Scaling invariant Lax pairs of nonlinear evolution 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 article, 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 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 Korteweg–de Vries (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.

Keywords:

• Lax pair
• Lax operator
• scaling symmetry
• complete integrability
• fifth-order KdV-type equations

AMS Subject Classifications::

• Primary: 37J35
• 37K40
• 35Q51
• Secondary: 68W30
• 47J35
• 70H06

Acknowledgements

This material is based in part upon research supported by the National Science Foundation (NSF) under Grant No. CCF-0830783. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF. WH is grateful for the hospitality and support of the Department of Computer Engineering at Turgut Özal University (Keçiören, Ankara, Turkey) where code for Lax pair computations was further developed. MH thanks the Department of Applied Mathematics and Statistics, Colorado School of Mines for their hospitality while this work was completed. Undergraduate students Oscar Aguilar, Sara Clifton, William ‘Tony’ McCollom, and graduate student Jacob Rezac are thanked for their help with this project.