A new integrable generalization of the Korteweg - de Vries equation
📝 Abstract
A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found of the new equation, and its travelling wave solutions and generalized symmetries are studied.
💡 Analysis
A new integrable sixth-order nonlinear wave equation is discovered by means of the Painleve analysis, which is equivalent to the Korteweg - de Vries equation with a source. A Lax representation and a Backlund self-transformation are found of the new equation, and its travelling wave solutions and generalized symmetries are studied.
📄 Content
In this paper we apply the Painlevé test for integrability of partial differential equations [1,2] to the class of sixth-order nonlinear wave equations
x u xx + du tt + eu xxxt + f u x u xt + gu t u xx = 0, (1) where a, b, c, d, e, f and g are arbitrary parameters. We show that there are four distinct cases of relations between the parameters when equation (1) passes the Painlevé test well. Three of those cases correspond to known integrable equations, whereas the fourth one turns out to be new. This new integrable case of equation ( 1) is equivalent to the Korteweg-de Vries equation with a source of a new type, and we find its Lax pair, Bäcklund self-transformation, travelling wave solutions and third-order generalized symmetries.
There are the following reasons to explore the class of equations (1) for integrability. Recently Dye and Parker [3] constructed and studied two integrable nonlinear integro-differential equations,
which describe the propagation of waves in two opposite directions and represent bidirectional versions of the Sawada-Kotera-Caudrey-Dodd-Gibbon equation [4,5] and Kaup-Kupershmidt equation [6,7], respectively. Equations ( 2) and
(3) possess Lax pairs due to their construction [3] and fall into class (1) after the potential transformation v = u x . There is one more well-known integrable equation in class (1), namely
This equation is equivalent to the Drinfel’d-Sokolov-Satsuma-Hirota system of coupled Korteweg-de Vries equations [8,9], of which a fourth-order recursion operator was found in [10]. A Bäcklund self-transformation of equation ( 4) was derived in [11] by the method of truncated singular expansion [1,12]. Multisoliton solutions of equations ( 3) and (4) were studied in [13,14,15]. Thus we have already known three interesting integrable equations of class (1), and it is natural to ask what are other integrable equations in this class and, if there are any, what are their properties. Solving problems of this kind is important for testing the reliability of integrability criteria and for discovering new interesting objects of soliton theory. The paper is organized as follows. In section 2 we perform the singularity analysis of equation ( 1) and find four distinct cases which possess the Painlevé property and correspond, up to scale transformations of variables, to equations (2)-( 4) and to the new integrable equation
In this part the present study is similar to the recent Painlevé classifications done in [16,17,18,19], where new integrable nonlinear wave equations were discovered as well. The method of truncated singular expansion is successfully used in section 3, where we derive a Lax pair for the new equation ( 5) and also obtain and study its Bäcklund self-transformation. The contents of section 4 is not related to the Painlevé property: there we find and discuss travelling wave solutions and third-order generalized symmetries of equation (5). Section 5 contains concluding remarks.
In order to select integrable cases of equation ( 1) we use the so-called Weiss-Kruskal algorithm for singularity analysis of partial differential equations [20], which is based on the Weiss-Tabor-Carnevale expansions of solutions near movable singularity manifolds [1], Ward’s requirement not to examine singularities of solutions at characteristics of equations [21,22] and Kruskal’s simplifying representation for singularity manifolds [23], and which follows step by step the Ablowitz-Ramani-Segur algorithm for ordinary differential equations [24]. Computations are made using the Mathematica computer algebra system [25], and we omit inessential details. Equation ( 1) is a sixth-order normal system, and its general solution must contain six arbitrary functions of one variable [26]. A hypersurface φ(x, t) = 0 is noncharacteristic for equation (1) if φ x = 0, and we set φ x = 1 without loss of generality. Substitution of the expansion u = u 0 (t) 1) determines branches of the dominant behavior of solutions near φ = 0, i.e. admissible choices of δ and u 0 , and corresponding positions r of the resonances, where arbitrary functions of t can enter the expansion. There are two singular branches, both with δ = -1, values of u 0 being the roots of a quadratic equation with constant coefficients. Without loss of generality we make u 0 = 1 for one of the two branches by a scale transformation of u, thus fixing the coefficient c of equation ( 1) as
and then we get u 0 = 60/(2a + b -60) for the other branch. We require that at least one of singular branches is a generic one, representing the general solution of equation (1), and without loss of generality we assume that we have set u 0 = 1 for the generic branch, whereas the branch with u 0 = 60/(2a + b -60) may be nongeneric. Positions r of the resonances are determined by the equation (r + 1)(r -1)(r -6) r 3 -15r 2 + (86 -a)r + (4a + 2b -240) = 0 (7) for the branch with u 0 = 1, and by the equation (r + 1)(r -1)(r -6) r 3 -15r 2 + 86 -60a/(2a + b -60
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