Elliptic equation's new solutions and their applications to two nonlinear partial differential equations

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of first-order elliptic equation ϕ′2 = a0 + a1ϕ + a2ϕ2 + a3ϕ3 + a4ϕ4 (where ϕ′=ddxϕ) are obtained. To our knowledge, these nontrivial solutions can not be found in [Chaos Solitons Fract. 26 (2005) 785-794] and [Phys. Lett. A 336 (2005) 463-476] by Yomba and other existent papers until now. By using these nontrivial solutions, a direct algebraic method is described to construct several kinds of exact non-travelling wave solutions for the (2 + 1)-dimensional Breaking soliton equations and the (2 + 1)-dimensional asymmetric Nizhnik-Novikov-Vesselov equation. By using this method, many other physically important nonlinear partial differential equations (NLPDEs) can be investigated and new non-travelling wave solutions can be explicitly obtained with the aid of symbolic computation system Maple.