uu x + a u xxx + a u + a u x (cid:1) − a xx F u u xxx − a x F xu u xxx − a F xxu u xxx − a x F uu u x u xxx − a F xuu u x u xxx + [ F x , F u ] a u xxx +[ F , F xu ] a u xxx + [ F , F u ] a x u xxx − K x − a x F uu u x u xxx − a F xuu u x u xxx − a F uuu u x u xxx + [ F , F uu ] a u x u xxx − K u u x − a F uu u xx u xxx − K u x u xx − K u xx u xxx + a x [ F , F u ] u xxx + a [ F , F xu ] u xxx + a [ F , F uu ] u x u xxx − [ F , [ F , F u ]] a u xxx + [ F , K ] = 0 (71)15ince F and K do not depend on u xxx we can equate the coefficient of the u xxx term to zero.This requires that we have − F u ( a u + a ) − a xx F u − a x F xu − a F xxu − a x F uu u x − a F xuu u x +[ F x , F u ] a + [ F , F xu ] a + [ F , F u ] a x − a x F uu u x − a F xuu u x − a F uuu u x +[ F , F uu ] a u x − a F uu u xx − K u xx + a x [ F , F u ] + a [ F , F xu ] + a [ F , F uu ] u x − [ F , [ F , F u ]] a = 0 (72)Integrating with respect to u xx and solving for K and collecting like terms we have K = − F u ( a u + a ) u xx − ( a F u ) xx u xx − a F uu ) x u x u xx + 2( a [ F , F u ]) x u xx − a F uuu u x u xx + 2 a [ F , F uu ] u x u xx − a F uu u xx − a [ F x , F u ] u xx − a [ F , [ F , F u ]] u xx + K ( x, t, u, u x ) (73)Plugging (73) into (71) and simplifying a little bit we obtain F t − F u ( a u x u xx + a u u x + a uu x + a u + a u x ) + ( a F u ) x uu xx + ( a F u ) x u xx +( a F u ) xxx u xx + 2( a F uu ) xx u x u xx − ( a [ F , F u ]) xx u xx + ( a F uuu ) x u x u xx + 12 ( a F uu ) x u xx − ([ F , ( a F u ) x ]) x u xx + ( a [ F , [ F , F u ]]) x u xx − K x + F uu a uu x u xx + F u a u x u xx + F uu a u x u xx + ( a F uu ) xx u x u xx + 2( a F uuu ) x u x u xx + a F uuuu u x u xx − a [ F u , F uu ] u x u xx − a [ F , F uuu ] u x u xx + 52 a F uuu u xx u x − [ F u , ( a F u ) x ] u x u xx − F , ( a F uu ) x ] u x u xx − a [ F u , F uu ] u x u xx − a [ F , F uuu ] u x u xx + a [ F u , [ F , F u ]] u x u xx + a [ F , [ F , F uu ]] u x u xx − K u u x + 2( a F uu ) x u xx − a [ F , F uu ] u xx − K u x u xx − a [ F , F u ] uu xx − a [ F , F u ] u xx − [ F , ( a F u ) xx ] u xx − [ F , ( a F uu ) x ] u x u xx + [ F , ( a [ F , F u ]) x ] u xx − a [ F , F uuu ] u x u xx + 2 a [ F , [ F , F uu ]] u x u xx + [ F , [ F , ( a F u ) x ]] u xx − a [ F , F uu ]) x u x u xx − a [ F , [ F , [ F , F u ]]] u xx + [ F , K ] = 0 (74)Now, since K and F do not depend on u xx we can start by setting the coefficients of the u xx and the u xx terms to zero. Note the difference here that we have multiple powers of u xx present inthe (74). Setting the O ( u xx ) term to zero requires32 ( a F uu ) x + 52 a F uuu u x − a [ F , F uu ] = 0 (75)Since F does not depend on u x we must have that the coefficient of the u x term in this previousexpression is zero. This is equivalent to F uuu = 0 ⇒ F = X ( x, t ) + X ( x, t ) u + 12 X ( x, t ) u Plugging this into (75) we obtain 16( a X ) x − a ([ X , X ] + [ X , X ] u ) = 0 (76)Now since the X i do not depend on u we can set the coefficient of the u to zero. That is, we requirethat X and X commute. We find now that (76) reduces to the condition( a X ) x − a [ X , X ] = 0 (77)For ease of computation and in order to immediately satisfy (77) we set X = 0. Plugging into (74)our expression for F we obtain X ,t + X ,t u − X ( a u x u xx + a u u x + a uu x + a u + a u x ) + ( a X ) x uu xx + ( a X ) x u xx +( a X ) xxx u xx − ( a [ X , X ]) xx u xx − K x + X a u x u xx − [ X , ( a X ) x ] u x u xx − ([ X , ( a X ) x ]) x u xx − ([ X , ( a X ) x ]) x uu xx + ( a [ X , [ X , X ]]) x u xx + ( a [ X , [ X , X ]]) x uu xx − K u u x − K u x u xx − a [ X , X ] uu xx + a [ X , [ X , X ]] u x u xx + [ X , ( a [ X , X ]) x ] uu xx − a [ X , X ] u xx − [ X , ( a X ) xx ] u xx − [ X , ( a X ) xx ] uu xx + [ X , ( a [ X , X ]) x ] u xx +[ X , [ X , ( a X ) x ]] u xx + [ X , [ X , ( a X ) x ]] uu xx + [ X , [ X , ( a X ) x ]] uu xx + [ X , K ] − a [ X , [ X , [ X , X ]]] u xx − a [ X , [ X , [ X , X ]]] uu xx − a [ X , [ X , [ X , X ]]] uu xx +[ X , [ X , ( a X ) x ]] u u xx − a [ X , [ X , [ X , X ]]] u u xx + [ X , K ] u = 0 (78)Now again using the fact that the X i and K do not depend on u xx we can set the coefficientof the u xx term in (78) to zero. This requires( a X ) xxx − ( a [ X , X ]) xx + a X u x − [ X , ( a X ) x ] u x − a X u x − ([ X , ( a X ) x ]) x − ([ X , ( a X ) x ]) x u + ( a [ X , [ X , X ]]) x + ( a [ X , [ X , X ]]) x u − K u x − a [ X , X ] u + a [ X , [ X , X ]] u x + [ X , ( a [ X , X ]) x ] u + ( a X ) x − a [ X , X ] − [ X , ( a X ) xx ] − [ X , ( a X ) xx ] u + [ X , ( a [ X , X ]) x ]+[ X , [ X , ( a X ) x ]] + [ X , [ X , ( a X ) x ]] u + [ X , [ X , ( a X ) x ]] u + ( a X ) x u − a [ X , [ X , [ X , X ]]] − a [ X , [ X , [ X , X ]]] u − a [ X , [ X , [ X , X ]]] u +[ X , [ X , ( a X ) x ]] u − a [ X , [ X , [ X , X ]]] u = 0 (79)Thus integrating with respect to u x and solving for K we have K = ( a X ) xxx u x + 12 a X u x −
12 [ X , ( a X ) x ] u x − a X u x + ( a [ X , [ X , X ]]) x uu x − ( a [ X , X ]) xx u x − ([ X , ( a X ) x ]) x u x − ([ X , ( a X ) x ]) x uu x + ( a [ X , [ X , X ]]) x u x − a [ X , X ] uu x + 12 a [ X , [ X , X ]] u x + [ X , ( a [ X , X ]) x ] uu x + ( a X ) x u x − a [ X , X ] u x − [ X , ( a X ) xx ] u x − [ X , ( a X ) xx ] uu x + [ X , ( a [ X , X ]) x ] u x +[ X , [ X , ( a X ) x ]] u x + [ X , [ X , ( a X ) x ]] uu x + [ X , [ X , ( a X ) x ]] uu x + ( a X ) x uu x − a [ X , [ X , [ X , X ]]] u x − a [ X , [ X , [ X , X ]]] uu x − a [ X , [ X , [ X , X ]]] uu x +[ X , [ X , ( a X ) x ]] u u x − a [ X , [ X , [ X , X ]]] u u x + K ( x, t, u ) (80)17t is helpful at this stage to define the following new matrices X = [ X , X ] , X = [ X , X ] , X = [ X , X ] (81) X = [ X , X ] , X = [ X , X ] , X = [ X , X ] , X = [ X , X ] (82)Now we update (78) by plugging in (80). This yields a long expression which is (C.1) inAppendix C.Now since K and the X i do not depend on u x we can set the coefficient of the u x term to zero(C.1). Therefore we require −
12 ( a X ) x + 12 ([ X , ( a X ) x ]) x + 12 ( a X ) x − ( a X ) x + a X −
12 ( a X ) x + ([ X , ( a X ) x ]) x + a X + [ X , ( a X ) xx ] + 12 a X u − [ X , [ X , ( a X ) x ]] − [ X , [ X , ( a X ) x ]] − ( a X ) x − [ X , ( a X ) x ] − X , [ X , ( a X ) x ]] u + 2 a X u − a X + a X + 12 a X −
