A Novel Approach to find Exact Solutions of Nonlinear ODE Systems: Applications to Coupled Non-Integrable and Integrable mKdV Equations

TL;DR

This paper introduces a comprehensive method combining series transformations, modified variation of parameters, and inverse power series techniques to derive exact solutions for nonlinear ODE systems, demonstrated on coupled mKdV equations.

Contribution

It presents a new multi-step approach that unifies series solutions, linearization, and inverse series methods to find exact solutions of nonlinear ODEs, applicable to both integrable and non-integrable systems.

Findings

Successfully derived exact solutions for coupled mKdV equations.

Developed a recursive series solution method with closed-form solutions.

Established a rational generating function for the series solutions.

Abstract

A novel approach is introduced for deriving exact solutions to nonlinear systems of ordinary differential equations. This method consists of four parts. In the initial part, the examined nonlinear differential equation system is transformed into a linear differential equation system using an infinite series sum transformation and Adomian polynomials. In the second part, we presented a modified method of variation of parameters utilizing a revised Cramer's rule for addressing a linear system of block matrices. In the following part, we merged the outcomes of the prior two parts and introduced a new recursive method for acquiring a series solution of the system established in the first section. In the final part, we introduce the fundamental ideas of the multiplicative inverse of power series and methods for summing infinite series. The method utilizes the property that the multiplicative inverse of the power series either terminates after a finite number of terms or simplifies to a geometric series, which can be easily summed to yield closed-form solutions for the series in question. The effectiveness of this method is shown by successfully deriving exact localized solutions for both integrable and non-integrable coupled modified Korteweg-de Vries equations. Using the multiplicative inverse technique, it is shown that a rational generating function can be created for the series solution derived from the proposed method for the specified system of nonlinear differential equations, regardless of whether it is integrable or non-integrable.