Qualitative analysis and new soliton solutions for the coupled nonlinear Schrödinger type equations

2021 ◽  
Author(s):  
M. E. Elbrolosy
Keyword(s):  
Qualitative Analysis ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Singular Solutions ◽  
Wave System ◽  
Nonlinear Schrödinger ◽  
Perturbed System ◽  
Wave Solutions

Abstract This work is interested in constructing new traveling wave solutions for the coupled nonlinear Schrödinger type equations. It is shown that the equations can be converted to a conservative Hamiltonian traveling wave system. By using the bifurcation theory and Qualitative analysis, we assign the permitted intervals of real propagation. The conserved quantity is utilized to construct sixteen traveling wave solutions; four periodic, two kink, and ten singular solutions. The periodic and kink solutions are analyzed numerically considering the affect of varying each parameter keeping the others fixed. The degeneracy of the solutions discussed through the transmission of the orbits illustrates the consistency of the solutions. The 3D and 2D graphical representations for solutions are presented. Finally, we investigate numerically the quasi-periodic behavior for the perturbed system after inserting a periodic term.

scholarly journals Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

2021 ◽  
Author(s):  
Mati Youssoufa ◽  
Ousmanou Dafounansou ◽  
Camus Gaston Latchio Tiofack ◽  
Alidou Mohamadou
Keyword(s):  
Power Law ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Chaotic Behavior ◽  
Traveling Wave Solutions ◽  
Chaotic Motions ◽  
Auxiliary Equation Method ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Quasiperiodic Route To Chaos

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima’s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.

Various Exact Soliton Solutions and Bifurcations of a Generalized Dullin–Gottwald–Holm Equation with a Power Law Nonlinearity

2017 ◽  
Vol 27 (08) ◽  
pp. 1750129 ◽  
Author(s):  
Temesgen Desta Leta ◽  
Jibin Li
Keyword(s):  
Power Law ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Wave System ◽  
Wave Solutions ◽  
Holm Equation ◽  
Exact Soliton Solutions ◽  
Planar Dynamical Systems

In this paper, we study a model of generalized Dullin–Gottwald–Holm equation, depending on the power law nonlinearity, that derives a series of planar dynamical systems. The study of the traveling wave solutions for this model derives a planar Hamiltonian system. By investigating the dynamical behavior and bifurcation of solutions of the traveling wave system, we derive all possible exact explicit traveling wave solutions, under different parametric conditions. These results completely improve the study of traveling wave solutions to the mentioned model stated in [Biswas & Kara, 2010].

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS FOR A GENERALIZED CAMASSA–HOLM EQUATION

2013 ◽  
Vol 23 (03) ◽  
pp. 1350057 ◽  
Author(s):  
JIBIN LI ◽  
ZHIJUN QIAO
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Breaking Wave ◽  
Wave System ◽  
Wave Solutions ◽  
Holm Equation ◽  
Kink Wave ◽  
Quadratic And Cubic Nonlinearities

In this paper, we study all possible traveling wave solutions of an integrable system with both quadratic and cubic nonlinearities: [Formula: see text], m = u-uxx, where b, k1 and k2 are arbitrary constants. We call this model a generalized Camassa–Holm equation since it is kind of a cubic generalization of the Camassa–Holm (CH) equation: mt + mxu + 2mux = 0. In the paper, we show that the traveling wave system of this generalized Camassa–Holm equation is actually a singular dynamical system of the second class. We apply the method of dynamical systems to analyze the dynamical behavior of the traveling wave solutions and their bifurcations depending on the parameters of the system. Some exact solutions such as smooth soliton solutions, kink and anti-kink wave solutions, M-shape and W-shape wave profiles of the breaking wave solutions are obtained. To guarantee the existence of those solutions, some constraint parameter conditions are given.

Traveling Wave Solutions of a Two-Component Dullin–Gottwald–Holm System

2016 ◽  
Vol 12 (3) ◽  
Author(s):  
Jiyu Zhong ◽  
Shengfu Deng
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Two Component ◽  
Planar Systems

In this paper, we investigate the traveling wave solutions of a two-component Dullin–Gottwald–Holm (DGH) system. By qualitative analysis methods of planar systems, we investigate completely the topological behavior of the solutions of the traveling wave system, which is derived from the two-component Dullin–Gottwald–Holm system, and show the corresponding phase portraits. We prove the topological types of degenerate equilibria by the technique of desingularization. According to the dynamical behaviors of the solutions, we give all the bounded exact traveling wave solutions of the system, including solitary wave solutions, periodic wave solutions, cusp solitary wave solutions, periodic cusp wave solutions, compactonlike wave solutions, and kinklike and antikinklike wave solutions. Furthermore, to verify the correctness of our results, we simulate these bounded wave solutions using the software maple version 18.

Explicit, periodic and dispersive soliton solutions to the conformable time-fractional Wu–Zhang system

2021 ◽  
pp. 2150417
Author(s):  
Kalim U. Tariq ◽  
Mostafa M. A. Khater ◽  
Muhammad Younis
Keyword(s):  
Fractional Derivative ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Nonlinear Evolution Equations ◽  
Conformable Fractional Derivative ◽  
Physical Behavior ◽  
Wave Solutions

In this paper, some new traveling wave solutions to the conformable time-fractional Wu–Zhang system are constructed with the help of the extended Fan sub-equation method. The conformable fractional derivative is employed to transform the fractional form of the system into ordinary differential system with an integer order. Some distinct types of figures are sketched to illustrate the physical behavior of the obtained solutions. The power and effective of the used method is shown and its ability for applying different forms of nonlinear evolution equations is also verified.

Bifurcations and Exact Traveling Wave Solutions for a Model of Nonlinear Pulse Propagation in Optical Fibers

2014 ◽  
Vol 24 (06) ◽  
pp. 1450088
Author(s):  
Jibin Li
Keyword(s):  
Optical Fibers ◽  
Traveling Wave ◽  
Pulse Propagation ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Wave System ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Nonlinear Pulse Propagation

In this paper, we consider a model of nonlinear pulse propagation in optical fibers. By investigating the dynamical behavior and bifurcations of solutions of the traveling wave system of PDE, we derive all possible exact explicit traveling wave solutions under different parameter conditions. These results completed the study of traveling wave solutions for the mentioned model posed by [Lenells, 2009].

scholarly journals Exact Traveling Wave Solutions and Bifurcation of a Generalized (3+1)-Dimensional Time-Fractional Camassa-Holm-Kadomtsev-Petviashvili Equation

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Zhigang Liu ◽  
Kelei Zhang ◽  
Mengyuan Li
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Phase Portraits ◽  
Wave System ◽  
Bifurcation Method ◽  
Exact Traveling Wave Solutions ◽  
Fractional Complex Transform ◽  
Wave Solutions ◽  
Petviashvili Equation

In this paper, we study the (3+1)-dimensional time-fractional Camassa-Holm-Kadomtsev-Petviashvili equation with a conformable fractional derivative. By the fractional complex transform and the bifurcation method for dynamical systems, we investigate the dynamical behavior and bifurcation of solutions of the traveling wave system and seek all possible exact traveling wave solutions of the equation. Furthermore, the phase portraits of the dynamical system and the remarkable features of the solutions are demonstrated via interesting figures.

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