12 [ X , [ X , ( a X ) x ]] + 12 a X −
12 [ X , [ X , ( a X ) x ]] u = 0 (83)Further since we know that the X i do not depend on u we can decouple this condition as follows.32 ([ X , ( a X ) x ]) x −
32 ( a X ) x + 12 ( a X ) x −
32 ( a X ) x + 32 a X + 32 a X −
32 [ X , [ X , ( a X ) x ]] − [ X , [ X , ( a X ) x ]] − [ X , ( a X ) x ] + [ X , ( a X ) xx ] − a X + a X = 0 (84) a X − [ X , [ X , ( a X ) x ]] = 0 (85)Taking these conditions into account and once again noting the fact that K and the X i are notindependent of u x we can simplify and equate the coefficient of the u x in (C.1) to zero. Thus wenow obtain the condition (C.2) in Appendix C.Now we update (C.1) by plugging in (C.3). Upon doing this we will have a rather largeexpression in which is no more than a algebraic equation in u . We will find our remaining constraintsby equating the coefficients of the different powers of u in this expression to zero. This updatedversion of (C.1) is very lengthy, and omitted here.Now, in the final step, as the X i do not depend on u we can set the coefficients of the differentpowers of u in this last, lengthy expression to zero. Thus we have18 (1) : X ,t − X ,x + [ X , X ] = 0 (86) O ( u ) : [ X , X ] − a [ X , [ X , X ]] + [ X , [ X , [ X , ( a X ) x ]]] + [ X , [ X , [ X , [ X , ( a X ) x ]]]]+[ X , [ X , ( a X ) x ]] − [ X , [ X , [ X , ( a X ) xx ]]] + [ X , ( a X ) x ] − [ X , [ X , ( a X ) xx ]] − a X − [ X , [ X , ([ X , ( a X ) x ]) x ]] + [ X , [ X , ( a X ) xxx ]] − [ X , ([ X , ( a X ) x ]) x ] − [ X , ([ X , [ X , ( a X ) x ]]) x ] + [ X , ( a X ) x ] + [ X , ([ X , ( a X ) xx ]) x ] − [ X , ( a X ) xx ] − [ X , ( a X ) xx ] + [ X , ( a X ) xxx ] + [ X , ([ X , ( a X ) x ]) xx ] − a X − [ X , ( a X ) xxxx ]+[ X , [ X , ( a X ) x ]] + ( a [ X , X ]) x − ([ X , [ X , [ X , ( a X ) x ]]]) x − ([ X , [ X , ( a X ) x ]]) x +( a X ) x + ([ X , [ X , ( a X ) xx ]]) x − ([ X , ( a X ) x ]) x − ( a X ) xx + ([ X , ( a X ) xx ]) x +([ X , ([ X , ( a X ) x ]) x ]) x − ([ X , ( a X ) xxx ]) x + ([ X , [ X , ( a X ) x ]]) xx + ( a X ) x + X ,t − ([ X , ( a X ) xx ]) xx − ( a X ) xx + ( a X ) xxx + ( a X ) xxx − ([ X , ( a X ) x ]) xxx − ([ X , ( a X ) x ]) x + ( a X ) xxxxx − ( a X ) xxxx − a X + ([ X , ( a X ) x ]) xx = 0 (87)19 ( u ) : − a X − a [ X , [ X , X ]] + [ X , [ X , [ X , [ X , ( a X ) x ]]]] + [ X , [ X , [ X , ( a X ) x ]]] − [ X , [ X , [ X , ( a X ) xx ]]] + [ X , [ X , ( a X ) x ]] − [ X , [ X , ([ X , ( a X ) x ]) x ]] − a X +[ X , ( a X ) x ] − [ X , [ X , ( a X ) xx ]] + [ X , [ X , ( a X ) xxx ]] − [ X , ([ X , ( a X ) x ]) x ] − [ X , ([ X , [ X , ( a X ) x ]]) x ] + [ X , ( a X ) x ] + [ X , ([ X , ( a X ) xx ]) x ] − [ X , ( a X ) xx ]+[ X , ( a X ) xxx ] + [ X , ([ X , ( a X ) x ]) xx ] − [ X , ( a X ) xxxx ] + [ X , [ X , ( a X ) x ]] − [ X , ( a X ) xx ] − a X + 12 [ X , [ X , ( a X ) x ]] + 12 [ X , [ X , [ X , [ X , ( a X ) x ]]]] −
12 [ X , [ X , [ X , ( a X ) xx ]]] + 12 [ X , [ X , ( a X ) x ]] −
12 [ X , [ X , ([ X , ( a X ) x ]) x ]]+ 12 [ X , [ X , [ X , [ X , ( a X ) x ]]]] −
12 [ X , [ X , ( a X ) xx ]] + 12 [ X , [ X , [ X , ( a X ) x ]]] − a [ X , [ X , X ]] + 12 [ X , [ X , ( a X ) xxx ]] + 12 [ X , [ X , [ X , [ X , ( a X ) x ]]]]+ 12 [ X , [ X , ( a X ) x ]] −
12 [ X , [ X , [ X , ( a X ) xx ]]] −
12 [ X , [ X , ([ X , ( a X ) x ]) x ]] − a [ X , [ X , X ]] + 12 [ X , [ X , [ X , ( a X ) x ]]] − a X + 12 [ X , [ X , ( a X ) x ]] −
12 [ X , ([ X , [ X , ( a X ) x ]]) x ] + 12 [ X , ( a X ) x ] + 12 [ X , ( a X ) x ] −
12 [ X , ( a X ) xx ] −
12 [ X , ([ X , [ X , ( a X ) x ]]) x ] + 12 [ X , ([ X , ( a X ) xx ]) x ] −
12 [ X , ([ X , ( a X ) x ]) x ] −
12 [ X , ( a X ) xx ] + 12 [ X , ( a X ) x ] + 12 [ X , ([ X , ( a X ) x ]) xx ] − a [ X , [ X , X ]] −
12 ([ X , ( a X ) x ]) x −
12 ([ X , [ X , [ X , ( a X ) x ]]]) x + 12 ([ X , [ X , ( a X ) xx ]]) x + 12 ( a X ) x + 12 ([ X , ([ X , ( a X ) x ]) x ]) x −
12 ([ X , ( a X ) x ]) x −
12 ([ X , ( a X ) x ]) x + 12 ([ X , ( a X ) xx ]) x + 12 ( a [ X , X ]) x + 12 ( a [ X , X ]) x −
12 ([ X , [ X , ( a X ) x ]]) x −
12 ([ X , ( a X ) xxx ]) x −
12 ([ X , [ X , [ X , ( a X ) x ]]]) x −
12 ([ X , [ X , [ X , ( a X ) x ]]]) x + 12 ([ X , [ X , ( a X ) xx ]]) x −
12 ([ X , [ X , ( a X ) x ]]) x + 12 ([ X , ([ X , ( a X ) x ]) x ]) x + 12 ( a X ) x −
12 ([ X , ( a X ) x ]) x −
12 ( a X ) xx −
12 ( a X ) xx + 12 ([ X , [ X , ( a X ) x ]]) xx + 12 ( a X ) xxx + 12 ([ X , [ X , ( a X ) x ]]) xx −
12 ([ X , ( a X ) xx ]) xx + 12 ([ X , ( a X ) x ]) xx + 12 ( a X ) xxx −
12 ( a X ) xx −
12 ([ X , ( a X ) x ]) xxx + 12 ( a [ X , X ]) x + 12 ( a X ) x = 0(88)20 ( u ) : −
13 ([ X , ( a X ) x ]) x + 12 [ X , [ X , [ X , [ X , ( a X ) x ]]]] −
12 [ X , [ X , [ X , ( a X ) xx ]]]+ 13 ( a X ) x −
12 [ X , [ X , ( a X ) xx ]] + 12 [ X , [ X , ( a X ) x ]] + 12 [ X , [ X , ( a X ) x ]] −
12 [ X , [ X , ([ X , ( a X ) x ]) x ]] − a [ X , [ X , X ]] + 12 [ X , [ X , [ X , [ X , ( a X ) x ]]]] − a [ X , [ X , X ]] + 12 [ X , [ X , [ X , ( a X ) x ]]] + 12 [ X , [ X , [ X , [ X , ( a X ) x ]]]]+ 12 [ X , [ X , ( a X ) xxx ]] −
12 [ X , [ X , [ X , ( a X ) xx ]]] −
12 [ X , [ X , ([ X , ( a X ) x ]) x ]] − a X + 12 [ X , [ X , [ X , ( a X ) x ]]] + 12 [ X , [ X , ( a X ) x ]] −
12 [ X , ( a X ) xx ] −
12 [ X , ([ X , [ X , ( a X ) x ]]) x ] + 12 [ X , ( a X ) x ] −
12 [ X , ([ X , [ X , ( a X ) x ]]) x ]+ 12 [ X , ([ X , ( a X ) xx ]) x ] −
12 [ X , ( a X ) xx ] + 12 [ X , ([ X , ( a X ) x ]) xx ] + 12 [ X , ( a X ) x ]+ 12 [ X , ( a X ) x ] −
12 [ X , ([ X , ( a X ) x ]) x ] − a [ X , [ X , X ]] − a [ X , [ X , X ]]+ 13 [ X , [ X , ( a X ) x ]] + 13 [ X , [ X , [ X , [ X , ( a X ) x ]]]] −
13 [ X , [ X , ([ X , ( a X ) x ]) x ]]+ 13 [ X , [ X , [ X , [ X , ( a X ) x ]]]] −
13 [ X , [ X , [ X , ( a X ) xx ]]] − a [ X , [ X , X ]]+ 13 [ X , [ X , ( a X ) x ]] + 13 [ X , [ X , [ X , ( a X ) x ]]] − a X −
13 ([ X , ( a X ) x ]) x − a X + 13 ( a [ X , X ]) x + 13 ( a [ X , X ]) x −
13 ([ X , [ X , [ X , ( a X ) x ]]]) x −
13 ([ X , [ X , [ X , ( a X ) x ]]]) x + 13 ([ X , [ X , ( a X ) xx ]]) x + 13 ([ X , ([ X , ( a X ) x ]) x ]) x −
