Solution Structures of Burgers and Burgers–Fisher Equations via Sine-Gordon expansion method

Abstract

Nonlinear partial differential equations such as the Burgers and Burgers–Fisher equations are fundamental models in fluid dynamics, traffic flow, and reaction–diffusion processes. In this work, the Sine–Gordon expansion method (SGEM) is employed to derive new exact traveling wave solutions of these equations. By applying a traveling wave transformation, the governing equations are reduced to ordinary differential equations, allowing systematic construction of closed-form solutions. The obtained solutions include kink type solitons, shock-like structures, bell-shaped solitary waves, and periodic wave solutions expressed in hyperbolic and trigonometric forms. The results demonstrate the effectiveness of SGEM in capturing the combined effects of convection, diffusion, and reaction, and provide deeper insight into the nonlinear wave dynamics of Burgers-type equations.