A Study of Nonlinear Riccati Equation and Its Applications to Multi-dimensional Nonlinear Evolution Equations

In this paper, an improved nonlinear Riccati equation method is established and then used to study nonlinear evolution equations with variable coefficients. The idea behind this current study is to use the nonlinear Riccati equation with constant coefficients and convert it to second-order linear ordinary differential equations using appropriate transformations. The solutions of these second-order linear ordinary differential equations are then used to propose the solutions to the nonlinear Riccati equation. This improved method consists of the well-known \((G^{\prime }/G)\) $(G^{\prime }/G)$ -expansion and \((1/G^{\prime })\) $(1/G^{\prime })$ -expansion methods as a special case. It also consists of a newly proposed (in this current study) \((G/G^{\prime })\) $(G/G^{\prime })$ -expansion method. As a result, this improved method is employed to investigate the traveling wave solutions of \((2+1)\) $(2+1)$ -dimensional models with variable coefficients, such as the extended Kadomtsev–Petviashvili equation, nonlinear Schrödinger equation, and Davey–Stewartson equations. Solitary waves are obtained by taking the parameters involving these derived traveling wave solutions as special values. Furthermore, the constraint conditions for the existence of solutions to these variable coefficient equations are also revealed. This improved method has a wider range of applications for addressing nonlinear wave equations.