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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 $C\left(l,n,p\right)$. 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 $C\left(l,n,p\right)$ 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 Schrödinger) equations 68W30 Symbolic computation and algebraic computation 81U15 Exactly and quasi-solvable systems (quantum theory) 81U30 Dispersion theory, dispersion relations (quantum theory)