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^pu_x + bu^2pu_x + δu_xxx)_x + su_yy = 0, and general two-dimensional KdV-Burgers-type equations with nonlinear terms of any order, (u_t + au^pu_x + bu^2pu_x + yu_xx + δu_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.