Application of the Sine-Gordon Expansion Method to Obtain Soliton Solutions of the KdV and mKdV Equations

Abstract

In this work, we apply the Sine-Gordon Expansion Method (SGEM) to construct explicit soliton solutions of the classical Korteweg–de Vries (KdV) and modified Korteweg–de Vries (mKdV) equations, which play fundamental roles in nonlinear science and mathematical physics. The SGEM, based on the expansion of solutions in terms of the sine–Gordon equation, provides an efficient and systematic framework to derive exact traveling wave solutions of nonlinear evolution equations. By employing suitable transformations, we reduce the governing partial differential equations into ordinary differential equations and then obtain analytical expressions for various types of solutions, including solitary wave and periodic solutions. The derived results demonstrate the capability of the SGEM in handling nonlinear dispersive equations and further highlight its effectiveness compared to other existing analytical methods. These exact solutions not only contribute to the theoretical study of nonlinear wave propagation but also have potential applications in fluid dynamics, plasma physics, and optical communications.