scholarly journals Exact Solutions of Travelling Wave Model via Dynamical System Method

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Heng Wang ◽  
Longwei Chen ◽  
Hongjiang Liu
Keyword(s):  
Dynamical System ◽  
Solitary Wave ◽  
Elliptic Functions ◽  
Wave Model ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Jacobian Elliptic Functions ◽  
Wave Solutions ◽  
Periodic Travelling Wave

By using the method of dynamical system, the exact travelling wave solutions of the coupled nonlinear Schrödinger-Boussinesq equations are studied. Based on this method, the bounded exact travelling wave solutions are obtained which contain solitary wave solutions and periodic travelling wave solutions. The solitary wave solutions and periodic travelling wave solutions are expressed by the hyperbolic functions and the Jacobian elliptic functions, respectively. The results show that the presented findings improve the related previous conclusions. Furthermore, the numerical simulations of the solitary wave solutions and the periodic travelling wave solutions are given to show the correctness of our results.

scholarly journals Existence and Orbital Stability of Cnoidal Waves for a 1D Boussinesq Equation

2007 ◽  
Vol 2007 ◽  
pp. 1-36 ◽  
Author(s):  
Jaime Angulo ◽  
Jose R. Quintero
Keyword(s):  
Orbital Stability ◽  
Type Equation ◽  
Elliptic Functions ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Jacobian Elliptic Functions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Complete Study ◽  
Set Up

We will study the existence and stability of periodic travelling-wave solutions of the nonlinear one-dimensional Boussinesq-type equationΦtt−Φxx+aΦxxxx−bΦxxtt+ΦtΦxx+2ΦxΦxt=0. Periodic travelling-wave solutions with an arbitrary fundamental periodT0will be built by using Jacobian elliptic functions. Stability (orbital) of these solutions by periodic disturbances with periodT0will be a consequence of the general stability criteria given by M. Grillakis, J. Shatah, and W. Strauss. A complete study of the periodic eigenvalue problem associated to the Lame equation is set up.

Exact single travelling wave solutions to the fractional perturbed Gerdjikov–Ivanov equation in nonlinear optics

2021 ◽  
pp. 2150377
Author(s):  
Xiang Xiao ◽  
Zhixiang Yin
Keyword(s):  
Solitary Wave ◽  
Periodic Solutions ◽  
Travelling Wave ◽  
Polynomial Method ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Trial Equation Method ◽  
Wave Solutions ◽  
Triangular Function ◽  
Almost All

In this paper, exact single travelling wave solutions to the nonlinear fractional perturbed Gerdjikov–Ivanov equation are captured by the complete discrimination system for polynomial method and the trial equation method. In the classification, we can find out the original equation has rational function solutions, solitary wave solutions, triangular function periodic solutions, and elliptic function periodic solutions, which are normally very difficult to be obtained by other methods. In particular, the concrete parameters are set to show that the solutions in the classification can be realized in almost all cases.

scholarly journals Periodic and Solitary Wave Solutions to the Fornberg-Whitham Equation

2009 ◽  
Vol 2009 ◽  
pp. 1-10 ◽  
Author(s):  
Jiangbo Zhou ◽  
Lixin Tian
Keyword(s):  
Solitary Wave ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Two Parameters ◽  
Whitham Equation

New travelling wave solutions to the Fornberg-Whitham equationut−uxxt+ux+uux=uuxxx+3uxuxxare investigated. They are characterized by two parameters. The expresssions for the periodic and solitary wave solutions are obtained.

scholarly journals Jacobian Elliptic Wave Solutions for the Wadati–Segur–Ablowitz Equation

1997 ◽  
Vol 11 (23) ◽  
pp. 2849-2854 ◽  
Author(s):  
C. G. R. Teh ◽  
W. K. Koo ◽  
B. S. Lee
Keyword(s):  
Solitary Wave ◽  
Wave Train ◽  
Travelling Wave ◽  
Amplitude Equation ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Wave Solutions

Jacobian elliptic travelling wave solutions for a new Hamiltonian amplitude equation determining some instabilities of modulated wave train are obtained. By a mere variation of the Jacobian elliptic parameter k2 from zero to one, these solutions are transformed from a trivial one to the known solitary wave solutions.1,2

A Variety of Exact Periodic Wave and Solitary Wave Solutions for the Coupled Higgs Equation

