Compacton equations and integrability: the Rosenau-Hyman and Cooper-Shepard-Sodano equations
Rafael Hernández Heredero, Marianna Euler, Norbert Euler, Enrique G. Reyes
aa r X i v : . [ n li n . S I] A p r Compacton equations and integrability: theRosenau-Hyman and Cooper-Shepard-Sodanoequations
R Hern´andez Heredero , M Euler , N Euler and E G Reyes Departamento de Matem´atica Aplicada a las TICUniversidad Polit´ecnica de Madrid. C. Nikola Tesla s/n. 28031 Madrid. Spain Division of Mathematics, Department of Engineering Sciences and MathematicsLule˚a University of Technology, SE-971 87 Lule˚a, Sweden Departamento de Matem´atica y Ciencia de la Computaci´onUniversidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile
April 2, 2019
Abstract
We study integrability –in the sense of admitting recursion operators– oftwo nonlinear equations which are known to possess compacton solutions: the K ( m, n ) equation introduced by Rosenau and Hyman D t ( u ) + D x ( u m ) + D x ( u n ) = 0 , and the CSS equation introduced by Coooper, Shepard, and Sodano, D t ( u ) + u l − D x ( u ) + αpD x ( u p − u x ) + 2 αD x ( u p u x ) = 0 . We obtain a full classification of integrable K ( m, n ) and CSS equations ; wepresent their recursion operators, and we prove that all of them are related(via nonlocal transformations) to the Korteweg-de Vries equation. As an ap-plication, we construct isochronous hierarchies of equations associated to theintegrable cases of
CSS . Mathematics Subject Classification:
Keywords:
Compacton, Rossenau-Hyman equation, Cooper-Shepard-Sodano equation,isochronous equation, formal integrability, recursion operator. Introduction
We begin by quoting Rosenau [30]: ‘’We define a compact wave as a robust solitarywave with compact support beyond which it vanishes identically. We then define a compacton as a compact wave that preserves its shape after interacting with othercompacton”. Rosenau and Hyman found examples of compactons while studyinggeneralizations of the Korteweg-de Vries equation for which the dispersion term isnonlinear. Their model equation is the so-called K ( m, n ) equation u t + ( u m ) x + ( u n ) xxx = 0 , (1.1)and an example of a compacton bearing equation within the family (1.1) is K (2 , u ( x, t ) = (4 c/
3) cos (( x − ct ) /
4) for | x − ct | ≤ π and u ( x, t ) = 0 otherwise, is a compacton solution. Further works on compactons are [21,22] and the comprehensive review [27].It turns out that solutions to equations within the K ( m, n ) can exhibit verycomplex behaviors; we refer the reader to [2–4], and to the papers [22, 27, 36] au-thored by Rosenau and his coworkers, for general discussions. Here, we just mentionone example: in [31] the authors present four local conservation laws of K (2 , ∗ . This (non)existence of conservation laws has an important analytic im-plication, see [36]: initially nonnegative, smooth and compactly supported solutionsto K ( m, n ) lose their smoothness within a finite time.We wonder if this complex behavior has to do with (lack of) integrability. In thiswork we present a detailed study of the integrability properties of K ( m, n ). We findthat, module a rather general space of allowable transformations, the only integrableequations belonging to the K ( m, n ) family are the KdV and modified KdV equations,and that integrable equations within the K ( m, n ) family cannot have compactonsolutions. In particular, we recover the observation in [18, 35] that K (2 ,
2) is notintegrable.In order to obtain this result we classify all integrable K ( m, n ) equations usingthe theory of formal symmetries (to be summarized in Section 2). The power ofthis approach has been amply demonstrated by the classification results for evolu-tion equations and systems of equations due to researchers such as Shabat, Fokas,Svinolupov, Sokolov, Mikhailov and others (see [15–17, 19, 24, 32]), and also by theimportant papers [33,34] on the classification of integrable scalar evolution equationssatisfying an homogeneity condition.Since our search for integrable compacton bearing equations within the K ( m, n )class does not yield examples, we also investigate a related family, the Cooper-Shepard-Sodano family of equations u t + u l − u x − αpD x (cid:0) u p − u x (cid:1) + 2 αD x ( u p u x ) = 0 , α = 0 , (1.2)introduced in [12]. We quote from this paper: “These equations have the same termsas the equations considered by Rosenau and Hyman, but the relative weights of the ∗ This observation has been proven rigorously by Vodov´a in 2013, see [35]. l = 3, p = 2. This family of equations is further studied in [14, 20].Encouraging properties of (1.2) are the facts that it admits a Hamiltonian for-mulation, and that it possesses three physically interesting conservation laws: area,mass and energy. Regretfully, we prove herein that they are not integrable in gen-eral. Using formal symmetries once more, we obtain six integrable equations withinthe (1.2) family. None of them can support compacton solutions.Since the existence of compacton solutions is a rather extraordinary occurrencein the nonlinear world, we believe that our results are not only important by them-selves, but also because they seem to express certain rigidity in our present alge-braic/geometric/analytic approach to integrability. In other words, K (2 ,
