Scaling Invariant Lax Pairs of Nonlinear Evolution Equations

TL;DR

This paper introduces a scalable method to compute Lax pairs for nonlinear PDEs, utilizing scaling symmetry to simplify the process, and demonstrates its effectiveness on well-known soliton equations.

Contribution

It presents a new, easily implementable approach to find Lax pairs for PDEs using scaling invariance, splitting the problem into kinematic and dynamical parts.

Findings

Successfully applied to fifth-order KdV-like equations.

Discovered a second Lax pair for the Sawada--Kotera equation.

Simplified computation of Lax pairs using algebraic equations.

Abstract

A completely integrable nonlinear partial differential equation (PDE) can be associated with a system of linear PDEs in an auxiliary function whose compatibility requires that the original PDE is satisfied. This associated system is called a Lax pair. Two equivalent representations are presented. The first uses a pair of differential operators which leads to a higher order linear system for the auxiliary function. The second uses a pair of matrices which leads to a first-order linear system. In this paper we present a method, which is easily implemented in Maple or Mathematica, to compute an operator Lax pair for a set of PDEs. In the operator representation, the determining equations for the Lax pair split into a set of kinematic constraints which are independent of the original equation and a set of dynamical equations which do depend on it. The kinematic constraints can be solved generically. We assume that the operators have a scaling symmetry. The dynamical equations are then reduced to a set of nonlinear algebraic equations. This approach is illustrated with well-known examples from soliton theory. In particular, it is applied to a three parameter class of fifth-order KdV-like evolution equations which includes the Lax fifth-order KdV, Sawada-Kotera and Kaup-Kuperschmidt equations. A second Lax pair was found for the Sawada--Kotera equation.