13 ([ X , [ X , ( a X ) x ]]) x + 13 ( a X ) x = 0 (89) O ( u ) : [ X , [ X , [ X , [ X , ( a X ) x ]]]] − a [ X , [ X , X ]] + [ X , [ X , [ X , [ X , ( a X ) x ]]]] − a [ X , [ X , X ]] − [ X , [ X , [ X , ( a X ) xx ]]] − [ X , [ X , ([ X , ( a X ) x ]) x ]]+[ X , [ X , ( a X ) x ]] + [ X , [ X , [ X , ( a X ) x ]]] = 0 (90)Note that if we decouple (84) into the following conditions([ X , ( a X ) x ]) x − ( a X ) x + a X − [ X , [ X , ( a X ) x ]] = 0 (91)[ X , [ X , ( a X ) x ]] + [ X , ( a X ) x ] − [ X , ( a X ) xx ] − a X = 0 (92)(( a − a ) X ) x − ( a − a ) X = 0 (93)then the O ( u ) equation is identically satisfied. To reduce the complexity of the O ( u ) equationwe can decouple it into the following equations 21 X , [ X , [ X , ( a X ) x ]]] − [ X , [ X , ( a X ) xx ]] − [ X , ( a X ) xx ] + [ X , ( a X ) x ]+[ X , ( a X ) x ] − [ X , ([ X , ( a X ) x ]) x ] + [ X , [ X , ( a X ) x ]] + [ X , ( a X ) xxx ] − a X − ( a X ) xx + ( a X ) x − a [ X , X ] = 0 (94)( a X ) x + [ X , [ X , ( a X ) x ]] − a X − ([ X , ( a X ) x ]) x − a X + ( a X ) x = 0 (95)From this last condition, we can use (91) − (95) to reduce the O ( u ) condition to the following − a X − a [ X , [ X , X ]] + [ X , [ X , [ X , [ X , ( a X ) x ]]]] + [ X , [ X , [ X , ( a X ) x ]]] − [ X , [ X , [ X , ( a X ) xx ]]] + [ X , [ X , ( a X ) x ]] − [ X , [ X , ([ X , ( a X ) x ]) x ]]+[ X , ( a X ) x ] − [ X , [ X , ( a X ) xx ]] + [ X , [ X , ( a X ) xxx ]] − [ X , ([ X , ( a X ) x ]) x ] − [ X , ([ X , [ X , ( a X ) x ]]) x ] + [ X , ( a X ) x ] + [ X , ([ X , ( a X ) xx ]) x ] − [ X , ( a X ) xx ]+[ X , ( a X ) xxx ] + [ X , ([ X , ( a X ) x ]) xx ] − [ X , ( a X ) xxxx ] + [ X , [ X , ( a X ) x ]] − [ X , ( a X ) xx ] − a X + 12 [ X , [ X , ( a X ) x ]] −
12 ([ X , ( a X ) x ]) x + 12 ( a X ) x − a X + 12 ( a X ) x = 0 (96)Decoupling this equation allows for the simplification of the O ( u ) equation. Thus we write theprevious condition as the following system of equations − [ X , ( a X ) x ] − [ X , [ X , ( a X ) x ]] + a X + [ X , ( a X ) xx ] − ( a X ) x +( a X ) xx + ([ X , ( a X ) x ]) x − ( a X ) xxx − ( a X ) x = 0 (97) − a X + [ X , [ X , ( a X ) x ]] − ([ X , ( a X ) x ]) x + ( a X ) x − a X + 12 ( a X ) x = 0 (98)Using this the O ( u ) equation is reduced to X ,t + [ X , X ] − a X + ( a X ) x − a X = 0 (99)We therefore find that the final, reduced constraints are given by (85) , (91) − (95) and (97) − (99). In order to satisfy these constraints we begin with the following rather simple forms for ourgenerators, X = (cid:20) g ( x, t ) g ( x, t ) g ( x, t ) g ( x, t ) (cid:21) , X = (cid:20) f ( x, t ) f ( x, t ) 0 (cid:21) , X = (cid:20) f ( x, t ) f ( x, t ) 0 (cid:21) To get more general results we will assume a = 3 a . Note that had we instead opted for theforms X = (cid:20) g ( x, t ) g ( x, t ) g ( x, t ) g ( x, t ) (cid:21) , X = (cid:20) f ( x, t ) f ( x, t ) f ( x, t ) f ( x, t ) (cid:21) , X = (cid:20) f ( x, t ) f ( x, t ) f ( x, t ) f ( x, t ) (cid:21) we would obtain an equivalent system to that obtained in [ ] . The additional unknown functionswhich appear in Khawaja’s method [ ] can be introduced with the proper substitutions via theirfunctional dependence on the twelve unknown functions given above. X , X , and X , becomes( a − a )( f f − f f ) = 0 (100)(( a − a ) f j ) x = 0 j = 3 , f , f and f yields f ( x, t ) = F ( t ) a ( x, t ) − a ( x, t ) (102) f ( x, t ) = F ( t ) a ( x, t ) − a ( x, t ) (103) f ( x, t ) = f ( x, t ) F ( t ) F ( t ) (104)(105)where F , ( t ) are arbitrary functions of t . With these choices we’ve elected to satisfy X = 0 ratherthan a = 3 a . Looking next at (99) we obtain the system (cid:18) F j a − a (cid:19) t − F j a a − a + (cid:18) F j a a − a (cid:19) x + 12 (cid:18) F j a a − a (cid:19) x +( − j F j ( g − g ) a − a = 0 j = 3 , F g a − a − F g a − a = 0 (107)Solving the second equation for g yields g = F ( t ) g ( x, t ) F ( t )Considering the O (1) equation next, we have the following system of equations g x = g x = 0 (108) f t − g x + f ( g − g ) = 0 (109) F ( F f ) t − f F F t − g x F F + F F f ( g − g ) = 0 (110)It follows that we must have g ( x, t ) = G ( t ) and g ( x, t ) = G ( t ) where G and G are arbitraryfunctions of t . Since (109) and (110) do not depend on the a i we will postpone solving them untilthe end. At this point the remaining conditions have been reduced to conditions involving soleythe a i and the previously introduced arbitrary functions of t . The remaining conditions are givenby 23 a a − a (cid:19) x + (cid:18) a a − a (cid:19) xxx = 0 (111) (cid:18) a a − a (cid:19) x = 0 (112) (cid:18) a a − a (cid:19) xx = 0 (113) (cid:18) a a − a (cid:19) x = 0 (114)One can easily solve the system of equations given by (106), (111) − (114) yielding F = c F e R ( G − G ) dt (115) g = Z ( f t + f ( G − G )) dx + F (116) a = − (3 F − − F x ) a F x − F (117) a = F a F x − F (118) a = F a F x − F (119) a = ( F + F x + F x ) a F x − F − Z x Z y a ( z, t ) dz dya ( z, t ) − a ( z, t ) (120) a = a − a F (cid:18) F a − a (cid:19) t + ( a − a ) (cid:18) a a − a (cid:19) x + G − G (121)where F − are arbitrary functions of t . Note that a , a and a have no restrictions beyond theappropriate differentiability and integrability conditions. The Lax pair for the generalized variable-coefficient KdV equation with the previous integrabilityconditions is therefore given by F = X + X u (122) G = − a X u xxxx + ( a X ) x u xxx − X ( a u + a ) u xx − ( a X ) xx u xx − a X u + 12 a X u x − a X u x + ( a X ) x uu x − a X u − a X u + X (123)This completes the extended EW analysis of the generalized fifth-order vcKdV equation. Next,we consider some solutions of these new integrable equations derived by two different methods inthe preceding sections. Next we shall consider methods to derive some solutions of the generalized integrable hierarchiesof NLPDEs derived in the previous sections. 