On Lie symmetry analysis, conservation laws and solitary waves to a longitudinal wave motion equation

Abstract

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.