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

Abstract

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of first-order elliptic equation ϕ′ 2 = a 0 + a 1 ϕ + a 2 ϕ 2 + a 3 ϕ 3 + a 4 ϕ 4 (where ϕ ′ = d d x ϕ ) 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.