Solutions for one class of nonlinear fourth-order partial differential equations

Abstract

Some solutions for one class of nonlinear fourth-order partial differential equations \[u_{tt} = ({κu + γu^2})_{xx} + νuu_{xxxx} + μu_{xxtt} + αu_x u_{xxx} + βu_{xx}^2 \] where $\alpha ,\beta ,\gamma ,\mu ,\nu$ and $\kappa$ are arbitrary constants are presented in the paper. This equation may be thought of as a fourth-order analogue of a generalization of the Camassa-Holm equation, about which there has been considerable recent interest. Furthermore, this equation is a Boussinesq-type equation which arises as a model of vibrations of harmonic mass-spring chain. The idea of travelling wave solutions and linearization criteria for fourth-order ordinary differential equations by point transformations are applied to this problem.

11 pages