Elliptic equation’s new solutions and their applications to two nonlinear partial differential equations. (English) Zbl 1118.65379

Wang, Deng-Shan; Li, Hong-Bo

Appl. Math. Comput. 188, No. 1, 762-771 (2007).

Summary: In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of first-order elliptic equation \(\varphi^{\prime 2}= a_0+ a_1\varphi+ a_2\varphi^2+ a_3\varphi^3+ a_4\varphi^4\) (where \(\varphi'= (\frac{d}{dx}\varphi)\)) are obtained. To our knowledge, these nontrivial solutions can not be found in [Chaos Solitons Fract. 26, No. 3, 785–794 (2005; Zbl 1080.35096) and Phys. Lett. A 336, 463–476 (2005)] by E. 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 can be investigated and new non-travelling wave solutions can be explicitly obtained with the aid of symbolic computation system Maple.

MSC:

65N22	Numerical solution of discretized equations for boundary value problems involving PDEs
65M70	Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q35	PDEs in connection with fluid mechanics
35J65	Nonlinear boundary value problems for linear elliptic equations

Keywords:

symbolic computation; elliptic equation; non-travelling wave solution; nonlinear partial differential equations; breaking soliton equations; asymmetric Nizhnik-Novikov-Vesselov equation

Citations:

Zbl 1080.35096

Software:

Maple

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