Factorization method for nonlinear evolution equations Factorization method for nonlinear evolution equations

TL;DR

This paper introduces a modified factorization method that constructs general solutions for nonlinear evolution equations, demonstrated on various equations including KdV, mKdV, RH, and NLS, revealing new solutions and structural differences.

Contribution

A novel modification of the factorization approach enabling the derivation of general solutions for nonlinear evolution equations.

Findings

Derived general solutions for KdV, mKdV, RH, and NLS equations.

Obtained a singular solution of the KdV equation via the Muira transform.

Found a soliton solution for the NLS equation in optical media.

Abstract

The traditional method of factorization can be used to obtain only the particular solutions of the Li\'enard type ordinary differential equations. We suggest a modification of the approach that can be used to construct general solutions . We first demonstrate the effectiveness of our method by dealing with a solvable form of the modified Emden-type equation and subsequently employ it to obtain the solitary wave solutions of the KdV, mKdV, Rosenau-Hyman (RH) and NLS equations. The solution of the mKdV equation, via the so-called Muira transform, leads to a singular solution of the KdV equation in addition to the well-known soliton solution supported by it. We obtain the solution of the non-integrable RH equation in terms of the Jacobi function and show that, although robust, it is structurally different from the KdV soliton. We also find the soliton solution of the NLS equation that accounts for the evolution of complex envelopes in an optical medium.