2012 ◽  
Vol 67 (10-11) ◽  
pp. 545-549 ◽  
Author(s):  
Houria Trikia ◽  
Abdul-Majid Wazwazb
Keyword(s):  
Solitary Wave ◽  
Soliton Solutions ◽  
Higgs Field ◽  
Periodic Wave ◽  
Travelling Wave ◽  
Jacobi Elliptic Function ◽  
Travelling Wave Solutions ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Jacobi Elliptic Function Expansion

In this work, the coupled Higgs field equation is studied. The extended Jacobi elliptic function expansion methods are efficiently employed to construct the exact periodic solutions of this model. As a result, many exact travelling wave solutions are obtained which include new shock wave solutions or kink-shaped soliton solutions, solitary wave solutions or bell-shaped soliton solutions, and combined solitary wave solutions are formally obtained.

scholarly journals On the Instability of a Class of Periodic Travelling Wave Solutions of the Modified Boussinesq Equation

2010 ◽  
Vol 2010 ◽  
pp. 1-21
Author(s):  
Lynnyngs Kelly Arruda
Keyword(s):  
Boussinesq Equation ◽  
Elliptic Functions ◽  
Travelling Wave ◽  
Energy Space ◽  
Fundamental Period ◽  
Travelling Wave Solutions ◽  
Wave Solutions ◽  
Periodic Travelling Wave ◽  
Modified Boussinesq Equation

This paper is concerned with instability of periodic travelling wave solutions of the modified Boussinesq equation. Periodic travelling wave solutions with a fixed fundamental periodLwill be constructed by using Jacobi's elliptic functions. It will be shown that these solutions, calleddnoidal waves, are nonlinearly unstable in the energy space for a range of their speeds of propagation and periods.

scholarly journals On some novel exact solutions to the time fractional (2 + 1) dimensional Konopelchenko–Dubrovsky system arising in physical science

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 806-819
Author(s):  
Junaid Akhtar ◽  
Aly R. Seadawy ◽  
Kalim U. Tariq ◽  
Dumitru Baleanu
Keyword(s):  
Solitary Wave ◽  
Physical Science ◽  
Nonlinear Partial Differential Equations ◽  
Sufficient Conditions ◽  
Elliptic Functions ◽  
Travelling Wave ◽  
Solitary Wave Solutions ◽  
Integration Schemes ◽  
Wave Solutions ◽  
Wide Range

AbstractThe purpose of this article is to construct some novel exact travelling and solitary wave solutions of the time fractional (2 + 1) dimensional Konopelchenko–Dubrovsky equation, and two different forms of integration schemes have been utilized in this context. As a result, a variety of bright and dark solitons, kink- and antikink-type solitons, hyperbolic functions, trigonometric functions, elliptic functions, periodic solitary wave solutions and travelling wave solutions are obtained, and the sufficient conditions for the existence of solution are also discussed. Moreover, some of the obtained solutions are illustrated as two- and three-dimensional graphical images by using computational software Mathematica. These types of solutions have a wide range of applications in applied sciences and mathematical physics. The proposed methods are very useful for solving nonlinear partial differential equations arising in physical science and engineering.

scholarly journals Construction of new solitary wave solutions of generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified form of Camassa-Holm equations

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 896-909 ◽  
Author(s):  
Dianchen Lu ◽  
Aly R. Seadawy ◽  
Mujahid Iqbal
Keyword(s):  
Solitary Wave ◽  
Nonlinear Partial Differential Equations ◽  
Research Work ◽  
Traveling Wave Solutions ◽  
Elliptic Functions ◽  
Mapping Method ◽  
New Method ◽  
Auxiliary Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions

AbstractIn this research work, for the first time we introduced and described the new method, which is modified extended auxiliary equation mapping method. We investigated the new exact traveling and families of solitary wave solutions of two well-known nonlinear evaluation equations, which are generalized Zakharov-Kuznetsov-Benjamin-Bona-Mahony and simplified modified forms of Camassa-Holm equations. We used a new technique and we successfully obtained the new families of solitary wave solutions. As a result, these new solutions are obtained in the form of elliptic functions, trigonometric functions, kink and antikink solitons, bright and dark solitons, periodic solitary wave and traveling wave solutions. These new solutions show the power and fruitfulness of this new method. We can solve other nonlinear partial differential equations with the use of this method.

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