Uniformly constructing a series of explicit exact solutions to nonlinear equations in mathematical physics.

*(English)* Zbl 1030.35136
Summary: We devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, 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 soliton solutions, (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, combined KdV-MKdV, Camassa-Holm, Kaup-Kupershmidt, Jaulent-Miodek, $(2+1)$-dimensional dispersive long wave, new $(2+1)$-dimensional generalized Hirota, $(2+1)$-dimensional breaking soliton and double sine-Gordon equations. 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:

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

35C10 | Series solutions of PDE |

35Q51 | Soliton-like equations |

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