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

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.