Kink-type solutions of the SIdV equation and their properties

We study the nonlinear integrable equation, u t + 2(( u x u xx )/ u ) = ϵu xxx , which is invariant under scaling of dependent variable and was called the SIdV equation (see Sen et al. 2012 Commun. Nonlinear Sci. Numer. Simul . 17 , 4115–4124 ( doi:10.1016/j.cnsns.2012.03.001 )). The order- n kink solution u [ n ] of the SIdV equation, which is associated with the n -soliton solution of the Korteweg–de Vries equation, is constructed by using the n -fold Darboux transformation (DT) from zero ‘seed’ solution. The kink-type solutions generated by the onefold, twofold and threefold DT are obtained analytically. The key features of these kink-type solutions are studied, namely their trajectories, phase shifts after collision and decomposition into separate single kink solitons.

The Application of the exp⁡(-Φ(ξ))-Expansion Method for Finding the Exact Solutions of Two Integrable Equations

In this article, the exp⁡-Φξ method is connected to search for new hyperbolic, periodic, and rational solutions of (1+1)-dimensional fifth-order nonlinear integrable equation and (2+1)-dimensional Date-Jimbo-Kashiwara-Miwa equation. The obtained solutions consist of trigonometric, hyperbolic, rational functions and W-shaped soliton. Furthermore, 3D and 2D graphs are plotted by choosing the suitable values of the parameters involved.

Riemann–Hilbert approach for the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation and its N-soliton solutions

On the basis of the spectral analysis for the Lax pair, a Riemann–Hilbert problem of the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation is established. Using the inverse scattering transformation and the Riemann–Hilbert approach, the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation is studied. As an application, N-soliton solutions of the combined nonlinear Schrödinger and Gerdjikov–Ivanov equation are obtained. In addition, some figures are given to illustrate the soliton characteristics of the nonlinear integrable equation.

New solutions for two integrable cases of a generalized fifth-order nonlinear equation

Multiple-complexiton solutions for a new generalized fifth-order nonlinear integrable equation are constructed with the help of the Hirota's method and the simplified Hirota's method. By extending the real parameters into complex parameters, nonsingular complexiton solutions are obtained for two specific coefficients of the new generalized equation.