Simple identification of fake Lax pairs

TL;DR

This paper introduces two straightforward methods to identify fake Lax pairs, which are useful in analyzing integrable systems, by examining gauge transformations and degrees of freedom, applicable to various types of Lax pairs.

Contribution

The paper presents two novel, simple, and complementary techniques for detecting fake Lax pairs, enhancing the analysis of integrability in differential and difference systems.

Findings

The gauge transformation method can remove dependent variables from fake Lax pairs.

Fake Lax pairs exhibit excess degrees of freedom.

The methods are applicable to scalar and matrix Lax pairs in differential and difference systems.

Abstract

Two simple ways to identify and explain fake Lax pairs are provided. The two methods are complementary, one involves finding a gauge transformation which can be used to remove the associated nonlinear system's dependent variable(s) from a fake Lax pair. The second method shows that excess degrees of freedom exist in fake Lax pairs. We provide several examples to illustrate both tests. The tests proposed here can be applied to all types of Lax pairs, including scalar or matrix linear problems associated with ordinary or partial difference and/or differential systems.