On a method for constructing the Lax pairs for integrable models via a quadratic ansatz

Abstract

A method for constructing the Lax pairs for nonlinear integrable models is suggested. First we look for a nonlinear invariant manifold of the linearization of the given equation. Examples show that such an invariant manifold does exist and can effectively be found. Actually, it is defined by a quadratic form. As a result we get a nonlinear Lax pair consisting of the linearized equation and the invariant manifold. Our second step consists of finding an appropriate change of the variables to linearize the found nonlinear Lax pair. The desired change of the variables is again defined by a quadratic form. The method is illustrated by the well-known KdV equation, the modified Volterra chain and a less studied coupled lattice connected to the affine Lie algebra . New Lax pairs are found. The formal asymptotic expansions for their eigenfunctions are constructed around the singular values of the spectral parameter. By applying the method of the formal diagonalization to these Lax pairs, the infinite series of the local conservation laws are obtained for the corresponding nonlinear models.

On description of generalized invariant manifolds for nonlinear equations

“…This follows the observation that the general solution to equation (6) can be expressed via the classical symmetry = of the KdV equation as = .…”

“…The experience suggests that such obtained Lax pair can be linearized by an appropriate change of variables, cf. [6]. To reduce this pair to a linear form, we express the variables and as some quadratic forms of new variables and so that the radicand in equation (36) becomes the perfect square.…”

An upper order bound of the invariant manifold in Lax pairs of a nonlinear evolution partial differential equation

“…We formulate the method in [4,5] in the context of generalized conditional symmetry and give an upper order bound of the derivatives appearing in the invariant manifold, which provides a theoretical basis for the complete classification of the given form invariant manifold and then for the Lax pair. We illustrate the results by three examples.…”

A Direct Algorithm for Constructing Recursion Operators and Lax Pairs for Integrable Models

“…In a series of our works we have suggested a method for construction the recursion operators and the Lax pairs for the nonlinear integrable equations (see [1]- [4]). Let us give a brief explanation of core of the method.…”