A generalized auxiliary equation method and its application to nonlinear Klein-Gordon and generalized nonlinear Camassa-Holm equations.

*(English)* Zbl 1217.81075
Summary: With the aid of symbolic computation, a generalized auxiliary equation method is proposed to construct more general exact solutions to two types of NLPDEs. First, we present new family of solutions to a nonlinear Klein-Gordon equation, by using this auxiliary equation method including a new first-order nonlinear ODE with six-degree nonlinear term proposed by Sirendaoreji. Then, we apply an indirect F-function method very close to the F-expansion method to solve the generalized Camassa-Holm equation with fully nonlinear dispersion and fully nonlinear convection $C(l,n,p)$. Taking advantage of the new first-order nonlinear ODE with six degree nonlinear term, this indirect F-function method is used to map the solutions of $C(l,n,p)$ equations to those of that nonlinear ODE. As a result, we can successfully obtain in a unified way, many exact solutions.

##### MSC:

81Q05 | Closed and approximate solutions to quantum-mechanical equations |

35Q55 | NLS-like (nonlinear SchrĂ¶dinger) equations |

68W30 | Symbolic computation and algebraic computation |

81U15 | Exactly and quasi-solvable systems (quantum theory) |

81U30 | Dispersion theory, dispersion relations (quantum theory) |