Linearisable Abel equations and the Gurevich--Pitaevskii problem

TL;DR

This paper uses symmetry reduction to solve a class of Abel equations related to the Gurevich--Pitaevskii problem, providing explicit solutions and asymptotic analysis in the oscillatory zone.

Contribution

It introduces a method to parametrize solutions of certain Abel equations via hypergeometric functions, solving a key equation in the Gurevich--Pitaevskii problem for the first time.

Findings

Explicit parametric solutions to the Kudashev equation.

First term of large-time asymptotic expansion in the oscillatory zone.

Connection between symmetry reduction and integrable Abel equations.

Abstract

Applying symmetry reduction to a class of $\mathrm{SL}(2,\mathbb R)$-invariant third-order ODEs, we obtain Abel equations whose general solution can be parametrised by hypergeometric functions. Particular case of this construction provides a general parametric solution to the Kudashev equation, an ODE arising in the Gurevich--Pitaevskii problem, thus giving the first term of a large-time asymptotic expansion of its solution in the oscillatory (Whitham) zone.