Wave Solutions for a (2 + 1)-Dimensional Burgers–KdV Equation with Variable Coefficients via the Functional Expansion Method

Abstract

A (2 + 1)-dimensional fourth order Burgers–KdV equation with variable coefficients (vcBKdV) is studied here and interesting wave-type solutions with variable amplitudes and velocities are reported. The model has been not previously studied in the chosen form and it presents a twofold interest: as a model describing a rich variety of phenomena and as a higher-order equation solving difficulties generated by the presence of the variable coefficients. The novelty of our approach is related to the use of the functional expansion, a solving method based on an auxiliary equation that generalizes other approaches, such as, for example, the G′G proved here. We use a similarity reduction with a nonlinear wave variable that leads to a determining system that it is not usually algebraic, but an over-determined system of partial differential equations. It depends on 14 constant or functional parameters and can generate much richer classes of solutions. Three such classes of solutions, corresponding to the case when a specific form of the generalized reaction–diffusion equation is used as auxiliary equation, are considered. The influence on the dynamical behavior of two important factors, the choices of the auxiliary equation and the form of solution, are studied by providing graphical representations of specific solutions for various values of the parameters.