Constraints for Evolution Equations with Some Special Form of Lax Pairs and Distinguishing Lax Pairs by Available Constraints

TL;DR

This paper investigates constraints for evolution equations with special Lax pair forms, illustrating how to identify genuine Lax pairs and revealing insights into the linearity of integrable PDEs.

Contribution

It introduces a method to impose and distinguish constraints for general Lax pairs, including nonlinear ones, enhancing understanding of integrability.

Findings

Constraints can be used to find nontrivial solutions

Special Lax pair forms can be identified via the EW method

Distinguishing true and fake Lax pairs is feasible

Abstract

The constraints for evolution equations with some special form of Lax pair are first investigated. We show by examples how the method is rooted in the classical literatures and how the ignored constraints provide nontrivial solutions. Then we showed, by the example of the KdV, how this special form of Lax pair may be found by the method of EW. At last we propose how constraints should be imposed for general Lax pairs including nonlinear ones, with which the true Lax pairs and the fake ones can be distinguished easily. The linearity nature in integrable partial differential equations seems to have been revealed.