Painlevé analysis, auto-Bäcklund transformation and analytic solutions for modified KdV equation with variable coefficients describing dust acoustic solitary structures in magnetized dusty plasmas

In this paper, variable coefficients mKdV equation is examined by using Painlevé analysis and auto-Bäcklund transformation method. The proposed equation is an important equation in magnetized dusty plasmas. The Painlevé analysis is used to determine the integrability whereas an auto-Bäcklund transformation technique is being explored to derive unique family of analytical solutions for variable coefficients mKdV equation. New kink–antikink and periodic-kink- type soliton solutions have been determined successfully for the considered equation. This paper shows that auto-Bäcklund transformation method is effective, direct and easy to use, and used to determine the analytic soliton solutions of various nonlinear evolution equations in the field of science and engineering. The results are plotted graphically to signify the potency and applicability of this proposed scheme for solving the above considered equation. The obtained results are in the form of soliton-like solutions, solitary wave solutions, exponential and trigonometric function solutions. Therefore, these solutions help us to understand the potential and physical behaviors of the proposed equation.

Painlevé analysis of Fokas–Lenells equation with fractal temporal evolution

This paper presents new optical solitons of a fractal Fokas–Lenells equation that models the dynamics of optical fibers. The Painlevé technique is employed to identify kink soliton solutions. The constraint conditions guarantee the existence of these solitons. The outcomes of this research give new solutions that are not achieved using some already defined algorithms. The derived method is efficient and its applications are promising for other nonlinear problems. The 3D graphical illustrations of obtained results are depicted for various values of the fractal parameter.