2) say, mustbe “special”, and so far we have not been able to uncover the deeper source of itsspecial character.Our paper is organized as follows. We review the theory of formal symmetriesand integrability in Section 2 after, essentially, [24], and in Section 3 we use thistheory to classify integrable K ( m, n ) equations. We note that a previous classifica-tion has appeared in [18]. One integrable case was missing therein and we single itout here. Fortunately, the missing case does not alter the conclusion in [18] that theonly K ( m, n ) integrable case are (essentially, module a class of allowable transfor-mations specified in Section 3) the KdV and mKdV equations. The present classi-fication also differs from the one appearing in [18] in that here we explain in detailhow to connect our integrable cases to KdV (or, to the linear equation) and becausein Section 5 we exhibit explicit recursion operators for all our integrable K ( m, n )equations. In Section 4 we study integrability of the Cooper-Shepard-Sodano familyand again we are able to explain how to connect its integrable cases to KdV (or,to the linear equation), and to exhibit recursion operators. Finally in Section 6 wepresent an application of our results: we construct integrable isochronous equations,after [9–11], starting from the equations in our classification of integrable CSS equa-tions, explain how to obtain their point symmetries, and present their correspondingrecursion operators. The formal symmetry approach to integrability [24, 25] begins with the observationthat standard (systems of) partial differential equations which are integrable (for in-stance, in the sense of Calogero, see [8]) usually admit an infinite set of (generalized)symmetries of arbitrarily large differential order. A.B. Shabat and his collaborators,see for instance [24, 25], realized that it is possible to weaken the notion of a (gen-eralized) symmetry to the notion of a formal symmetry —to be defined preciselybelow— and that this new concept provides a computationally efficient tool fordefining integrability and classifying integrable equations. We recall from [24, 26]3hat G = ( G α ) is a symmetry of a system of partial differential equations of theform ∆ a ( x i , u α , u αx i , . . . ) = 0, if ∆ ∗ ( G ) = 0 (2.1)whenever u α ( x i ) is a solution to ∆ a = 0, where ∆ ∗ is the formal linearization of thesystem ∆ a = 0, that is, ∆ ∗ = (cid:16)P L ∂ ∆ a ∂u αL D L (cid:17) . If the system ∆ a = 0 consists of justone scalar evolution equation, ∆ = u t − F , (2.2)then equation (2.1) becomes D t G = F ∗ ( G ) or, equivalently, D t G = D τ F , (2.3)where D τ = X K ≥ D K ( G ) ∂∂u K . Note: Here and henceforth we use standard notation from the geometric theoryof differential equations as presented in [26] , see also [24] . Following [24, 25], we apply a second linearization to formula (2.3). We obtain,using some formulae appearing in [24],( D t G ) ∗ = ( D τ F ) ∗ ⇔ D t ( G ∗ ) + G ∗ ◦ F ∗ = D τ ( F ∗ ) + F ∗ ◦ G ∗ , (2.4)in which D t ( P L a L D L ) = P L D t ( a L ) D L if G ∗ = P L a L D L , and the last equalityholding on solutions to (2.2). The expression D τ ( F ∗ ) is defined analogously. Weinterpret our symmetry condition (2.4) using commutators: D t ( G ∗ ) − [ F ∗ , G ∗ ] = D τ ( F ∗ ) . (2.5)Let us consider the degree of the operators appearing in (2.5). The degree of F ∗ as a differential operator —let us denote it by deg( F ∗ )— is the differential orderof F , i.e. the order of the differential equation (2.2) and thus, it is fixed. The degreeof the left hand side of (2.5) depends on G : D t ( G ∗ ) − [ F ∗ , G ∗ ] is a differential operatorgenerically of degree deg( G ∗ ) plus deg( F ∗ ) minus 1, much higher than that of theoperator in the right hand side, of degree deg( F ∗ ), if there are high order symme-tries G . Thus, it is not clear at all that non-trivial solutions to (2.5) should exist: theexistence of (generalized) symmetries G of arbitrarily high differential order mustimpose extremely strong constraints on the function F .Following [24, 25], and partially motivated by the theory of recursion operators,see [26], we define formal symmetries using the left hand side of Equation (2.5): Definition 1.