24iven a nonlinear partial differential equation in ( n + 1)-dimensions, without specifying initialor boundary conditions, we may find a solution about a movable singular manifold φ − φ = 0as an infinite series given by u ( x , . . . , x n , t ) = φ − α ∞ X m =0 u m φ m (124)Note that when m ∈ ( Q − Z ) (124) is more commonly known as a Puiseux series. One canavoid dealing with Puiseux series if proper substitutions are made, as we will see a little later on.Plugging this infinite series into the NLPDE yields a recurrence relation for the u m ’s. As with mostseries-type solution methods for NLPDEs we will seek a solution to our NLPDE as (124) truncatedat the constant term. Plugging this truncated series into our original NLPDE and collecting termsin decreasing order of φ will give us a set of determining equations for our unknown coefficients u , . . . , u α known as the Painleve-Backlund equations. We now define new functions C ( x , . . . , x n , t ) = φ t φ x (125) C ( x , . . . , x n , t ) = φ x φ x (126)... (127) C n ( x , . . . , x n , t ) = φ x n φ x (128) V ( x , . . . , x n , t ) = φ x x φ x (129)which will allow us to eliminate all derivatives of φ other than φ x . For simplicity it is commonto allow C i ( x , . . . , x n , t ) and V ( x , . . . , x n , t ) to be constants, thereby reducing a system of PDEs(more than likely nonlinear) in { C i ( x, t ) , V ( x, t ) } to an algebraic system in { C i , V } for ( i = 0 , . . . , n ). (65) Consider the following example u t + H F ( xC + t ) uu xxx + (cid:18) c H − H + 1 F ( xC + t ) (cid:19) u x u xx + H F ( xC + t ) u u x + H ( t ) F ( xC + t ) uu x + (cid:18) c ( c H ( t ) + 2 H − H )8 F ( xC + t ) (cid:19) u xxx + (cid:18) c ( c H − F ( xC + t ) (cid:19) u xxxxx + (cid:18) (5 c ( H − H ) − c − c H − F ( xC + t ) C F ( xC + t ) (cid:19) u x (130)Note that in this example we have a = 0. The leading order analysis yields α = 2. Thereforewe seek a solution of the form u ( x, t ) = u φ ( x, t ) + u φ ( x, t ) + u ( x, t ) (131)25s this forms an auto-Backlund transformation for our For simplicity we will allow our initialsolution u ( x, t ) to be 0. Plugging this in to our pde yields the following determining equations for φ ( x, t ) , u ( x, t ) , u ( x, t ) , V and C : O ( φ − ) : 2 u φ x [(9 c H + 18 c ) φ x + (6 H c + 6) u φ x + H u ] = 0 (132) O ( φ − ) : − H u u φ x − H u u φ x + 60 c u φ x φ xx + 15 c H u x φ x − c H u φ x +10 H u φ x φ xx − H u u x φ x + 30 c H u φ x φ xx + 14 c H u u x φ x − c H u u φ x + 30 c u x φ x − c u φ x + 14 u u x φ x + 4 u φ x φ xx − u u φ x +4 c H u φ x φ xx + H u u x = 0 (133) O ( φ − ) : H u u x − H u φ x − u φ x ( c H − H + 1) + 3 u φ x + 2 H u φ x − c H u u φ x φ xx − H H u u φ x φ xx + 9 c H u φ x φ xx + 4 H u x φ x − H H u x u φ x + 14 H H u u x φ x − c H u u x φ x − c H u x u φ x +12 c H u φ x φ xxx + 2 H H u u x φ xx − H u u φ x φ xx − c H u φ x φ xx +18 c H u x φ x φ xx + 2 c H u u x φ xx + 2 H H u u xx φ x − c H u φ x φ xx +18 c H u φ x φ xx + 2 H H u φ x + 4 H u u φ x + 2 H H u φ x − c H u x φ x +36 c H u x φ x φ xx + 6 c H u φ x φ xxx + 2 c H u u xx φ x − c H u x φ x − H H u x φ x − H u u x φ x + 2 c H u φ x − H u x u φ x + 2 H u xx φ x +6 H u φ x ( c H − H + 1) + 2 H H u φ xxx + 12 c H u xx φ x + 6 c H u xx φ x +12 H H u φ x + 2 H u u x φ xx + 3 u φ x ( c H − H + 1) − H u φ x )+6 H u φ x ( c H − H + 1) − H u u x u + 4 c H u x φ x (134)Upon solving the O ( φ − ) and O ( φ − ) equations for u and u respectively we find that u ( x, t ) = − c φ x (135) u ( x, t ) = 3 c φ xx (136)which lends itself nicely to a representation of the solution as u ( x, t ) = 3 c log[ φ ( x, t )] xx . Furtherwith the choice V = 1 the choices for coefficients the remaining orders of φ are identically satisfied.Now solving the system for φ given in the previous section we find that φ ( x, t ) = c + c e x + Ct .Therefore we have the solution u ( x, t ) = 3 c c c e x + Ct ( c + c e x + Ct ) (137)which for the selection c = c reduces to the solution u ( x, t ) = c sech (cid:0) x + Ct (cid:1) The next example is similar to the first however in this case we don’t have a = 0 and we will26ot force the u term to be the trivial solution. We thus consider the following example u t + 10 H ξ ( t ) F ( R η ( t ) dt + x ) uu xxx + 2(3 + 2 H ) ξ ( t ) F ( R η ( t ) dt + x ) u x u xx + 6 H − F ( R η ( t ) dt + x ) u u x + 10 H ξ ( t ) F ( R η ( t ) dt + x ) u xxx + 4(3 H + 2) ξ ( t ) F ( R η ( t ) dt + x ) u xxxxx − (cid:18) ξ ( t ) (cid:19) ′ F (cid:18)Z η ( t ) dt + x (cid:19) u + (cid:18) H ( t ) + ξ ( t ) ( c (8 H − − c ( H + 30 c H − c ))5 F ( R η ( t ) dt + x ) (cid:19) u x + 10 H ξ ( t ) F ( R η ( t ) dt + x ) uu x = 0 (138)where ξ ( t ) = H c H − c + H and H ( t ) , H ( t ) , H ( t ) and η ( t ) are arbitrary functions of t and c , c are arbitrary constants. As with our last example the leading order analysis yields α = 2. Unlikeour last example we will not force the u term to be 0 initially. The first orders of φ which determinethe u i are as follows : O ( φ − ) : − H + 2) ξ ( t ) u φ x − H − u φ x − H u φ x − H ) ξ ( t ) u φ x = 0 (139) O ( φ − ) : 1440 H H φ x u x − H H φ x u + 600 c H u u x − c H u u x +6 H H u u x + 500 c φ x u u + 20 c H u u x + 5 H φ x u u − c H φ x u u x +84 H H φ x u u x + 600 c H φ x u u − H H φ x u u − c H φ x u u +6500 c H φ x u u + 120 c H H u u x − c H φ x u u − c H H u u x − H H φ x u u − c H φ x u u − H φ x u − c u u x − H u u x +960 H φ x u x + 1920 H φ x φ xx u + 2880 H H φ x φ xx u − c H φ x φ xx u +24 H H φ x φ xx u + 1960 c H H φ x φ xx u − c H H φ x φ xx u +2360 c H H φ x u u x − c H H φ x u u x + 236 H H H φ x u u x +2800 c H H φ x u u − H H H φ x u u − c H H φ x u u +700 c H H φ x u u + 800 c H u u x + 196 H H H φ x φ xx u − c H H φ x u u = 0 (140)27 ( φ − ) : − c H φ x u u x + 60 H H φ x u u x + 200 c H H φ x u − c u u x − H u u x − H H H φ x u − c H H φ x u x + 204 H H H φ x u x u +192 H φ x u x − H φ x u xx − c H H φ x u + 560 c H H φ x u − c H φ x u u + 5200 c H φ x u u − c H φ x u u − c H φ x u u − c H u u x u + 5200 c H φ x u u − H H φ x u u − c H φ x u u − c H H φ x u − c H H φ x φ xx u u + 256 H H H φ x φ xx u u +2560 c H H φ x φ xx u u − c H H φ x u u x + 100 H H H φ x u u x − H H H φ x u x − c H φ x u x u + 120 c H H u u x − c H H u u x +1200 c H u u x u + 1600 c H u u x u − c H φ x u u + 12 H H u u x u − H H φ x u u + 40 c H u u x u − c H φ x u u + 36 H H φ x u x u − c H H φ x u x + 200 c H H φ xxx u − H H H φ xxx u − c H H φ xxx u + 560 c H H φ x u u − H H φ x φ xxx u +120 c H φ x u u xx − H H φ x φ xx u x + 576 H H φ x φ xx u − H H φ x φ xx u + 120 c H φ xx u u x − H H φ xx u u x +240 c H H u u x u − c H H φ x u u − c H H u u x u − c H H φ x u u + 1000 c H H φ x u u x − c H H φ x u x u − H H φ x u u xx − c u u x u + 240 c H φ x u x + 2400 c H H φ x u u − c H H φ x u u − c H H H φ x u + 2040 c H H φ x u x u +400 c φ x u u − H u u x u + 4 H φ x u u + 400 c φ x u u + 4 H φ x u u +2400 c H φ x u − H H φ x u + 120 c H φ x u − H H φ x u − H H φ x u x + 600 c H u u x − c H u u x + 800 c H u u x +6 H H u u x − H φ x φ xxx u + 24 H H φ x φ xx u u +20 c H u u x − H H φ x u + 288 H H φ x u x + 384 H φ x φ xx u − H φ x φ xx u − H H φ x u xx − H φ x φ xx u x − H H H φ xx u u x − c H H φ x u u xx − c H H φ xx u u x − c H φ x φ xx u u +560 c H H φ x u u xx − H H H φ x u u xx + 560 c H H φ xx u u x − H H H φ x u u + 560 c H H φ x u u = 0 (141)Upon solving the O ( φ − ) , O ( φ − ) and O ( φ − ) equations for u , u and u respectively we findthat u ( x, t ) = − H φ x c H − c + H = − ξ ( t ) φ x (142) u ( x, t ) = 12 H φ xx c H − c + H = 12 ξ ( t ) φ xx (143) u ( x, t ) = − (4 φ x φ xxx − c φ x − φ xx ) H (10 c H − c + H ) φ x = (4 φ x φ xxx − c φ x − φ xx ) ξ ( t ) φ x (144)28hich similarly lends itself nicely to a representation of the solution as u ( x, t ) = 12 ξ ( t )[log( φ ( x, t ))] xx + u ( x, t )Further, if we let C ( x, t ) = B ( t ) and once again V ( x, t ) = 1 the choices for coefficients reducethe remaining orders of φ to an identically satisfied system. Solving the determining equations for φ ( x, t ) we have that φ ( x, t ) = c + c e R B ( t ) dt + x . Therefore we have the solution u ( x, t ) = − ξ ( t ) (cid:16) c (1 − c ) − c c (1 + 10 c ) e R B ( t ) dt + x + c (1 − c ) e R B ( t ) dt +2 x (cid:17)(cid:16) c + c e R B ( t ) dt + x (cid:17) (145) (4) The a i , ( i = 1 , a ( x, t ) = F ( x ) G ( t ) (separable) and using the results of Khawaja’s method we findthat a takes the form a ( x, t ) = (cid:18) F ( x ) Z x M ( x, t ) dx (cid:19) − (cid:18) xF ( x ) Z xM ( x, t ) dx (cid:19) + (cid:18) x F ( x ) Z M ( x, t ) dx (cid:19) + F ( x )( G ( t ) + G ( t ) x + G ( t ) x ) (146)where M ( x, t ) = H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) F ( x ) and C, F and G − are aribtrary functions in their respectivevariables. Letting F ( x ) = e − x , G − ( t ) = 0 and keeping all other functions arbitrary we have thefollowing vcMKDV u t + (cid:18) H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) u xxx + e − x G ( t ) u u x = 0 (147)Leading order analysis yields α = 1. Therefore we seek a solution of the form u ( x, t ) = u ( x,t ) φ ( x,t ) + u ( x, t ). Plugging this into (147) and collecting orders of φ we have O ( φ − ) : − u φ x (cid:18) e − x G u + 6 (cid:18) H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) φ x (cid:19) (148) O ( φ − ) : e − x G u u x + 6 (cid:18) H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) ( u φ x φ xx + u x φ x ) − e − x G u u φ x (149) O ( φ − ) : e − x G u u x + 2 e − x G u u x u − e − x G u u φ x − u φ t (150) − (cid:18) H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) (3 u x φ xx + 3 u xx φ x + u φ xxx ) (151) O ( φ − ) : (cid:18) H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) u xxx + 2 e − x G u u u x + e − x G u x u + u t (152) O ( φ ) : u t + (cid:18) H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) u xxx + e − x G ( t ) u u x (153)29olving the O ( φ − ) and O ( φ − ) equations for u and u respectively we obtain the following results u ( x, t ) = (cid:18) − e x H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) / φ x (154) u ( x, t ) = − (cid:18) − e x H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) G ( t ) (cid:19) / φ x (cid:18) φ xx φ x (cid:19) (155)Substituting the equations for C ( x, t ) and V ( x, t ) into the remaining orders of φ , solving the newsystem for C ( x, t ) and V ( x, t ), and mandating that H ( t ) satisfy q − G ′ ( t ) G ( t ) M ( t ) + 6 G ( t ) M ( t ) + 24 G M ′ ( t ) = M ( t ) G ( t )where M ( t ) = H ( t ) G ′ ( t ) − H ′ ( t ) G ( t ) H ( t ) we therefore have H ( t ) = c G ( t ) (cid:0) R G ( t ) dt + c (cid:1) / (156) C ( x, t ) = − G ( t )15 R G ( t ) dt + 72 c (157) V ( x, t ) = −
16 tanh (cid:18) x − (cid:18) Z G ( t ) dt + c (cid:19)(cid:19) (158)Solving the coupled pde system for φ ( x, t ) we get φ ( x, t ) = tanh (cid:18) x − (cid:18) Z G ( t ) dt + c (cid:19)(cid:19) (159)Therefore, after a bit of simplification, we have the solution u ( x, t ) = ie x/ q R G ( t ) dt + 24 c coth (cid:18) x − (cid:18) Z G ( t ) dt + c (cid:19)(cid:19) (160) We have used two direct methods to obtain very significantly extended Lax- or S-integrable familiesof generalized KdV and MKdV equations with coefficients which may in general vary in both spaceand time. Of these, the second technique which was developed here is a new, significantly extendedversion of the well-known Estabrook-Wahlquist technique for Lax-integrable systems with constantcoefficients. Some solutions for the generalized inhomogeneous KdV equations and one family ofLax-integrable generalized MKdV equations have also been presented here.Future work will address the derivation of additional solutions by various methods, as well asdetailed investigations of other integrability properties of these novel integrable inhomogeneousNLPDEs such as Backlund Transformations and conservation laws.30
Appendix: Lax integrability for generalized MKdV equations
The Lax pair is expanded in powers of u and its derivatives as follows: U = (cid:20) f + f v f + f vf + f v f + f v (cid:21) (A.1) V = (cid:20) V V V V (cid:21) (A.2) V = (cid:20) V V V V (cid:21) (A.3) V = (cid:20) V V V V (cid:21) (A.4)where V = g + g v + g v V = g + g v + g v + g v + g v x + g v xx V = g + g v + g v + g v + g v x + g v xx V = g + g v + g v V = g v x + g v + g vv xx + g v + g + g v xxxx + g vv x + g v v xx + g v + g v xxx + g v v x + g v xx + g v + g v x + g vV = g v x + g v + g vv xx + g v + g + g v xxxx + g vv x + g v v xx + g v + g v xxx + g v v x + g v xx + g v + g v x + g vV = g v x + g v + g vv xx + g v + g + g v xxxx + g vv x + g v v xx + g v + g v xxx + g v v x + g v xx + g v + g v x + g vV = g v x + g v + g vv xx + g v + g + g v xxxx + g vv x + g v v xx + g v + g v xxx + g v v x + g v xx + g v + g v x + g v = g + g v + g vv xx + g v x v xxx + g v v x + g v v xx + g v xx + g vv xxxx + g v x + g v + g v V = g v v xxx + g vv x + g v v xx + g v v x + g v v x + g v x v xx + g vv xx + g v v xxxx + g v v x + g v v xx + g vv x v xx + g vv x v xxx + g v + g v x + g v xx + g v xxx + g v xxxx + g v xxxxx + g v xxxxxx + g v x + g v + g v + g v + g V = g v v xxx + g vv x + g v v xx + g v v x + g v v x + g v x v xx + g vv xx + g v v xxxx + g v v x + g v v xx + g vv x v xx + g vv x v xxx + g v + g v x + g v xx + g v xxx + g v xxxx + g v xxxxx + g v xxxxxx + g v x + g v + g v + g v + g V = g + g v + g vv xx + g v x v xxx + g v v x + g v v xx + g v xx + g vv xxxx + g v x + g v + g v The compatibility condition gives U t − V x + [ U, V ] = ˙0 = (cid:20) p ( x, t ) F i [ v ] p ( x, t ) F i [ v ] 0 (cid:21) (A.5)where F i [ v ] represents the i th equation in the MKDV hierarchy ( i = 1 − A.1 Determining Equations for the First Equation
