Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation

TL;DR

This paper derives a general solution for a specific first order ODE related to the asymptotic behavior of solutions to the Korteweg-de Vries equation, linking integrability and asymptotics in nonlinear wave equations.

Contribution

It introduces a new general solution for an Abel-type ODE arising in KdV asymptotics, involving hypergeometric functions, and connects it to self-similar solutions of Whitham equations.

Findings

Solution depends on an arbitrary parameter and is expressed via hypergeometric functions.

Special case yields self-similar solutions of Whitham equations from 1988.

Supports the idea that integrable equations tend to produce other integrable equations in limits.

Abstract

We provide a general solution for a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary part of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this article supports once again an empirical rule saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the method of inverse scattering problem.