Let u t = F be an evolution equation with F a function of two inde-pendent variables x , t , one dependent variable u and a finite number of derivativesof u with respect to x . A formal symmetry of rank k of this partial differentialequation is a formal pseudo-differential operatorΛ = l r D r + l r − D r − + · · · + l + l − D − + l − D − + · · · , D = D x (2.6)4ith l i being functions of t , x , u and finite numbers of x -derivatives of u , that satisfiesthe equation D t (Λ) = [ F ∗ , Λ] (2.7)whenever u is a solution to u t = F , up to a pseudo-differential operator of degree r +deg( F ∗ ) − k . A formal symmetry of infinite rank is a pseudo-differential operator (2.6)such that (2.7) holds identically whenever u is a solution to u t = F .Note that if G is a symmetry of order p of u t = F , then (2.5) implies that G ∗ is a formal symmetry of rank p . We also remark that a formal symmetry of infiniterank is a recursion operator, see [26]. Thus, it generates, in principle, an infinitenumber of generalized symmetries of the equation at hand. For example, see [13],it can be proven that application of a quasilocal recursion operator (in the senseof [13, Section 1]) to a given symmetry yields a (generalized) symmetry, and so suchan operator could indeed generate an infinite chain of (generalized) symmetries.The main technical point behind Definition 1 is that the space of solutions ofequation (2.7) is much richer and structured than that of equation (2.5) or even (2.1).For example, powers and roots of formal symmetries (computed using the standardtheory of formal pseudo-differential operators, see [24, 26]) are also formal symme-tries. In fact, this observation was one of the original motivations for the use of formalpseudo-differential operators in Definition 1, because the r th root of a differentialoperator (2.6) is usually a pseudo -differential operator.Now we explain why Definition 1 restricts the function F . A theorem due toM. Adler, see [1], states that the residue (the coefficient of D − ) of a commutatorof formal pseudo-differential operators is always a total derivative. If we apply thisresult to different powers Λ i/r of a generic formal symmetry † Λ of rank k insertedinto (2.7), we obtain D t (residue(Λ i/r )) = residue D t (Λ i/r ) = residue[ F ∗ , Λ i/r ] = Dσ i for some differential functions σ i , i.e. a sequence of conservation laws D t ρ i . = D x σ i , i = − , , . . . , (2.8)which are, together with the special case D t ρ = D t ( l r /l r − ) = D x σ , the socalled canonical conservation laws . The symbol . = means that equations (2.8) musthold on solutions of (2.2), i.e. all derivatives with respect to t must be substitutedusing the equation and its differential consequences.As observed in [24], the canonical densities ρ i and conserved fluxes σ i are differ-ential functions which can be recursively written in terms of the right hand side F ofthe equation and its derivatives. The fact that the left hand side of (2.8) must be atotal derivative with respect to x for all i = − , , · · · , produces obstructions thatare necessary conditions for the existence of (generalized/formal) symmetries G ,i.e. for integrability. † The foregoing discussion implies that instead of a general formal symmetry Λ of degree r , wecan consider its r th root Λ /r of degree 1 without loss of generality, see [24] for details. u t = F ( x, u, u x , u xx , u xxx ) , (2.9)the first canonical density is ρ − = (cid:0) ∂F (cid:14) ∂u xxx (cid:1) − / , see [24]. Therefore, a firstintegrability condition is requiring D t ρ − to be the total derivative of a local func-tion σ − . The second canonical density imposes further differential restrictions on F ,and so forth. Usually, after a small number of steps our family of equations eitherfails to satisfy the integrability conditions, or the right hand side F becomes so spe-cific that we are able to produce a formal symmetry of infinite rank and therefore,in principle, a sequence of generalized symmetries of u t = F . If u t = F representsa family of equations, this procedure allows us to find all integrable cases in thefamily. We are led to the following precise definition of integrability, after [24, 26]: Definition 2.
A system of evolution equations is integrable if and only if it possessesa formal symmetry of infinite rank.Let us we write down the first five canonical densities for a third order equa-tion (2.9) following [24]. We will use them in the next section to study the integra-bility of the Rosenau-Hyman and Cooper-Shepard-Sodano equations:
Proposition 1.
Let u t = F ( x, u, u x , u xx , u xxx ) be an arbitrary third order evolutionequation. The first five canonical conserved densities can be written explicitly as ρ − = (cid:18) ∂F∂u xxx (cid:19) − / , (2.10) ρ = ρ − ∂F∂u xx , (2.11) ρ = D x (cid:18) ρ − − u x + ρ − ∂F∂u xx (cid:19) + ρ − − ( D x ρ − ) + 13 ρ − (cid:18) ∂F∂u xx (cid:19) + ρ − ( D x ρ − ) ∂F∂u xx − ρ − ∂F∂u x + ρ − σ − , (2.12) ρ = −
13 ( D x ρ − ) ∂F∂u xx − ( D x ρ − ) ∂F∂u x + ρ − ∂F∂u + ρ − − ( D x ρ − ) ∂F∂u xx − ρ − ∂F∂u x ∂F∂u xx + 13 ρ − ( D x ρ − ) (cid:18) ∂F∂u xx (cid:19) + 227 ρ − (cid:18) ∂F∂u xx (cid:19) + 13 ρ − σ , (2.13) ρ = ρ − σ − ρ σ − . (2.14) Remark 1.