Requiring the compatability condition yield the F [ v ] gives f = p , f = p , g = − p a , g = − p a , g = − p a , g = − p a , f = f = g = g = 0 f p − f p = 0 (A.6) g x + p g − p g = 0 (A.7) g x + p g − p g = 0 (A.8) g + f g − f g = 0 (A.9)2 g + p g − p g = 0 (A.10) g + f g − f g = 0 (A.11)2 g + p g − p g = 0 (A.12) f ( g − g ) + p ( g − g ) = 0 (A.13) f ( g − g ) + p ( g − g ) = 0 (A.14) f t − g x + f g − f g = 0 (A.15) f t − g x − f g + f g = 0 (A.16) g + g x − g ( f − f ) = 0 (A.17)32 + g x + g ( f − f ) = 0 (A.18) g − ( p a ) x − p a ( f − f ) = 0 (A.19) g − ( p a ) x + p a ( f − f ) = 0 (A.20) g x + f g − f g + p g − p g = 0 (A.21) g x − f g + f g − p g + p g = 0 (A.22)13 ( p a ) x + 13 p a ( f − f ) + p ( g − g ) = 0 (A.23)13 ( p a ) x − p a ( f − f ) − p ( g − g ) = 0 (A.24) f t − g x − g ( f − f ) − f ( g − g ) = 0 (A.25) f t − g x + g ( f − f ) + f ( g − g ) = 0 (A.26) p t − g x − g ( f − f ) − p ( g − g ) − f ( g − g ) = 0 (A.27) p t − g x + g ( f − f ) + p ( g − g ) + f ( g − g ) = 0 (A.28) A.2 Deriving a relation between the a i In this section we reduce the previous system down to equations which depend solely on the a i ’s.We find that g = g = f = f = g = g = g = g = f = f = 0 , p = − p = − C ( t ) a g = − g = − C ( t ) g a g = − g = g x g = − g = − C ( t )( a a ) x which leads to 6 a a x − a a a x a xx + a a a xxx − K t K a + a a t − a a xxx +3 a xx a a x − a x a a x + 3 a x a a xx = 0 (A.29)where K ( t ) and C ( t ) are arbitrary functions of t . A.3 Determining Equations for the Second Equation
Requiring the compatability condition yield the F [ v ] gives p = p , g = g = g = g = g = g = g = g = 0 , g = − g , g = − g , g = − g , g = g = − p b , g = g = − p b , g = g = − p b , g = g = − p b = 2 b + 2 b (A.30) g x + f g − f g = 0 (A.31) g + p b ( f − f ) = 0 (A.32) g x + f g − f g = 0 (A.33)2 g − g + g = 0 (A.34) g + p b ( f − f ) = 0 (A.35) g + 4 g − g = 0 (A.36) g − g − g = 0 (A.37) g − g − g = 0 (A.38) g + 2 g − g = 0 (A.39) g − g − g = 0 (A.40) g − g − g = 0 (A.41) g − g − g = 0 (A.42) g − g − g = 0 (A.43) g + 2 g − g = 0 (A.44) g − g − g = 0 (A.45) g + 2 g − g = 0 (A.46) f t − g x + f g − f g = 0 (A.47) f t − g x + f g − f g = 0 (A.48) g + g x + f g − f g = 0 (A.49) g + g x + f g − f g = 0 (A.50) g + g x + f g − f g = 0 (A.51) g + g x + f g − f g = 0 (A.52)3 g + g x + f g − f g = 0 (A.53) g + g x + f g − f g = 0 (A.54)3 g + g x + f g − f g = 0 (A.55) g − g + 15 p b ( f − f ) = 0 (A.56) g + g x + f g − f g = 0 (A.57) g − g + 2 g x + 2 f g − f g = 0 (A.58) g − g − g x + f g − f g = 0 (A.59)34 − g − g x + f g − f g = 0 (A.60) g x + g ( f − f ) + f ( g − g ) = 0 (A.61) g + 2 g x − g + 2 f g − f g = 0 (A.62) g − g x − g + f g − f g = 0 (A.63) g + g x − g + f g − f g = 0 (A.64) g − g x − g + f g − f g = 0 (A.65) g x + g ( f − f ) + f ( g − g ) = 0 (A.66) g − ( p b ) x − p b ( f − f ) = 0 (A.67) g − ( p b ) x + p b ( f − f ) = 0 (A.68) g − g + 2 g + p b ( f − f ) = 0 (A.69)2 g + g − g + p b ( f − f ) = 0 (A.70) g − g x − g + f g − f g = 0 (A.71) g − g x − g + f g − f g = 0 (A.72) g + 2 g − g − p b ( f − f ) = 0 (A.73)2 g − g − g + p b ( f − f ) = 0 (A.74) g − g x − g + f g − f g = 0 (A.75) g − g + 15 ( p b ) x − p b ( f − f ) = 0 (A.76) g + g x − g ( f − f ) + f ( g − g ) = 0 (A.77) g + g x − g ( f − f ) − f ( g − g ) = 0 (A.78) g + g x + g ( f − f ) + f ( g − g ) = 0 (A.79) g + g x + g ( f − f ) + f ( g − g ) = 0 (A.80) f t − g x − g ( f − f ) − f ( g − g ) = 0 (A.81) f t − g x + g ( f − f ) + f ( g − g ) = 0 (A.82)3 g + g x − g ( f − f ) − f ( g − g ) = 0 (A.83) g + g x − g ( f − f ) − f ( g − g ) = 0 (A.84)3 g + g x + g ( f − f ) + f ( g − g ) = 0 (A.85) g + g x + g ( f − f ) + f ( g − g ) = 0 (A.86) g − g + 15 ( p b ) x + 15 p b ( f − f ) = 0 (A.87) g + 2 g − g − ( p b ) x − p b ( f − f ) = 0 (A.88) g + 2 g x − g − g ( f − f ) − f ( g − g ) = 0 (A.89)35 − g x − g − g ( f − f ) − f ( g − g ) = 0 (A.90) g + g x − g − g ( f − f ) − f ( g − g ) = 0 (A.91) g − g x − g − g ( f − f ) − f ( g − g ) = 0 (A.92) g − g x − g − g ( f − f ) − f ( g − g ) = 0 (A.93) g + g x − g − g ( f − f ) − f ( g − g ) = 0 (A.94) g − g − g + ( p b ) x + p b ( f − f ) = 0 (A.95)(A.96) g + 2 g − g − ( p b ) x + p b ( f − f ) = 0 (A.97)2 g − g − g + ( p b ) x − p b ( f − f ) = 0 (A.98) g + 2 g x − g + 2 g ( f − f ) − f ( g − g ) = 0 (A.99) g − g x − g − g ( f − f ) − f ( g − g ) = 0 (A.100) g + g x − g − g ( f − f ) − f ( g − g ) = 0 (A.101) g + g x − g − g ( f − f ) − f ( g − g ) = 0 (A.102) A.4 Deriving a relation between the b i In this section we reduce the previous system down to equations which depend solely on the b i ’s.We find that g = − g , g = g , g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = g = f = f = 0 , f = f , p = H ( t ) b g = 2 g g = g g = g g = g + 2 g f g = 12 f ( g − g ) g = g − f g = g x g = H ( t ) (cid:18) b b (cid:19) x g = H ( t ) (cid:18) b b (cid:19) x g = − H ( t ) (cid:18) b b (cid:19) xx g = 43 H ( t ) (cid:18) b b (cid:19) x − H ( t ) (cid:18) b b (cid:19) xx g = − (cid:18) b b (cid:19) xxxx g = − H ( t ) (cid:18) b b (cid:19) xx which leads to b = 2 b + 2 b b = C ( t )(2 b − b )and the long equation previously presented. Here H ( t ) and C ( t ) are arbitrary functions of t . A.5 Determining Equations for the Third Equation
Requiring the compatability condition yield the F [ v ] gives f = f = 0 , g = − p a , g = − p a , f = p , g = − p a , g = − p a , g = − p a ,g = p a , g = 17 p a , g = 13 p a , g = p a , g = p a , f = − p , g = − g = − g = g = G ( t ) , g = g = G ( t ) , g = − g = g = G ( t ) , f = f , f = f , p = − p f ( g i +24 − g i ) = 0 ( i = 12 − , , , − , −
