Exact solutions and invariant subspaces to the nonlinear dissipative–dispersive equation

Abstract

In this paper, we performed Lie symmetry analysis and applied [Formula: see text] expansion method on the nonlinear dissipative–dispersive equation. The purpose of this research is to find the vector fields and transform the nonlinear dissipative–dispersive equation into simpler forms. The Maple software was used to obtain the vector field and similarity reductions for nonlinear dissipative–dispersive equations. In addition, we obtained exact solutions based on the [Formula: see text] expansion method and power series method, including the hyperbolic functions, the trigonometric functions and the rational functions. The method we used is direct, concise, elementary and effective, and can be used for many other nonlinear evolution equations. Furthermore, the invariant subspaces of the nonlinear dissipative–dispersive equation were identified using the refined invariant subspaces method. The invariant subspaces of solutions to linear ordinary differential equations were used to prove that nonlinear dissipative–dispersive equation admits subspaces. The exact solutions were obtained by using generalized separated variables.