The condition of existence of a formal symmetry of infinite rank isstrictly weaker than the condition of existence of an infinite number of generalizedsymmetries. Indeed, the 2 component system (cid:26) u t = u xxxx + v v t = v xxxx , one generalized symmetry, as provedby Beukers, Sanders and Wang [6]. On the other hand, it does possess a recursionoperator [7].Now, a very important remark, see [24] and also [26], is that the use of trans-formations between equations is a very convenient way to proceed when seekingclassifications. In this paper we deal with integrable equations of the type u t = f ( u ) u xxx + g ( u, u x , u xx ) , (2.15)see [12, 31]. The general strategy we use to classify these equations consists in per-forming a sequence of convenient point transformations and differential substitutionsthat preserve integrability, and then apply and compute the integrability conditionsassociated to (2.10)–(2.14). Our general procedure is as follows:First, if f ′ ( u ) = 0 the point transformation u → f ( u ) − / converts Equa-tion (2.15) into another one of the form u t = D x h u xx u + f ( u, u x ) i + f ( u, u x , u xx ) , (2.16)where f ( u, u x , u xx ) is not a total x -derivative. This form is very convenient becauseit follows from (2.10) that the integrability condition D t ρ − = D x σ − is equivalentto requiring that f = 0. This condition greatly restricts the form of the equation.Once f = 0, the equation admits a potentiation u → u x that brings it into theform u t = u xxx /u x + f ( u x , u xx ). A subsequent hodograph transformation x → u , u → x simplifies it to one of the form u t = u xxx + h ( u x , u xx ) . This equation can be “antipotentiated” ( u x → u ) to get u t = u xxx + D x h ( u, u x ) . Our integrability conditions imply that the integrable cases of this equation are allof the form u t = u xxx + D x (cid:2) h ( u ) u x + ( a + bu ) u x + h ( u ) (cid:3) . (2.17)If h ( u ) = 0 a further point transformation R exp (cid:2) R h ( u ) du (cid:3) du → u transformsthis equation into another one of the form u t = u xxx + ( a + bu ) u xx + f ( u ) u x + f ( u ) u x + f ( u ) u x . (2.18)We will show that all the integrable cases of the RH and CSS equations can bewritten in the form (2.18), as linear equations, KdV or mKdV, or the Calogero-Degasperis-Fokas (CDF) equation (see below; the CDF is Miura-transformable toKdV). Thus, if we “pullback” the recursion operator of KdV (or, the linear equation)by the foregoing transformations, we can construct recursion operators of the originalequations, and therefore we obtain an explicit proof of integrability in terms ofDefinition 2. In actual fact, we seldom perform this pullback operation explicitly.Once we know that a given equation is integrable, it is usually straightforward tocompute its recursion operator from first principles, as in [29].Now we carry out this plan. 7 Integrability of the Rosenau-Hyman equation
We consider the compacton equation of Rosenau and Hyman (see [31]) D t ( u ) + D x ( u m ) + D x ( u n ) = 0 , n = 0 . (3.1)As we informed in Section 2, D t and D x are total derivatives with respect to inde-pendent variables t and x . For simplicity, we use the subindex notation u t = D t u , u x = D x u , u xx = D x but we prefer to use the total derivative notation when appliedto a more complicated differential function, e.g. D x ( u m ) = mu m − u x .The case n = 1 i.e. u t + mu m − u x + u xxx = 0 , is well-known, see [24, Section 4.1]: the only integrable cases are m = 0 , , ,
3, i.e. thelinear equation, the KdV and the modified KdV equations. We write them as (usingthe point transformation x → − x ) u t = u xxx + α u x + β, α, β ∈ C , (3.2) u t = u xxx + 2 uu x , (3.3) u t = u xxx + 3 u u x . (3.4)If n = 1, the point transformation x → − x , t → t/n , u → u / (1 − n ) changes (3.1)into equations of the form (2.16), namely: u t = D x (cid:20) u xx u − m ( n − n (3 m − n − u − m + n − n − n n − u x u (cid:21) + ( n + 2)(2 n + 1) u x ( n − u (3.5)if 3 m − n − = 0, and u t = D x (cid:20) u xx u − n n − u x u + ( n + 2)3 n log u (cid:21) + ( n + 2)(2 n + 1) u x ( n − u (3.6)if 3 m − n − D t ρ − = D x σ − or, equivalently, u t ∈ Im D x implies that either n = − n = − / u t = u xxx + D x (cid:20) n + 2 n − u x u + m ( n − n (3 m − n − u m − n − (cid:21) , (3.7) u t = u xxx + D x (cid:20) (cid:18) n + 2 n − (cid:19) u x u + n + 23 n u log u (cid:21) . (3.8)If n = −
2, Equation (3.7) becomes u t = u xxx − m − u − m u x m = 1 , , − , −
2, corresponding to linear equations,KdV and mKdV. On the other hand, if n = −
2, Equation (3.8) becomes a linearequation included in case (3.2).If n = − /
2, Equation (3.7) can be written in the form (2.18), this is, u t = u xxx − u x − m ( m − m − (1 − m ) u u x , after a point transformation u → e u . This family of equations satisfies the first twointegrability conditions. The third integrability condition is D t ρ ∈ Im D x , and thecanonical conserved density ρ satisfies D t ( ρ ) ∼ − m − m (2 m − m + 1) e u − mu u x , in which the symbol ∼ denotes equality except for the addition of a total x -derivative.Thus, integrability can be achieved only in the cases m = − /
2, 0, 1, 3 / ‡ whichare all subcases of the Calogero-Degasperis-Fokas (CDF) equation u t = u xxx − u x + (cid:0) α e u + β e − u + γ (cid:1) u x . (3.9)Finally, when n = − / u t = u xxx − u x u xx u + 32 u x u − (1 + log u ) u x and for it D t ( ρ ) ∼ − u x u so this case is not integrable. Remark 2.
