The Gibbons--Tsarev equation: symmetries, invariant solutions, and applications

TL;DR

This paper classifies symmetry-invariant solutions of the Gibbons--Tsarev equation and applies them to derive explicit solutions for related integrable systems, enhancing understanding of their structure and solutions.

Contribution

It provides a complete classification of symmetry-invariant solutions for the Gibbons--Tsarev equation and demonstrates their applications to several important integrable systems.

Findings

Explicit solutions for reductions of Benney's moments equations

Solutions of Pavlov's equation derived from symmetry-invariant solutions

Identification of integrable reductions of the Ferapontov--Huard--Zhang system

Abstract

In this paper we present the full classification of the symmetry-invariant solutions for the Gibbons--Tsarev equation. Then we use these solutions to construct explicit expressions for reductions of Benney's moments equations, to get solutions of Pavlov's equation, and to find integrable reductions of the Ferapontov--Huard--Zhang system, which describes implicit two-phase solutions of the dKP equation.