Travelling wave solutions, symmetry reductions and conserved vectors of a generalized hyper-elastic rod wave equation

Abstract

This work presents a generalized hyper-elastic rod wave (gHRW) equation from the Lie symmetry method’s standpoint. The equation illustrates dispersive waves generating in hyper-elastic rods. Using multiplier approach we find conserved vectors of the underlying equation. We subsequently obtain first integrals of the conserved vectors under the time–space group invariant u(t,x)=H(x−νt). Finally, by analysing various attainable instances of the arbitrary coefficient function g(u), we perform symmetry reductions of gHRW equation to lower order ordinary differential equations and in some instances obtain analytic solutions for special values of arbitrary constants.

Keywords

• Hyper-elastic rod wave equation
• Symmetry reduction
• Group invariant
• Conservation laws
• First integral
• Multiplier approach