We note that the CDF equation (3.9) can be related to the KdV equa-tion through the Miura transformation32 u xx − u x − p β e − u u x − αe u − βe − u − γ → u . We summarize the integrable cases of the Rosenau-Hyman family (3.1) in thefollowing theorem. We make the point transformation x → − x , t → t/n and write u t = n D x ( u m ) + n D x ( u n ) , n = 0 (3.10)instead of (3.1). This transformation is invertible and does not affect integrability. ‡ We note that the case n = − / m = 3 / heorem 1. The integrable cases of the Rosenau-Hyman family (3.10) are1. n = 1 , m = 0 , , , , corresponding to Equations (3.2) , (3.3) and (3.4) ,namely, u t = u xxx + α u x + β, α, β ∈ C , (3.11) u t = u xxx + 2 uu x , (3.12) u t = u xxx + 3 u u x . (3.13) n = − , m = − , − , , , corresponding to Equations u t = D x (cid:20) D x (cid:16) u x u (cid:17) − u (cid:21) , (3.14) u t = D x (cid:20) D x (cid:16) u x u (cid:17) − u (cid:21) , (3.15) u t = − D x (cid:2) u − (cid:3) , (3.16) u t = D x (cid:20) D x (cid:16) u x u (cid:17) − u (cid:21) (3.17) respectively.3. n = − , m = , , , − , corresponding to Equations u t = D x h D x (cid:16) u x u / (cid:17) − u / i , (3.18) u t = D x h D x (cid:16) u x u / (cid:17) − u i , (3.19) u t = D x h u x u / i , (3.20) u t = D x (cid:20) D x (cid:16) u x u / (cid:17) − u / (cid:21) (3.21) respectively.All these equations are related to the linear equation or to the KdV equation throughdifferential substitutions. Remark 3.
It is clear that the equations appearing above cannot admit solutionswith compact support, let alone compactons. As explained in Section 1, this the-orem extends and enriches the discussion on K ( m, n ) appearing in [18]. We men-tion that J. Vodov´a classified conservation laws of the K ( m, m ) equations and ob-served that K ( − , − K ( − / , − /
2) are integrable; integrability of K ( − , − K ( − / , − / K ( − , − Integrability of Cooper-Shepard-Sodano
In this section we study equations of the form u t + u l − u x − αpD x (cid:0) u p − u x (cid:1) + 2 αD x ( u p u x ) = 0 , α = 0 . (4.1)We consider the case p = 0 first. Equation (4.1) becomes u t + u l − u x + 2 αu xxx = 0,which is integrable if and only if l = 2 , ,
4, i.e. in the linear, KdV and mKdV case,as observed in [24]. Let us now consider p = 0. We use the same strategy as with theRosenau-Hyman case: first we apply the change t → − t , u → (2 αu ) − /p to obtain u t = D x " u xx u − p + 3)2 p u x u − p (2 α ) − lp l − p − u − l + pp + ( p + 3)( p + 6) u x p u (4.2)if 3 l − p − = 0, and u t = D x (cid:20) u xx u − p + 3)2 p u x u + 1 √ α log u (cid:21) + ( p + 3)( p + 6) u x p u (4.3)if 3 l − p − p = − p = −
6. If p = −
3, a combination of potentiation, hodograph, antipotentia-tion and exponential point transformation u → e u , with a scaling to absorb theconstant α , change (4.2) and (4.3) into the equations u t = u xxx − u x l − e u − lu u x l − u t = u xxx − u x uu x respectively. On the other hand, if p = −
6, the same combination of transformations,using u → u instead of the exponential, changes (4.2) and (4.3) into u t = u xxx + ( l − l u x u l and u t = u xxx + log( u ) u x respectively.Let us assume that 3 l − p − = 0. The integrability condition in ρ implies thatif p = − l = − , ,
3, and if p = −
6, then l = − , − , l − p − u t = au u xxx − au u x u xx + 21 au u x + u x u ( p = − l = 0) and u t = au u xxx − au u x u xx + 6 au u x + u x u , ( p = − l = 1), in which a = 2 α . The integrability condition for ρ implies thatboth equations are not integrable.Summarizing, we have the following theorem.11 heorem 2. The integrable equations of family (4.1) are1. p = 0 , l = 2 , , , corresponding to the linear equation, KdV equation, mKdVequation;2. p = − , l = − , − , , corresponding to the Equations u t = au u xxx − au u x u xx + 21 au u x + u x u , (4.4) u t = au u xxx − au u x u xx + 21 au u x + u x u , (4.5) u t = au u xxx − au u x u xx + 21 au u x + u x . (4.6) respectively.3. p = − , l = − , , , corresponding to the Equations u t = au u xxx − au u x u xx + 6 au u x + u x u , (4.7) u t = au u xxx − au u x u xx + 6 au u x + u x , (4.8) u t = au u xxx − au u x u xx + 6 au u x + uu x . (4.9) respectively.All these equations are related to the linear equation or to the KdV equation throughdifferential substitutions. As in Section 3, this theorem implies that no integrable CSS equation admitssolutions with compact support.
