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

TL;DR

This paper presents solutions for a class of nonlinear fourth-order PDEs, related to models of vibrations and generalizations of the Camassa-Holm equation, using traveling wave and linearization methods.

Contribution

It introduces explicit solutions for a specific nonlinear fourth-order PDE class, extending understanding of related vibration and wave equations.

Findings

Solutions for the PDE class are derived.

Traveling wave solutions are constructed.

Linearization criteria for associated ODEs are established.

Abstract

Some solutions for one class of nonlinear fourth-order partial differential equations \[u_{tt} = ({\kappa u + \gamma u^2})_{xx} + \nu uu_{xxxx} + \mu u_{xxtt} + \alpha u_x u_{xxx} + \beta u_{xx}^2 \] 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.