Abundant explicit and exact solutions for the space–time fractional Vakhnenko–Parkes model in the relaxing medium with stability analysis

Abstract

In this work, we have considered the beta-fractional derivative form of the Vakhnenko–Parkes equation (VPE) to study its novel analytical solutions. To achieve the required new distinct traveling wave solutions, the new sub-equation method is used for the space–time fractional VPE. The dynamics of these solutions are analyzed under different parametric and fractional conditions by the graphical view. The produced wave patterns include combined bright–dark, w-shape, u-shape, v-shape, bright, propagation of two solitons, combined dark–bright, bell shape, a dark, periodic wave, breather, v-shape with two bright spots and combined anti-peakon–bright solutions. In recognizing the physical significance of the obtained wave solutions at each fraction of an interval, these solutions are highly commendable. In addition, to show the stability of the considered model, the stability analysis of the governing model is discussed by the linear stability analysis method which states the behavior of the model in the corresponding medium. These solutions are useful to study the proliferation of high-frequency waves in the relaxing medium.