A geometric formulation of Lax integrability for nonlinear equationsin two independent variables

Paul Bracken

doi:10.3934/math.2017.4.610

Abstract

A geometric formulation of Lax integrability is introduced which makes use of a Pfaffianformulation of Lax integrability. The Frobenius theorem gives a necessary and sufficient conditionfor the complete integrability of a distribution, andprovides a powerful way to study nonlinear evolution equations. This permits an examination of the relationbetween complete integrability and Lax integrability. The prolongation method is formulated in this contextand gauge transformations can be examined in terms ofdifferential forms as well as the Frobenius theorem.

Highlights

The Frobenius theorem gives a necessary and sufficient condition for the complete integrability of a distribution, and provides a powerful way to study nonlinear evolution equations. This permits an examination of the relation between complete integrability and Lax integrability
The prolongation method is formulated in this context and gauge transformations can be examined in terms of differential forms as well as the Frobenius theorem
When a nonlinear evolution equation can be generated from a pair of linear partial differential equations of first order by means of a compatibility condition, the nonlinear evolution equation is said to be Lax integrable and the nonlinear system is called a Lax pair [7, 8]

Summary

Introduction

When a nonlinear evolution equation can be generated from a pair of linear partial differential equations of first order by means of a compatibility condition, the nonlinear evolution equation is said to be Lax integrable and the nonlinear system is called a Lax pair [7, 8] If such a pair of equations can be determined, such objects such as gauge and Darboux transformations as well as an infinite number of conservation laws can be constructed for the equation. It will be seen that the relation between complete and Lax integrability can be formulated in this way when the Lax integrability of the nonlinear evolution equation is given in terms of exterior differential forms [11]. The terminology nonlinear evolution equation will be understood to signify an equation in one space and one time variable everywhere, that is, in one+one dimensions [2,3,4]

Lax Representations in Two Independent Variables

Prolongation and Differential Systems

Lax Systems and Gauge Transformations in Terms of Differential Systems