An algebraic method for finding a series of exact solutions to integrable and nonintegrable nonlinear evolution equations.

*(English)* Zbl 1167.35324
Summary: An algebraic method is devised to uniformly construct a series of exact solutions for general integrable and nonintegrable nonlinear evolution equations. Compared with most existing tanh methods, the Jacobi function expansion method or other sophisticated methods, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic and solitary wave solutions and (f) Jacobi and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, new (2+1)-dimensional Calogero-KdV equation, (3+1)-dimensional Jimbo-Miwa equation, symmetric regular long wave equation, Drinfel’d-Sokolov-Wilson equation, (2+1)-dimensional generalized dispersive long wave equation, double sine-Gordon equation, Calogero-Degasperis-Fokas equation and coupled Schrödinger-Boussinesq equation. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.

##### MSC:

35C05 | Solutions of PDE in closed form |

35G20 | General theory of nonlinear higher-order PDE |

35Q53 | KdV-like (Korteweg-de Vries) equations |

35Q58 | Other completely integrable PDE (MSC2000) |

37K10 | Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies |