On a method for constructing the Lax pairs for nonlinear integrable equations

Abstract

Abstract.We suggest a direct algorithm for searching the Lax pairs for nonlinear integrable equations. It is effective for both continuous and discrete models. The first operator of the Lax pair corresponding to a given nonlinear equation is found immediately, coinciding with the linearization of the considered nonlinear equation. The second one is obtained as an invariant manifold to the linearized equation. A surprisingly simple relation between the second operator of the Lax pair and the recursion operator is discussed: the recursion operator can immediately be found from the Lax pair. Examples considered in the article are convincing evidence that the found Lax pairs differ from the classical ones. The examples also show that the suggested objects are true Lax pairs which allow the construction of infinite series of conservation laws and hierarchies of higher symmetries. In the case of the hyperbolic type partial differential equation our algorithm is slightly modified; in order to construct the Lax pairs from the invariant manifolds we use the cutting off conditions for the corresponding infinite Laplace sequence. The efficiency of the method is illustrated by application to some equations given in the Svinolupov-Sokolov classification list for which the Lax pairs and the recursion operators have not been found earlier.

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.…”

“…by the method in [4], it means Proof: Since X = H∂ v with H given in (11) is a generalized conditional symmetry of Eqs. (1) and (3), then…”

“…However, the invariant manifold (4) in the Lax pair of Eq. (1) is linear in v i and with the assumption that the order in α i is bounded by s ≤ n, then the order bound in Theorem 2.7 is suitable for the invariant manifold with the form (4).…”

Invariant manifolds and Lax pairs for integrable nonlinear chains

“…We notice that in fact the system defines a nonlinear Lax pair with two arbitrary constant parameters λ and c for the Volterra chain. By applying the operator D n − 1 to the equation (38) we immediately obtain a linear equation (see [1]) u n+1 (P n+2 + P n+1 ) − u n (P n + P n−1 ) = λ(P n+1 − P n ) which also defines a generalized invariant manifold since it is compatible, as it is easily checked, with the equations (1), (7).…”

“…According to the scheme proposed in [1], we add to the pair of equations (2), (3) an ordinary difference equation of the form U n+k = G(U n+k−1 , U n+k−2 , ...U n−k 1 , [u n ]), k ≥ 1, k 1 ≥ 0 (4) compatible with the linearized equation (3) for each solution of the equation (2). The notation [u n ] indicates that function G might depend on the variable u n and its several shifts u n±1 , u n±2 , .…”