Modified Vakhnenko–Parkes equation with power law nonlinearity: Painlevé analysis, analytic solutions and conservation laws

Abstract

The Vakhnenko equation governs the propagation of high-frequency waves in a relaxation medium. The Painleve analysis of modified Vakhnenko–Parkes equation is performed by using the Kruskal approach. The Lie symmetry formalism is also applied to derive symmetries of generalised modified Vakhnenko–Parkes equation with power law nonlinearity. The exact solutions of this equation are obtained in the form of power series; and other explicit exact solutions in the form of trigonometric, hyperbolic and Jacobi elliptic functions are obtained for modified Vakhnenko–Parkes equation. The bright soliton solutions of generalised modified Vakhnenko–Parkes equation are also obtained. The conservation laws are obtained with the help of multiplier approach. The 3D plots and contour plots are shown by considering the possibilities of arbitrary function occurring in the solutions.