Accelerating solutions of the Korteweg-de Vries equation

TL;DR

This paper demonstrates that the Korteweg-de Vries equation admits accelerated wavepacket solutions by transforming it into the Painlevé I equation, with numerical analysis confirming their accelerated behavior.

Contribution

It introduces a novel approach by linking the Korteweg-de Vries equation to the Painlevé I equation to find accelerated soliton solutions.

Findings

Numerical solutions show explicit acceleration of wavepackets.

The transformation to Painlevé I enables new solution types.

Accelerated solutions differ from classical solitons.

Abstract

The Korteweg-de Vries equation is a fundamental nonlinear equation that describes solitons with constant velocity. On the contrary, here we show that this equation also presents accelerated wavepacket solutions. This behavior is achieved by putting the Korteweg-de Vries equation in terms of the Painlev\'e I equation. The accelerated waveform solutions are explored numerically showing their accelerated behavior explicitly.