Solitary wave solutions along with Painleve analysis for the Ablowitz–Kaup–Newell–Segur water waves equation

This study focuses on the Ablowitz–Kaup–Newell–Segur (AKNS) water waves equation. Painleve test (P-test) will be implemented to check the integrability of AKNS equation and an extended modified auxiliary equation mapping (EMAEM) architectonic is implemented to get a new set of traveling wave solutions like periodic and doubly periodic, bell type, kink, singular kink, anti-kink, trigonometric, singular, rational, combined soliton like solutions and hyperbolic solutions. Furthermore, it is analyzed that the implemented algorithm is efficient and accurate for solving nonlinear evolution equations (NLEEs). Finally, graphical simulations (2D, 3D and contours) are also provided to illustrate the detailed behavior of the solution and effectiveness of the proposed method.

Exact solutions of (2 + 1)-Ablowitz-Kaup-Newell-Segur equation

The aim of the present study is to obtain different types of hyperbolic type solutions of the (2+1)-Ablowitz-Kaup-Newell-Segur (AKNS) equation. In order to construction exact solutions of AKNS equation, (1/G′)-expansion method is successfully applied. At the end of this application, singular soliton wave with considerable importance for the shock wave structure and asymptotic behavior employees have emerged. By giving arbitrary values to the constants in the solutions obtained, 3D, 2D and contour graphics are presented. The method used in this article can be used in other nonlinear differential equations (NPDEs) as it is reliable, easy and effective. Ready package programs are used to solve complex and difficult processes in this study.

Local and Nonlocal Reductions of Two Nonisospectral Ablowitz-Kaup-Newell-Segur Equations and Solutions

In this paper, local and nonlocal reductions of two nonisospectral Ablowitz-Kaup-Newell-Segur equations, the third order nonisospectral AKNS equation and the negative order nonisospectral AKNS equation, are studied. By imposing constraint conditions on the double Wronskian solutions of the aforesaid nonisospectral AKNS equations, various solutions for the local and nonlocal nonisospectral modified Korteweg-de Vries equation and local and nonlocal nonisospectral sine-Gordon equation are derived, including soliton solutions and Jordan block solutions. Dynamics of some obtained solutions are analyzed and illustrated by asymptotic analysis.

Dynamical Behaviour of the Light Pulses through the Optical Fiber: Two Nonlinear Atangana Conformable Fractional Evolution Equations

This study, using the extended simplest method of equation, examines the explicit movement solutions of both the Schwarzian Korteweg-de Vries (SKdV) and (2 + 1)-Ablowitz-Kaup-Newell-Segur (AKNS.) equation. These models show the movement of the waves in optical fiber mathematically. The SKdV equation explains the movement of the isolated waves in diverse fields and on the site in a small space microsection. Some solutions obtained have been developed to show the physical and dynamic behaviors of these solutions in the obtained wave.

Optimal System and Invariant Solutions of a New AKNS Equation with Time-Dependent Coefficients

The Lie point symmetries are reported by performing the Lie symmetry analysis to the Ablowitz-Kaup-Newell-Suger (AKNS) equation with time-dependent coefficients. In addition, the optimal system of one-dimensional subalgebras is constructed. Based on this optimal system, several categories of similarity reduction and some new invariant solutions for the equation are obtained, which include power series solutions and travelling and non-traveling wave solutions.

Noncommutative negative order AKNS equation and its soliton solutions

A noncommutative negative order AKNS (NC-AKNS(-1)) equation is studied. To show the integrability of the system, we present explicitly the underlying integrable structure such as Lax pair, zero-curvature condition, an infinite sequence of conserved densities, Darboux transformation (DT) and quasideterminant soliton solutions. Moreover, the NC-AKNS(-1) equation is compared with its commutative counterpart not only on the level of nonlinear evolution equation but also for the explicit solutions.