Travelling-wave solutions and solitons of KdV, mKdV and NLS equations

TL;DR

This paper explores soliton solutions of integrable nonlinear PDEs like KdV, mKdV, and NLS, using inverse spectral methods and Jacobi elliptic functions to derive various types of solitons.

Contribution

It provides a pedagogic method to find traveling wave soliton solutions for KdV, mKdV, and NLS equations, including elliptic function representations and kink solutions.

Findings

Derivation of bell-type KdV solitons

Construction of kink and anti-kink solutions for mKdV and NLS

Representation of solutions using Jacobi elliptic functions

Abstract

We introduce the concept of soliton solutions of integrable nonlinear partial differential equations and point out that the inverse spectral method represents the rigorous mathematical formalism to construct such solutions. We work with the travelling waves of the KdV, mKdV and nonlinear Schr\"{o}dinger (NLS) equations and derive a pedagogic method to find their soliton solutions. The travelling wave of the KdV equation leads directly to the well known bell type KdV soliton while the mKdV equation needs some additional consideration in respect of this. The travelling waves of a generalized mKdV and NLS equations are obtained in terms of $sn(u,m)$, the so called Jacobi elliptic sine function. The choice $m=1$ provides a constraint on the parameters of the equations and gives their kink and anti-kink soliton solutions. We further show that by expressing the travelling waves of these equations in terms of Jacobi elliptic cosine functions, $cn(u,m)$, it is possible to construct bell-type soliton solutions.