Collisionless shock region of the KdV equation and an entry in Gradshteyn and Ryzhik

TL;DR

This paper analyzes the long-time asymptotic behavior of solutions to the KdV equation, identifying five regions with distinct behaviors, and confirms a classical elliptic integral entry in Gradshteyn and Ryzhik related to the phase representation.

Contribution

It introduces a new asymptotic region between the self-similar and similarity regions described by elliptic functions and confirms a classical integral formula in Gradshteyn and Ryzhik.

Findings

Five asymptotic regions for KdV solutions are described.

A new elliptic function region is identified.

A classical elliptic integral entry is validated.

Abstract

The long-time behavior of solutions to the initial value problem for the Korteweg-de Vries equation on the whole line, with general initial conditions has been described uniformly using five different asymptotic forms. Four of these asymptotic forms were expected: the quiescent behavior (for |x| very large), a soliton region (in which the solution behaves as a collection of isolated solitary waves), a self-similar region (in which the solution is described via a Painlev\'e transcendent), and a similarity region (where the solution behaves as a simple trigonometric function of the quantities t and x/t). A fifth asymptotic form, lying between the self-similar (Painlev\'e) and the similarity one, has been described in terms of classical elliptic functions. An integral of elliptic type, giving an explicit representation of the phase, has appeared in this context. The same integral has appeared in the table of integrals by Gradshteyn and Ryzhik. Our goal here is to confirm the validity of this entry.