Negaton and Positon Solutions of the KDV Equation

TL;DR

This paper classifies and analyzes negaton and positon solutions of the KdV equation, revealing their structures, motions, scattering behaviors, and connections to the modified KdV equation.

Contribution

It provides a systematic classification of negaton and positon solutions, detailing their structures, dynamics, and phase shifts, and extends these solutions to the modified KdV equation.

Findings

Negaton solutions have finite singularities and move in the positive x direction.

Positon solutions exhibit periodic oscillations and move in the negative x direction.

Negatons and positons retain their identities during scattering, with phase shifts explained via generalized mass concepts.

Abstract

We give a systematic classification and a detailed discussion of the structure, motion and scattering of the recently discovered negaton and positon solutions of the Korteweg-de Vries equation. There are two distinct types of negaton solutions which we label $[S^{n}]$ and $[C^{n}]$, where $(n+1)$ is the order of the Wronskian used in the derivation. For negatons, the number of singularities and zeros is finite and they show very interesting time dependence. The general motion is in the positive $x$ direction, except for certain negatons which exhibit one oscillation around the origin. In contrast, there is just one type of positon solution, which we label $[\tilde C^n]$. For positons, one gets a finite number of singularities for $n$ odd, but an infinite number for even values of $n$. The general motion of positons is in the negative $x$ direction with periodic oscillations. Negatons and positons retain their identities in a scattering process and their phase shifts are discussed. We obtain a simple explanation of all phase shifts by generalizing the notions of ``mass" and ``center of mass" to singular solutions. Finally, it is shown that negaton and positon solutions of the KdV equation can be used to obtain corresponding new solutions of the modified KdV equation.