Soliton solutions of non-linear Schrodinger (NLS) and Korteweg de Vries (KdV) equations related to zero curvature in the x,t plane

TL;DR

This paper explores soliton solutions of the NLS and KdV equations through the zero curvature condition, deriving explicit solutions via auxiliary matrix compatibility in the x,t plane.

Contribution

It introduces a method linking soliton solutions to the zero curvature condition using matrix compatibility, providing explicit solutions for NLS and KdV equations.

Findings

Explicit soliton solutions for NLS and KdV equations derived.

Connection established between zero curvature condition and soliton solutions.

Method demonstrated with two-dimensional and one-dimensional wave functions.

Abstract

Soliton solutions of non-linear NLS and KdV equations are related to compatibility condition between matrices M and H describing the movement of an auxilary function Psi in the x,t plane with a zero curvature condition. Non-linear equation for a function u is obtained by the compatibility equation where the matrix elements of M and H include only functions of u and its derivatives. By solving the equations of motion for Psi a soliton solution for u is obtained. Explicit calculations are made with two-dimensional and one-dimensional wave functions Psi for the NLS and KdV solitons, correspondingly.