The Laplace-Adomian Decomposition Method Applied to the Kundu-Eckhaus Equation

TL;DR

This paper introduces a novel combined Laplace-Adomian Decomposition Method to solve the Kundu-Eckhaus equation, a nonlinear PDE relevant in physics, providing a new approach that is effective and applicable to other nonlinear equations.

Contribution

The paper presents a new combined method using Laplace Transform and Adomian Decomposition to solve the Kundu-Eckhaus equation, demonstrating its effectiveness and potential for broader applications.

Findings

Successfully solves the Kundu-Eckhaus equation

Shows the method's solutions align with known exact solutions

Proposes the method as a versatile tool for nonlinear PDEs

Abstract

The Kundu-Eckhaus equation is a nonlinear partial differential equation which seems in the quantum field theory, weakly nonlinear dispersive water waves and nonlinear optics. In spite of its importance, exact solution to this nonlinear equation are rarely found in literature. In this work, we solve this equation and present a new approach to obtain the solution by means of the combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). Besides, we compare the behaviour of the solutions obtained with the only exact solutions given in the literature through fractional calculus. Moreover, it is shown that the proposed method is direct, effective and can be used for many other nonlinear evolution equations in mathematical physics.