Sub-ODE's New Solutions and Their Applications to Two Nonlinear Partial Differential Equations with Higher-Order Nonlinear Terms

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of the first-order sub-ODE F'2 = AF2 + BF2+p + CF2+2p (where F' = dF/dξ, p > 0) are obtained. To our best knowledge, these nontrivial solutions have not been found in [X.Z. Li and M.L. Wang, Phys. Lett. A 361 (2007) 115] and [S. Zhang, W. Wang, and J.L. Tong, Phys. Lett. A 372 (2008) 3808] and other existent papers until now. Using these nontrivial solutions, the sub-ODE method is described to construct several kinds of exact travelling wave solutions for the generalized KdV-mKdV equation with higher-order nonlinear terms and the generalized ZK equation with higher-order nonlinear terms. By means of this method, many other physically important nonlinear partial differential equations with nonlinear terms of any order can be investigated and new nontrivial solutions can be explicitly obtained with the help of symbolic computation system Maple or Mathematica.