Elliptic function soliton solutions of the higher-order nonlinear dispersive Kundu–Eckhaus dynamical equation with applications and stability

In this paper, we employ the four mathematical methods, namely improved simple equation method (SEM), modified extended SEM, generalized exp(-Φ(ξ))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Phi (\xi ))$$\end{document}-expansion technique and modified extended mapping technique, and attain exact solutions in various forms of Kundu–Eckhaus (K–E) equation. The wave solutions in the forms of bright and dark solitons, periodic solitary wave, kink and anti-kink soliton, trigonometric and hyperbolic trigonometric function and rational function are achieved. The movements of few results are presented graphically, which are useful to mathematician and physician for understanding the complex phenomena. The standard linear stability analysis is employed to examine the stability of the equation. This shows that exact solutions are stable. The results obtained are novel in nature and have good potential application values. The results demonstrate that these techniques are simple, effective and more powerful to solve nonlinear evolution equations.