Explicit exact solutions for new general two-dimensional KdV-type and two-dimensional KdV Burgers-type equations with nonlinear terms of any order

In this paper, we have improved the tanh-method by means of a proper transformation and a general ansatz. Applying the improved method and direct assumption method with symbolic computation, we consider two kinds of equations, general two-dimensional KdV-type equations with nonlinear terms of any order, ( u t + au p u x + bu 2 p u x + δ u xxx ) x + su yy = 0, and general two-dimensional KdV–Burgers-type equations with nonlinear terms of any order, ( u t + au p u x + bu 2 p u x + γ u xxx + δ u xxx ) x + su yy = 0. As a result, rich explicit exact solutions for these two equations, which contain kink-profile solitary-wave solutions, bell-profile solitary-wave solutions, rational solutions, periodic solutions and combined formal solutions, are obtained.