In this section we construct explicit recursion operators for the equations appearingin the above theorems using the work [29]. First of all, we note that these equationsare all in [24]. We have (when we write (4.x.xx) we are referring to the correspondingequation in [24]):1. Equation (3.14) is a special case of (4.1.27), namely, u t = D x (cid:18) u xx u − u x u + 12 u (cid:19) = − u x u xx u + 12 u x u − u x u + u xxx u . (5.1)2. Equation (3.15) is equivalent to a subcase of (4.1.25) u t = D x (cid:18) u xx u − u x u − u + cu (cid:19) = − u x u xx u +12 u x u + 3 u x u + cu x + u xxx u . (5.2)12. Equations (3.16) and (3.17) are equivalent to subcases of (4.1.34) u t = D x (cid:18) u xx u − u x u + c u x u + c u (cid:19) = − u x u xx u + 12 u x u − c u x u + c u x + c u xx u + u xxx u . (5.3)4. Equations (3.18)-(3.21) are all equivalent to subcases of (4.1.30) u t = D x (cid:20) u xx u − u x u − λu x u ( λu + 1) + c ( λu + 1) u + c u λu + 1 + c u (cid:21) with λ = 0, that is, u t = − u x u xx u + 6 u x u − c u x u + 2 c uu x + c u x + u xxx u , (5.4)after applying the point transformation u → u .Now, Equations (5.1), (5.2) and (5.3) are special cases (for the values of theconstant numbers c , c relevant to us) of Equation (81) in [29], namely, u t = u xxx u − u x u xx u + 12 u x u + 23 λ u x u + 12 λ u x u − cu x , (5.5)while Equation (5.4) is Equation (85) in [29], see below. The recursion operator forEquation (5.5) is as follows: R [ u ] = u − D x − u − u x D x − u − u xx + 12 u − u x + 2 λ u − + 2 λ u − − u t D − x ◦ u x D − x ◦ (cid:18) λ u − − c (cid:19) . Acting R [ u ] on the t -translation symmetry u t ∂∂u , yields a corresponding symmetry-integrable hierarchy of order 2 m + 3, namely u t = R m [ u ] ◦ (cid:18) u xxx u − u x u xx u + 12 u x u + 23 λ u x u + 12 λ u x u − cu x (cid:19) ,m = 0 , , , . . . , and we note that for the x -translation symmetry we obtain R [ u ] ◦ u x = 0 . Let us now consider the integrable cases of the CSS equations. We see thatEquations (4 . .
5) and (4 .
6) are special cases of Equation (90) in [29], namely u t = au xxx u − au x u xx u + 21 au x u + β u x u + β u x u + β u x , (5.6)13here a , β , β and β are arbitrary constants and a = 0. This equation admits thefollowing recursion operator: R [ u ] = 1 u D x − u x u D x − u xx u + 22 u x u + 2 β a u + 4 β a u − a u t D − x ◦ u + u x D − x ◦ (cid:18) − u xx u + 6 u x u + β a u + 2 β a u + 2 β a u (cid:19) . (5.7)Acting R [ u ] on the t -translation symmetry u t ∂∂u , we obtain a corresponding symmetry-integrable hierarchy of order 2 m + 3, namely u t = R m [ u ] ◦ (cid:18) au xxx u − au x u xx u + 21 au x u + β u x u + β u x u + β u x (cid:19) , (5.8) m = 0 , , , . . . , and we note that for the x -translation symmetry we obtain R [ u ] ◦ u x = 0 . On the other hand, Equations (4.7), (4.8) and (4.9) are special cases of Equa-tion (85) in [29], namely u t = au xxx u − au x u xx u + 6 au x u + β u x u + β uu x + β u x , (5.9)where a , β , β and β are arbitrary constants and a = 0. This equation admits thefollowing recursion operator: R [ u ] = 1 u D x − u x u D x − u xx u + 6 u x u + β a u + β a u − a u t D − x ◦ β a u x D − x ◦ u + β a u x D − x ◦ . (5.10)Acting R [ u ] on the t -translation symmetry u t ∂∂u , we obtain a symmetry-integrablehierarchy of order 2 m + 3, namely u t = R m [ u ] ◦ (cid:18) au xxx u − au x u xx u + 6 au x u + β u x u + β uu x + β u x (cid:19) , (5.11) m = 0 , , , . . . , and we note that for the x -translation symmetry we obtain R [ u ] ◦ u x = 0 . The isochronous equations for (5.6) and (5.9)
In this section we construct new integrable evolution equations starting from whatwe will call the Cooper-Shepard-Sodano model equations (5.6) and (5.9). Our newequations are isochronous in the sense of Calogero, see [9, Chapter 7] and [10,11,23]:they are autonomous evolution PDEs which depend on a positive parameter ω andpossess many solutions which are time-periodic with period T = 2 π/ω . For com-pleteness, we also explain how to obtain the Lie point symmetries of our equationsand present their recursion operators. Following [9–11, 23], we introduce a new dependent variable v ( r, s ), where r and s are new independent variables, as follows: u ( x, t ) = e − iλωs v ( r, s ) (6.1a) x = re iµωs (6.1b) t = 1 iω (cid:0) e iωs − (cid:1) . (6.1c)The prolongations are u t = e − i ( λ +1) ωs [ v s − iλω v − iµωr v r ] u nx = e − i ( λ + nµ ) ωs v nr , n = 1 , , , . . . , where u nx = ∂ n u∂x n , v nr = ∂ n v∂r n . With the change of variables (6.1a) – (6.1c), equation (5.6) takes the form v s − iλωv − iµωrv r = e i (6 λ − µ +1) ωs (cid:18) av rrr v − av r v rr v + 21 av r v (cid:19) + β e i (4 λ − µ +1) ωs (cid:16) v r v (cid:17) + β e i (3 λ − µ +1) ωs (cid:16) v r v (cid:17) + β e i ( − µ +1) ωs v r . (6.3)This equation can become autonomous for a = 0 only if µ = 2 λ + 13 , so that (6.3) then takes the form v s − iλωv − i (cid:18) λ + 13 (cid:19) ωrv r = av rrr v − av r v rr v + 21 av r v + β e i (2 λ + ) ωs (cid:16) v r v (cid:17) + β e i ( λ + ) ωs (cid:16) v r v (cid:17) + β e i (2 λ + ) ωs v r . (6.4)15learly (6.4), and therefore (6.3), becomes autonomous in the following three cases: Case 1.1: λ = −
13 and µ = −
13 with β = β = 0. Then (6.3) becomes v s + i ωv + i ωrv r = av rrr v − av r v rr v + 21 av r v + β v r v . (6.5) Case 1.2: λ = −
23 and µ = − β = β = 0. Then (6.3) becomes v s + i ωv + iωrv r = av rrr v − av r v rr v + 21 av r v + β v r v . (6.6) Case 1.3: λ = 13 and µ = 1 with β = β = 0. Then (6.3) becomes v s − i ωv − iωrv r = av rrr v − av r v rr v + 21 av r v + β v r . (6.7)Now we consider (5.9). With the change of variables (6.1a) – (6.1c), Equa-tion (5.9) takes the form v s − iλωv − iµωrv r = e i (3 λ − µ +1) ωs (cid:18) av rrr v − av r v rr v + 6 av r v (cid:19) + β e i (3 λ − µ +1) ωs (cid:16) v r v (cid:17) + β e i ( − λ − µ +1) ωs vv r + β e i ( − µ +1) ωs v r . (6.8)This equation can become autonomous for a = 0 only if µ = λ + 13 , so that (6.8) then takes the form v s − iλωv − i (cid:18) λ + 13 (cid:19) ωrv r = av rrr v − av r v rr v + 6 av r v + β e i (2 λ + ) ωs (cid:16) v r v (cid:17) + β e i ( − λ + ) ωs vv r + β e i ( − λ + ) ωs v r . (6.9)Clearly (6.9), and therefore (6.8), becomes autonomous in the following three cases: Case 2.1: λ = −
13 and µ = 0 with β = β = 0. Then (6.8) becomes v s + i ωv = av rrr v − av r v rr v + 6 av r v + β v r v . (6.10)16 ase 2.2: λ = 13 and µ = 23 with β = β = 0. Then (6.8) becomes v s − i ωv − i ωrv r = av rrr v − av r v rr v + 6 av r v + β vv r . (6.11) Case 2.3: λ = 23 and µ = 1 with β = β = 0. Then (6.8) becomes v s − i ωv − iωrv r = av rrr v − av r v rr v + 6 av r v + β v r . (6.12)Our isochronous equations are (6.5)–(6.7) and (6.10)–(6.12). We list the Lie point symmetries of equation (5.6), that is u t = au xxx u − au x u xx u + 21 au x u + β u x u + β u x u + β u x . Besides the obvious x -translation, Z x = ∂∂x , and t -translation symmetry, Z t = ∂∂t ,Equation (5.6) also admits the following point symmetries:a) For β = β = β = 0: Z = x ∂∂x + 3 t ∂∂t , Z = u ∂∂u + 6 t ∂∂t . b) For β = β = 0 and β = 0: Z = ( x − tβ ) ∂∂x + 3 t ∂∂t , Z = u ∂∂u + 6 t ∂∂t − tβ ∂∂x . c) For β = β = 0 and β = 0: Z = − u ∂∂u + x ∂∂x − t ∂∂t . d) For β = β = 0 and β = 0: Z = − u ∂∂u + x ∂∂x − t ∂∂t . We list the Lie point symmetries of equation (5.9), that is u t = au xxx u − au x u xx u + 6 au x u + β u x u + β uu x + β u x . Besides the obvious x -translation symmetry and t -translation symmetry, (5.9) alsoadmits the following point symmetries: 17) For β = β = β = 0: Z = x ∂∂x + 3 t ∂∂t , Z = u ∂∂u + 3 t ∂∂t , Z = xu ∂∂u − x ∂∂x . b) For β = β = 0 and β = 0: Z = ( x − tβ ) ∂∂x + 3 t ∂∂t , Z = u ∂∂u + 3 t ∂∂t − tβ ∂∂xZ = u ( x + β t ) ∂∂u −
12 ( x + β t ) ∂∂x . c) For β = β = 0 and β = 0: Z = u ∂∂u − x ∂∂x − t ∂∂t . d) For β = β = 0 and β = 0: Z = u ∂∂u + 3 t ∂∂tZ = u sin (cid:16) a − / β / x (cid:17) ∂∂u + a / β − / cos (cid:16) a − / β / x (cid:17) ∂∂xZ = u cos (cid:16) a − / β / x (cid:17) ∂∂u − a / β − / sin (cid:16) a − / β / x (cid:17) ∂∂x . We can now obviously map the symmetries of (5.6) and (5.9) with the (6.1a) –(6.1c) to symmetries of the isochronous equations (6.3) and (6.8). For example, invertical form the x -translation symmetry u x ∂∂u , then takes the form e − iµωs v r ∂∂v , for (6.3) and (6.8), whereas the t -translation symmetry u t ∂∂u , becomes the symmetry e − iωs ( v s − iλωv − iµωrv r ) ∂∂v for (6.3) and (6.8). 18 .3 The isochronous hierarchies for (5.6) and (5.9) For equation (5.6) we have the hierarchy (5.8), namely u t = R m [ u ] ◦ (cid:18) au xxx u − au x u xx u + 21 au x u + β u x u + β u x u + β u x , (cid:19) m = 0 , , , . . . , where R [ u ] is given by (5.7). Corresponding to the above Case 1.1, Case 1.2 andCase 1.3, the isochronous hierarchies are the following: Case 1.1:
