A sub-ODE method for finding exact solutions of a generalized KdV–mKdV equation with high-order nonlinear terms

The bell type solitary wave solution, the kink type solitary wave solution, the algebraic solitary wave solution and the sinusoidal traveling wave solution of a generalized KdV–mKdV equation (GKdV–mKdV) with high-order nonlinear terms are obtained by a subsidiary ordinary differential equation method (sub-ODE method for short). The key ideas of the sub-ODE method are that the traveling wave solutions of a complicated nonlinear wave equation can be constructed by means of the solutions of some simple and solvable ODEs which are called sub-ODEs.