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

Abstract

In this study, we apply the Sine-Gordon Expansion Method (SGEM) to obtain exact travelling wave solutions of KdV and mKdV equations, which model dispersive nonlinear wave phenomena in fluids, plasma, and optical systems. By introducing an auxiliary Sin-Gordon type equation and assuming a travelling wave transform, the nonlinear partial differential equations are reduced to algebraic systems. Consequently, hyperbolic, trigonometric, and exponential explicit solutions are obtained. The mKdV equation produces bright, dark, and periodic soliton structures, whereas the KdV equation admits localized solitary wave solutions. Graphical analyses in two and three dimensions show how model parameters affect the amplitude, width, and velocity of solitons. The findings show that SGEM may be successfully extended to other nonlinear dispersive systems and provides an effective and unified analytical framework for building accurate soliton solutions.