33) (A.103) f ( g i +59 − g i ) = 0 ( i = 2 , , , , , ,
11) (A.104)23 p c + 4 p a = p c (A.105) g − p (2 c − c ) = 0 (A.106) g − p (2 c − c ) = 0 (A.107) G = − G (A.108)37 = − G (A.109) G = − G (A.110) g + g = − p c (A.111) g + 2 g = − p c (A.112) g + g = − p c (A.113) g + 2 g = − p c (A.114) g + 2 g = − p c (A.115) g + 2 g = − p c (A.116) g − p ( g − g ) = 0 (A.117) g − p ( g − g ) = 0 (A.118) g − p ( g − g ) = 0 (A.119) g − p ( g − g ) = 0 (A.120) g − p ( g − g ) = 0 (A.121) g − p ( g − g ) = 0 (A.122) g − p ( g − g ) = 0 (A.123) g + 12 p ( g − g ) = 0 (A.124) g + p ( g − g ) = 0 (A.125) g + 12 p ( g − g ) = 0 (A.126) g + p ( g − g ) = 0 (A.127) g + p ( g − g ) = 0 (A.128) g + 14 p ( g − g ) = 0 (A.129) g + 16 p ( g − g ) = 0 (A.130) g x − p ( g − g ) = 0 (A.131) g x − p ( g − g ) = 0 (A.132) g x − p ( g − g ) = 0 (A.133) g x − p ( g − g ) = 0 (A.134) g x − p ( g − g ) = 0 (A.135) g x − p ( g − g ) = 0 (A.136) g x − p ( g − g ) = 0 (A.137) g x + p ( g − g ) = 0 (A.138)38 x + p ( g − g ) = 0 (A.139) g x + p ( g − g ) = 0 (A.140) g x + p ( g − g ) = 0 (A.141) g x + p ( g − g ) = 0 (A.142) g x + p ( g − g ) = 0 (A.143) g x + p ( g − g ) = 0 (A.144) g − ( p c ) x − p ( g − g ) = 0 (A.145)2 g + g x + g = 0 (A.146) g + g x = 0 (A.147) g + g x = 0 (A.148) g + g x + p ( g − g ) = 0 (A.149)2 g + g x = 0 (A.150) g − ( p c ) x − p ( g − g ) = 0 (A.151)4 g −
13 ( p c ) x − p ( g − g ) = 0 (A.152)4 g −
13 ( p c ) x + p ( g − g ) = 0 (A.153) g + g x = 0 (A.154) g + g x = 0 ( i = 24 −
29) (A.155)3 g + g x = 0 (A.156) g + g x = 0 (A.157) g + g x = 0 (A.158) g + g x − p ( g − g ) = 0 (A.159) g − ( p c ) x + p ( g − g ) = 0 (A.160) g + g x = 0 (A.161) g + g x + 2 g = 0 (A.162) g i + g i +1 ,x = 0 ( i = 48 −
52) (A.163)3 g + g x = 0 (A.164)5 g + g x = 0 (A.165) p ( g − g ) + f ( g − g ) = 0 (A.166)2 g + g x + 2 g = 0 (A.167) g + g x + 5 g = 0 (A.168)39 + g x + 3 g = 0 (A.169)2 g + g x + 2 g = 0 (A.170) g + g x + 3 g = 0 (A.171)2 g + 3 g − p ( g − g ) = 0 (A.172)2 g + 3 g + p ( g − g ) = 0 (A.173) f t − g x − f ( g + g ) = 0 (A.174) f t − g x + f ( g − g ) = 0 (A.175) f t − g x + f ( g − g ) = 0 (A.176) f t − g x − f ( g − g ) = 0 (A.177) g x + p ( g − g ) = 0 (A.178) g x + p ( g − g ) = 0 (A.179) g x − p ( g − g ) = 0 (A.180) g x − p ( g − g ) = 0 (A.181) g − ( p c ) x = 0 (A.182) g − ( p c ) x + p ( g − g ) = 0 (A.183) g − ( p c ) x = 0 (A.184)2 g + g x = 0 (A.185) ±
17 ( p c ) x + p ( g − g ) = 0 (A.186) p t − g x + p ( g − g ) = 0 (A.187) p t − g x − p ( g − g ) = 0 (A.188) A.6 Deriving a relation between the c i In this section we reduce the previous system down to equations which depend solely on the c i ’s.We find that g = g = g = g = g = g = g = g = g = g = G ( t ) = G ( t ) = 0 , g = g , g = g , p = H ( t ) c = − p c − g g = − p c − g g = g = − p c − g g = g = ( p c ) x g = g x − g + ( p c ) x g = g x − g + ( p c ) x g = g x − g xx − g −
12 ( p c ) xx g = g = − ( p c ) xx g = g = ( p c ) x g = g = ( p c ) x g = g = 13 (2 g − g x − ( p c ) x g = − g + g x − g xx −
12 ( p c ) xx g = g = p ( c − c − c ) g i = − g ix ( i = 24 − , − g = g = − g x g = g = − g x g = g = G ( t ) g = g = 13 ( p (2 c − c − c )) xx which leads to c = − c + 5 c + 2 c − c c = 4 c + 23 c and the set of equations presented earlier. Here H ( t ) is an arbitrary function of t .41 Appendix: Lax integrability conditions for Generalized KdVEquation
As mentioned in the text, the notation and calculations here refer to the treatment of the generalizedvcKdV equation of Section 3 ONLY.
The Lax pair for the generalized vcKdV equation is expanded in powers of u and its derivativesas follows: U = (cid:20) f + f u f + f uf + f u f + f u (cid:21) (B.1) V = (cid:20) V V V V (cid:21) (B.2)where V i = g k + g k +1 u + g k +2 u + g k +3 u + g k +4 u x + g k +5 u x + g k +6 u xx + g k +7 uu xx + g k +8 u xxx + g k +9 uu xxx + g k +10 u xxxx , k = 11( i −
1) + 1 and f − ( x, t ) and g − ( x, t ) are unknown functions.The compatibility condition U t − V x + [ U, V ] = ˙0 = (cid:20) p ( x, t ) F [ u ] p ( x, t ) F [ u ] 0 (cid:21) (B.3)where F [ u ] represents the (65) and p − ( x, t ) are unknown functions, require g = g = f = g = g = g = f = g = g = g = 0 , f = p , g = − p a ,g = − p a , f = p , g = − p a , g = − p a ,g − g = 0 (B.4) g − g = 0 (B.5) p g − p g = 0 (B.6) p g − p g = 0 (B.7) g + 2 g = − p a (B.8) g + 2 g = − p a (B.9) g + 2 g = 0 (B.10) g + 2 g = 0 (B.11) g x + f g − f g = 0 (B.12) g x + f g − f g = 0 (B.13)2 g + p ( g − g ) = − p a (B.14)(B.15)42 + p ( g − g ) = − p a (B.16) g + p g − p g = 0 (B.17) g + p g − p g = 0 (B.18) g + p ( g − g ) = − p a (B.19) g + a ( p f − p f ) = 0 (B.20) g + a ( p f − p f ) = 0 (B.21)2 g + p ( g − g ) = − p a (B.22)2 g + p g − p g = 0 (B.23)2 g + p g − p g = 0 (B.24) g + g x + f g − f g = 0 (B.25) g + g x + f g − f g = 0 (B.26) g + g x + f g − f g = 0 (B.27)(B.28) p (cid:18) g + 13 f a (cid:19) − p (cid:18) g + 13 f a (cid:19) = 0 (B.29) f t − g x + f g − f g = 0 (B.30) f t − g x + f g − f g = 0 (B.31) g + g x + f g − f g = 0 (B.32) g + g x + f g − f g = 0 (B.33) g + g x + f g − f g = 0 (B.34)( p a ) x − g + p a ( f − f ) = 0 (B.35) g x + p g + f g − p g − f g = 0 (B.36) g x + p g + f g − p g − f g = 0 (B.37) g x + p g + f g − p g − f g = 0 (B.38) g x + p g + f g − p g − f g = 0 (B.39) g x + p g + f g − p g − f g = 0 (B.40) g x + p g + f g − p g − f g = 0 (B.41) g x + g ( f − f ) + f ( g − g ) = 0 (B.42) g x + g ( f − f ) + f ( g − g ) = 0 (B.43) g − ( p a ) x + p a ( f − f ) = 0 (B.44)13 ( p a ) x + 13 p a ( f − f ) + p ( g − g ) = 0 (B.45) f t − g x + g ( f − f ) + f ( g − g ) = 0 (B.46)(B.47)43 + g x + g ( f − f ) + f ( g − g ) = − p a (B.48) f t − g x + g ( f − f ) + f ( g − g ) = 0 (B.49) g + g x + g ( f − f ) + f ( g − g ) = − p a (B.50)13 ( p a ) x + 13 p a ( f − f ) + p ( g − g ) = 0 (B.51) g + g x + g ( f − f ) + f ( g − g ) = − p a (B.52) g + g x + g ( f − f ) + f ( g − g ) = − p a (B.53) g + g x + g ( f − f ) + f ( g − g ) = 0 (B.54) g + g x + g ( f − f ) + f ( g − g ) = 0 (B.55)(B.56) g x + g ( f − f ) + p ( g − g ) + f ( g − g ) = 0 (B.57) g x + g ( f − f ) + p ( g − g ) + f ( g − g ) = 0 (B.58) g x + g ( f − f ) + p ( g − g ) + f ( g − g ) = 0 (B.59) g x + g ( f − f ) + p ( g − g ) + f ( g − g ) = 0 (B.60) p t − g x + g ( f − f ) + p ( g − g ) + f ( g − g ) = p a (B.61) p t − g x + g ( f − f ) + p ( g − g ) + f ( g − g ) = p a (B.62) B.0.1 Deriving a relation between the a i In this section we reduce the previous system down to equations which depend solely on the a i ’s.We find that g = g = g = g = g = g = g = g = g = g = g = g = g = g = 0 , f = f , f = f , g = g , g = g , g = − g = p a , g = g = − p a , g = − g = − ( p a ) x , g = − g = − g x , p = p g = − g = − p a − ( p a ) xx g = − g = −
