Isa Aliyu

Dark optical, singular solitons and conservation laws to the nonlinear Schrödinger’s equation with spatio-temporal dispersion

This paper studies the dynamics of solitons to the nonlinear Schrodinger’s equation (NLSE) with spatio-temporal dispersion (STD). The integration algorithm that is employed in this paper is the Riccati–Bernoulli sub-ODE method. This leads to dark and singular soliton solutions that are important in the field of optoelectronics and fiber optics. The soliton solutions appear with all necessary constraint conditions that are necessary for them to exist. There are four types of nonlinear media studied in this paper. They are Kerr law, power law, parabolic law and dual law. The conservation laws (Cls) for the Kerr law and parabolic law nonlinear media are constructed using the conservation theorem presented by Ibragimov.

Optical solitons to the nonlinear Shrödinger’s equation with spatio-temporal dispersion using complex amplitude ansatz

Abstract In this work, we investigate the optical solitons to the non-linear Shrödinger’s equation with spatio-temporal dispersion. There are two types of non-linear media studied in this paper. They are Kerr law and parabolic law. By adopting a complex amplitude ansatz method composed of the addition of bright and dark optical solitons, we present the exact dark, bright and dark-bright or combined optical solitons to the model. Numerical results and discussions are also presented.

Traveling wave solutions and conservation laws for nonlinear evolution equation

In this work, the Riccati-Bernoulli sub-ordinary differential equation and modified tanh-coth methods are used to reach soliton solutions of the nonlinear evolution equation. We acquire new types of traveling wave solutions for the governing equation. We show that the equation is nonlinear self-adjoint by obtaining suitable substitution. Therefore, we construct conservation laws for the equation using new conservation theorem. The obtained solutions in this work may be used to explain and understand the physical nature of the wave spreads in the most dispersive medium. The constraint condition for the existence of solitons is stated. Some three dimensional figures for some of the acquired results are illustrated.

Optical solitons, conservation laws and modulation instability analysis for the modified nonlinear Schrödinger’s equation for Davydov solitons

Abstract In this paper, the optical solitons to the modified nonlinear Schrödinger’s equation for davydov solitons are investigate. The modified F-expansion method is the integration technique employed to achieve this task. This yielded a combined and other soliton solutions. The Lie point symmetry generators of a system of partial differential equations acquired by decomposing the equation into real and imaginary components are derived. We prove that the system is nonlinearly self-adjoint with an explicit form of a differential substitution satisfying the nonlinear self-adjoint condition. Then we use these facts to construct a set of local conservation laws (Cls) for the system using the general Cls theorem presented by Ibragimov. Furthermore, the modulation instability (MI) is analyzed based on the standard linear-stability analysis and the MI gain spectrum is got. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.

Dark–bright optical solitary waves and modulation instability analysis with (2 + 1)-dimensional cubic-quintic nonlinear Schrödinger equation

ABSTRACT This paper addresses the (2+1)-dimensional cubic-quintic nonlinear Schrödinger equation (CQNLS) that serves as the model to study the light propagation through nonlinear optical media and non-Kerr crystals. A dark–bright optical solitary wave solution of this equation is retrieved by adopting the complex envelope function ansatz. This type of solitary wave describes the properties of bright and dark optical solitary waves in the same expression. The integration naturally lead to a constraint condition placed on the solitary wave parameters which must hold for the solitary waves to exist. Additionally, the modulation instability (MI) analysis of the model is studied based on the standard linear stability analysis and the MI gain spectrum is got. Numerical simulation and physical interpretations of the obtained results are demonstrated. It is hoped that the results reported in this paper can enrich the nonlinear dynamical behaviors of the CQNLS.

Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation

Abstract In this work, Lie symmetry analysis for the time fractional simplified modified Kawahara (SMK) equation with Riemann-Liouville (RL) derivative, is analyzed. We transform the time fractional SMK equation to nonlinear ordinary differential equation (ODE) of fractional order using its Lie point symmetries with a new dependent variable. In the reduced equation, the derivative is in the Erdelyi-Kober (EK) sense. We solve the reduced fractional ODE using a power series technique. Using Ibragimov’s nonlocal conservation method to time fractional partial differential equations, we compute conservation laws (Cls) for the time fractional SMK equation. Some figures of the obtained explicit solution are presented.

Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation

In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results.

Complexiton and solitary wave solutions of the coupled nonlinear Maccari’s system using two integration schemes

In this paper, we consider a coupled nonlinear Maccari’s system (CNMS) which describes the motion of isolated waves localized in a small part of space. There are some integration tools that are adopted to retrieve the solitary wave solutions. They are the modified F-Expansion and the generalized projective Riccati equation methods. Topological, non-topological, complexiton, singular and trigonometric function solutions are derived. A comparison between the results in this paper and the well-known results in the literature is also given. The derived structures of the obtained solutions offer a rich platform to study the nonlinear CNMS. Numerical simulation of the obtained solutions are presented with interesting figures showing the physical meaning of the solutions.

Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation

Abstract In this article, the generalized shallow water wave (GSWW) equation is studied from the perspective of one dimensional optimal systems and their conservation laws (Cls). Some reduction and a new exact solution are obtained from known solutions to one dimensional optimal systems. Some of the solutions obtained involve expressions with Bessel function and Airy function [1,2,3]. The GSWW is a nonlinear self-adjoint (NSA) with the suitable differential substitution, Cls are constructed using the new conservation theorem.

The investigation of soliton solutions and conservation laws to the coupled generalized Schrödinger–Boussinesq system

Abstract This paper employed the principle of undetermined coefficients and Bernoulli sub-ODE methods to acquire the topological, non-topological, periodic wave and algebraic solutions of the coupled generalized Schrödinger–Boussinesq system (CGSBs). The concept of Lie point symmetry is applied to derive the point symmetries of the CSGE. The problem on nonlinear self-adjointness of the CSGE has not been solved in previous time. In the present paper, we solve this problem and find an explicit form of the differential substitution providing the nonlinear self-adjointness. Then we use this fact to construct a set of conserved vectors using the classical symmetries admitted by the equation and the general conservation laws (Cls) theorem presented by Ibragimov. Numerical simulation of the obtained results are analyzed with interesting figures showing the physical meaning of the solutions.