We consider the hierarchy (5.8) with β = β = 0. This leads to thefollowing isochronous hierarchy v s + i (cid:18) m + 3 (cid:19) ωv + i (cid:18) m + 3 (cid:19) ωrv r = R m [ v ] ◦ (cid:18) av rrr v − av r v rr v + 21 av r v + β v r v (cid:19) , m = 0 , , , . . . , (6.13)where R [ v ] = 1 v D r − v r v D r − v rr v + 22 v r v + 4 β a v − a (cid:18) av rrr v − av r v rr v + 21 av r v + β v r v (cid:19) D − r ◦ v + v r D − r ◦ (cid:18) − v rr v + 6 v r v + 2 β a v (cid:19) . The first member of the hierarchy (6.13) for m = 0 is the equation (6.5). Case 1.2:
We consider the hierarchy (5.8) with β = β = 0. This leads to thefollowing isochronous hierarchy v s + i (cid:18) m + 3 (cid:19) ωv + i (cid:18) m + 3 (cid:19) ωrv r = R m [ v ] ◦ (cid:18) av rrr v − av r v rr v + 21 av r v + β v r v (cid:19) , m = 0 , , , . . . , (6.14)where R [ v ] = 1 v D r − v r v D r − v rr v + 22 v r v + 2 β a v − a (cid:18) av rrr v − av r v rr v + 21 av r v + β v r v (cid:19) D − r ◦ v + v r D − r ◦ (cid:18) − v rr v + 6 v r v + β a v (cid:19) . m = 0 is the equation (6.6). Case 1.3:
We consider the hierarchy (5.8) with β = β = 0. This leads to thefollowing isochronous hierarchy v s + iλωv + i (cid:18) λ + 12 m + 3 (cid:19) ωrv r = R m [ v ] ◦ (cid:18) av rrr v − av r v rr v + 21 av r v + β v r (cid:19) , m = 1 , , . . . (6.15)where λ is arbitrary and R [ v ] = 1 v D r − v r v D r − v rr v + 22 v r v − a (cid:18) av rrr v − av r v rr v + 21 av r v + β v r (cid:19) D − r ◦ v + v r D − r ◦ (cid:18) − v rr v + 6 v r v + 2 β a v r (cid:19) . Note that equation (6.7) does not correspond to m = 0 in (6.15). The reason israther obvious: since R m [ v ] ◦ ( β v r ) = 0for all m = 1 , , . . . , the β term disappears in (6.15) and there remains only oneconstraint on λ and µ to assure that the hierarchy does not depend explicitly on s ,namely µ − λ − m + 3 = 0 . For the equation (5.9) we have the hierarchy (5.11), namely u t = R m [ u ] ◦ (cid:18) au xxx u − au x u xx u + 6 au x u + β u x u + β uu x + β u x (cid:19) ,m = 0 , , , . . . , where R [ u ] is given by (5.10). Corresponding to the above Case 2.1, Case 2.2 andCase 2.3, the isochronous hierarchies are the following: Case 2.1:
We consider the hierarchy (5.11) with β = β = 0. This leads to thefollowing isochronous hierarchy v s + i (cid:18) m + 3 (cid:19) ωv = R m [ v ] ◦ (cid:18) av rrr v − av r v rr v + 6 av r v + β v r v (cid:19) (6.16) m = 0 , , , . . . , R [ v ] = 1 v D r − v r v D r − v rr v + 6 v r v + β a v − a (cid:18) av rrr v − av r v rr v + 6 av r v + β v r v (cid:19) D − r ◦ . Case 2.2:
We consider the hierarchy (5.11) with β = β = 0. This leads to thefollowing isochronous hierarchy v s − i (cid:18) m + 3 (cid:19) ωv − i (cid:18) m + 3 (cid:19) ωrv r == R m [ v ] ◦ (cid:18) av rrr v − av r v rr v + 6 av r v + β vv r (cid:19) (6.17) m = 0 , , , . . . , where R [ v ] = 1 v D r − v r v D r − v rr v + 6 v r v + β a v − a (cid:18) av rrr v − av r v rr v + 6 av r v + β vv r (cid:19) D − r ◦ β a v r D − r ◦ v. (6.18) Case 2.3:
We consider the hierarchy (5.11) with β = β = 0. This leads to thefollowing isochronous hierarchy v s − iλωv − i (cid:18) λ + 12 m + 3 (cid:19) ωrv r = R m [ v ] ◦ (cid:18) av rrr v − av r v rr v + 6 av r v + β v r (cid:19) (6.19) m = 1 , , , . . . , where R [ v ] = 1 v D r − v r v D r − v rr v + 6 v r v − a (cid:18) av rrr v − av r v rr v + 6 av r v + β v r (cid:19) D − r ◦ β a v r D − r ◦ . Note that equation (6.12) does not correspond to m = 0 in (6.19) for the samereason as in Case 1.3. That is, since R m [ v ] ◦ ( β v r ) = 021or all m = 1 , , . . . , the β term disappears in (6.19) and there remains only oneconstraint on λ and µ to assure that the hierarchy does not depend explicitly on s ,namely µ − λ − m + 3 = 0 . Acknowledgements:
E.G.R. has been partially supported by the FONDECYToperating grant
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