12 ( H ( t ) − H ( t )) g = − g = − ( p a ) xx − ( p a ) xxxx − p a p = H ( t ) a a − = H − ( t ) a which leads to the PDE (cid:18) H a (cid:19) t + (cid:18) H a a (cid:19) xxx + (cid:18) H a a (cid:19) xxxxx + (cid:18) H a a (cid:19) x = H a a (B.63)for which one clear solution (for H = 0) is a = a H (cid:18)(cid:18) H a (cid:19) t + (cid:18) H a a (cid:19) xxx + (cid:18) H a a (cid:19) xxxxx + (cid:18) H a a (cid:19) x (cid:19) (B.64)where a , a , a , a and H − are arbitrary functions in their respective variables.44 Appendix: Intermediate Results for Fifth-Order Equation
The intermediate results mentioned at the appropriate places in Section 3 are given here, with thederivation and the use of each detailed there. These intermediate results are: X ,t + X ,t u − X ( a u u x + a uu x + a u + a u x ) − ( a X ) x u x + a X u x − ( a X ) xxxx u x −
12 ( a X ) x u x + 12 ([ X , ( a X ) x ]) x u x + 12 ( a X ) x u x +( a X ) xxx u x + ([ X , ( a X ) x ]) xx u x + ([ X , ( a X ) x ]) xx uu x − ( a X ) xx u x +( a X ) x uu x −
12 ( a X ) x u x − ([ X , ( a X ) x ]) x uu x − ( a X ) xx u x − ( a X ) xx uu x +( a X ) x u x + ([ X , ( a X ) xx ]) x u x + ([ X , ( a X ) xx ]) x uu x − ([ X , ( a X ) x ]) x u x − ([ X , [ X , ( a X ) x ]]) x u x − ([ X , [ X , ( a X ) x ]]) x uu x − ([ X , [ X , ( a X ) x ]]) x uu x − ([ X , [ X , ( a X ) x ]]) x u u x + ( a X ) x u u x − K x + ([ X , ( a X ) x ]) x u x + a X u x − [ X , ( a X ) x ] u x + [ X , ( a X ) xx ] u x + a X u x + ( a X ) x uu x + ( a X ) x uu x − [ X , [ X , ( a X ) x ]] u x − [ X , [ X , ( a X ) x ]] u x − ( a X ) x u x − ( a X ) xx uu x − X , [ X , ( a X ) x ]] uu x + 2 a X uu x − K u u x + [ X , ( a X ) x ] u x − a X u x +[ X , ( a X ) xxx ] u x + 12 a X u x −
12 [ X , [ X , ( a X ) x ]] u x + [ X , ( a X ) x ] uu x − [ X , ( a X ) xx ] u x − [ X , ([ X , ( a X ) x ]) x ] u x − [ X , ([ X , ( a X ) x ]) x ] uu x − a X uu x + 12 a X u x + [ X , [ X , ( a X ) x ]] uu x + [ X , ( a X ) x ] u x + ( a X ) x u x − a X u x − [ X , [ X , ( a X ) xx ]] u x − [ X , [ X , ( a X ) xx ]] uu x + [ X , [ X , ( a X ) x ]] u x +[ X , [ X , [ X , ( a X ) x ]]] u x + [ X , [ X , [ X , ( a X ) x ]]] uu x + [ X , [ X , [ X , ( a X ) x ]]] uu x − a [ X , X ] u x − a [ X , X ] uu x − a [ X , X ] uu x + [ X , [ X , ( a X ) x ]] uu x +[ X , [ X , [ X , ( a X ) x ]]] u u x − a [ X , X ] u u x + [ X , K ] + [ X , ( a X ) x ] uu x +[ X , ( a X ) xxx ] uu x −
12 [ X , [ X , ( a X ) x ]] uu x + [ X , ( a X ) x ] u u x + [ X , ( a X ) x ] uu x − [ X , ( a X ) xx ] uu x − [ X , ([ X , ( a X ) x ]) x ] uu x − [ X , ([ X , ( a X ) x ]) x ] u u x − a X u u x + 12 a X uu x + [ X , [ X , ( a X ) x ]] u u x + [ X , ( a X ) x ] uu x − a X uu x − [ X , [ X , ( a X ) xx ]] uu x − [ X , [ X , ( a X ) xx ]] u u x +[ X , [ X , [ X , ( a X ) x ]]] uu x + [ X , [ X , [ X , ( a X ) x ]]] u u x + [ X , ( a X ) x ] u u x +[ X , [ X , [ X , ( a X ) x ]]] u u x − a [ X , X ] uu x − a [ X , X ] u u x − a [ X , X ] u u x +[ X , [ X , [ X , ( a X ) x ]]] u u x − a [ X , X ] u u x + [ X , K ] u = 0 , (C.1)45 X ( a u + a u + a ) − ( a X ) xxxx − K u + [ X , ( a X ) x ]+( a X ) xxx + ([ X , ( a X ) x ]) xx + ([ X , ( a X ) x ]) xx u − ( a X ) xx +( a X ) x u − ([ X , ( a X ) x ]) x u − ( a X ) xx − ( a X ) xx u − a X u +( a X ) x + ([ X , ( a X ) xx ]) x + ([ X , ( a X ) xx ]) x u − ([ X , ( a X ) x ]) x − ([ X , [ X , ( a X ) x ]]) x − ([ X , [ X , ( a X ) x ]]) x u − ([ X , [ X , ( a X ) x ]]) x u +( a X ) x u + ( a X ) x u − ( a X ) xx u + [ X , ( a X ) xxx ] + [ X , ( a X ) x ] u − [ X , ( a X ) xx ] − [ X , ([ X , ( a X ) x ]) x ] − [ X , ([ X , ( a X ) x ]) x ] u − a X u + [ X , [ X , ( a X ) x ]] u + [ X , ( a X ) x ] + ( a X ) x + [ X , [ X , ( a X ) x ]] u − a X − [ X , [ X , ( a X ) xx ]] − [ X , [ X , ( a X ) xx ]] u + [ X , [ X , ( a X ) x ]]+[ X , [ X , [ X , ( a X ) x ]]] + [ X , [ X , [ X , ( a X ) x ]]] u + [ X , [ X , [ X , ( a X ) x ]]] u − a [ X , X ] − a [ X , X ] u − a [ X , X ] u + [ X , [ X , ( a X ) x ]] u +[ X , ( a X ) xxx ] u + [ X , ( a X ) x ] u + [ X , ( a X ) x ] u + [ X , ( a X ) x ] u − [ X , ( a X ) xx ] u − [ X , ([ X , ( a X ) x ]) x ] u − [ X , ([ X , ( a X ) x ]) x ] u − a X u − [ X , [ X , ( a X ) xx ]] u − [ X , [ X , ( a X ) xx ]] u + [ X , ( a X ) x ] u +[ X , [ X , [ X , ( a X ) x ]]] u + [ X , [ X , [ X , ( a X ) x ]]] u + [ X , ( a X ) x ] u +[ X , [ X , [ X , ( a X ) x ]]] u − a [ X , X ] u − a [ X , X ] u − a [ X , X ] u = 0 , (C.2)and 46 = − a X u − a X u − a X u − ( a X ) xxxx u + [ X , ( a X ) x ] u − a [ X , X ] u +( a X ) xxx u + ([ X , ( a X ) x ]) xx u + 12 ([ X , ( a X ) x ]) xx u − ( a X ) xx u + 12 ( a X ) x u −
12 ([ X , ( a X ) x ]) x u − ( a X ) xx u −
12 ( a X ) xx u − a X u +( a X ) x u + ([ X , ( a X ) xx ]) x u + 12 ([ X , ( a X ) xx ]) x u − ([ X , ( a X ) x ]) x u − ([ X , [ X , ( a X ) x ]]) x u −
12 ([ X , [ X , ( a X ) x ]]) x u −
12 ([ X , [ X , ( a X ) x ]]) x u + 12 ( a X ) x u + 12 ( a X ) x u −
12 ( a X ) xx u + [ X , ( a X ) xxx ] u + 12 [ X , ( a X ) x ] u − [ X , ( a X ) xx ] u − [ X , ([ X , ( a X ) x ]) x ] u −
12 [ X , ([ X , ( a X ) x ]) x ] u − a X u + 12 [ X , [ X , ( a X ) x ]] u + [ X , ( a X ) x ] u + ( a X ) x u + 13 [ X , [ X , ( a X ) x ]] u − a X u − [ X , [ X , ( a X ) xx ]] u −
12 [ X , [ X , ( a X ) xx ]] u + [ X , [ X , ( a X ) x ]] u +[ X , [ X , [ X , ( a X ) x ]]] u + 12 [ X , [ X , [ X , ( a X ) x ]]] u + 12 [ X , [ X , [ X , ( a X ) x ]]] u − a [ X , X ] u − a [ X , X ] u − a [ X , X ] u + 12 [ X , [ X , ( a X ) x ]] u + 12 [ X , ( a X ) xxx ] u + 13 [ X , ( a X ) x ] u + 12 [ X , ( a X ) x ] u + 12 [ X , ( a X ) x ] u −
12 [ X , ( a X ) xx ] u −
12 [ X , ([ X , ( a X ) x ]) x ] u −
13 [ X , ([ X , ( a X ) x ]) x ] u − a X u −
12 [ X , [ X , ( a X ) xx ]] u −