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

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.


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


Zbl 1080.35096


Full Text: DOI


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