Exact and soliton solutions of nonlinear evolution equations in mathematical physics using the generalized (G′/ G)-expansion approach

Abstract The nonlinear evolution equations play a significant role in applied mathematics, including ordinary and partial differential equations, which are frequently used in many disciplines of applied sciences. The Klein–Gordon equation, a second-order equation in both space and time, is closely related to the Schrödinger equation and describes the behavior of spinless particles in theoretical physics. The equation is utilized extensively in many areas related to modern physics, including astrophysics, cosmology, and the theory of classic mechanics. The uKdV equation is suitable for use in various fields, including fluid dynamics, oceanography, and optical solitons research. It is useful for simulating events with wave-like characteristics and soliton dynamics, which contributes to the theoretical development of mathematical physics. This article addresses the exact and soliton solutions of nonlinear evolution equations in mathematical physics and engineering using the new generalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">G</mml:mi> <mml:mo accent="false">′</mml:mo> <mml:mo>/</mml:mo> <mml:mi mathvariant="normal">G</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> -expansion approach. We examine a large number of soliton solutions, including kink-shaped, singular-kink, single soliton, singular-soliton, periodic, and others. To better understand the characteristics of the solutions, some three-dimensional plots are set up for visualization. It has been demonstrated that the suggested approaches are more attractive as well as effective for getting single solutions to nonlinear evolution equations.