Contact Lax pairs and associated (3+1)-dimensional integrable dispersionless systems

TL;DR

This paper reviews the construction of (3+1)-dimensional integrable dispersionless systems using contact Lax pairs and R-matrix theory, highlighting various Lax representations and reductions to lower dimensions.

Contribution

It introduces a comprehensive approach to building higher-dimensional integrable systems via contact Lax pairs and explores their diverse representations and reductions.

Findings

Develops contact Lax pair framework for (3+1)-D systems

Classifies linear and nonlinear contact Lax pairs

Provides numerous examples and reductions to lower dimensions

Abstract

We review the recent approach to the construction of (3+1)-dimensional integrable dispersionless partial differential systems based on their contact Lax pairs and the related $R$-matrix theory for the Lie algebra of functions with respect to the contact bracket. We discuss various kinds of Lax representations for such systems, in particular, linear nonisospectral contact Lax pairs and nonlinear contact Lax pairs as well as the relations among the two. Finally, we present a large number of examples with finite and infinite number of dependent variables, as well as the reductions of these examples to lower-dimensional integrable dispersionless systems.