scholarly journals Kink-Like Wave and Compacton-Like Wave Solutions for a Two-Component Fornberg-Whitham Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Shaoyong Li ◽  
Ming Song
Keyword(s):  
Wave Solution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Wave Solutions ◽  
Two Component ◽  
New Phenomena ◽  
Whitham Equation

Using bifurcation method and numerical simulation approach of dynamical systems, we study a two-component Fornberg-Whitham equation. Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, two new phenomena are revealed; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. Exact implicit or parameter expressions of these solutions are given.

scholarly journals The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(m,n)Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Shaoyong Li ◽  
Zhengrong Liu
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Simulation Approach ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We investigate the traveling wave solutions and their bifurcations for the BBM-likeB(m,n)equationsut+αux+β(um)x−γ(un)xxt=0by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-likeB(3,2)equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-likeB(4,2)equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.

Bifurcations and Exact Solutions of Generalized Two-Component Peakon Type Dual Systems

2019 ◽  
Vol 29 (09) ◽  
pp. 1950128
Author(s):  
Jianli Liang ◽  
Jibin Li ◽  
Yi Zhang
Keyword(s):  
Traveling Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Wave Solution ◽  
Wave System ◽  
Dual Systems ◽  
Two Component

This paper investigates two generalized two-component peakon type dual systems, which can be reduced to the same planar dynamical systems via the dynamical system approach and the theory of singular traveling wave systems, where one of them contains the two-component Camassa–Holm system. By bifurcation analysis on the corresponding traveling wave system, we obtain the phase portraits and derive possible exact traveling wave solutions that include solitary wave solution, peakon and anti-peakon, pseudo-peakon, periodic peakon, compacton and periodic wave solution. Our results are also applicable to the two-component Camassa–Holm equation.

Cross-kink wave, solitary, dark, and periodic wave solutions by bilinear and He’s variational direct methods for the KP–BBM equation

2021 ◽  
Author(s):  
Baolin Feng ◽  
Jalil Manafian ◽  
Onur Alp Ilhan ◽  
Amitha Manmohan Rao ◽  
Anand H. Agadi
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Wave Solution ◽  
Solitary Wave Solution ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Wave Solutions ◽  
Ito Equation ◽  
Bbm Equation

This paper deals with cross-kink waves in the (2+1)-dimensional KP–BBM equation in the incompressible fluid. Based on Hirota’s bilinear technique, cross-kink solutions related to KP–BBM equation are constructed. Taking the special reduction, the exact expression of different types of solutions comprising exponential, trigonometric and hyperbolic functions is obtained. Moreover, He’s variational direct method (HVDM) based on the variational theory and Ritz-like method is employed to construct the abundant traveling wave solutions of the (2+1)-dimensional generalized Hirota–Satsuma–Ito equation. These traveling wave solutions include kinky dark solitary wave solution, dark solitary wave solution, bright solitary wave solution, periodic wave solution and so on, which are all depending on the initial hypothesis for the Ritz-like method. In continuation, the modulation instability is engaged to discuss the stability of the obtained solutions. Moreover, the rational [Formula: see text] method on the generalized Hirota–Satsuma–Ito equation is investigated. The applicability and effectiveness of the acquired solutions are presented through the numerical results in the form of 3D and 2D graphs. A variety of interactions are illustrated analytically and graphically. The influence of parameters on propagation is analyzed and summarized. The results and phenomena obtained in this paper enrich the dynamic behavior of the evolution of nonlinear waves.

scholarly journals Integral Bifurcation Method together with a Translation-Dilation Transformation for Solving an Integrable 2-Component Camassa-Holm Shallow Water System

2012 ◽  
Vol 2012 ◽  
pp. 1-21 ◽  
Author(s):  
Weiguo Rui ◽  
Yao Long
Keyword(s):  
Shallow Water ◽  
Traveling Wave ◽  
Wave Solution ◽  
Water System ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Bifurcation Method ◽  
Wave Solutions ◽  
Kink Wave ◽  
Shallow Water System

An integrable 2-component Camassa-Holm (2-CH) shallow water system is studied by using integral bifurcation method together with a translation-dilation transformation. Many traveling wave solutions of nonsingular type and singular type, such as solitary wave solutions, kink wave solutions, loop soliton solutions, compacton solutions, smooth periodic wave solutions, periodic kink wave solution, singular wave solution, and singular periodic wave solution are obtained. Further more, their dynamic behaviors are investigated. It is found that the waveforms of some traveling wave solutions vary with the changes of parameter, that is to say, the dynamic behavior of these waves partly depends on the relation of the amplitude of wave and the level of water.

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.

COEXISTENCE OF MULTIFARIOUS EXACT NONLINEAR WAVE SOLUTIONS FOR GENERALIZED b-EQUATION

2010 ◽  
Vol 20 (10) ◽  
pp. 3193-3208 ◽  
Author(s):  
RUI LIU
Keyword(s):  
Nonlinear Wave ◽  
Wave Solution ◽  
Wave Speed ◽  
Solitary Wave Solution ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Wave Solutions ◽  
Special Cases ◽  
Singular Wave ◽  
Nonlinear Wave Solutions

In this paper, we consider the generalized b-equation ut - uxxt + (b + 1)u2ux = buxuxx + uxxx. For a given constant wave speed, we investigate the coexistence of multifarious exact nonlinear wave solutions including smooth solitary wave solution, peakon wave solution, smooth periodic wave solution, single singular wave solution and periodic singular wave solution. Not only is the coexistence shown, but the concrete expressions are given via phase analysis and special integrals. From our work, it can be seen that the types of exact nonlinear wave solutions of the generalized b-equation are more than that of the b-equation. Many previous results are turned to our special cases. Also, some conjectures and questions are presented.

BIFURCATIONS AND EXACT TRAVELING WAVE SOLUTIONS OF THE GENERALIZED TWO-COMPONENT CAMASSA–HOLM EQUATION

2012 ◽  
Vol 22 (12) ◽  
pp. 1250305 ◽  
Author(s):  
JIBIN LI ◽  
ZHIJUN QIAO
Keyword(s):  
Solitary Wave ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Breaking Wave ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Two Component ◽  
Kink Wave

In this paper, we apply the method of dynamical systems to a generalized two-component Camassa–Holm system. Through analysis, we obtain solitary wave solutions, kink and anti-kink wave solutions, cusp wave solutions, breaking wave solutions, and smooth and nonsmooth periodic wave solutions. To guarantee the existence of these solutions, we give constraint conditions among the parameters associated with the generalized Camassa–Holm system. Choosing some special parameters, we obtain exact parametric representations of the traveling wave solutions.

scholarly journals Bounded Traveling Wave Solutions and Their Relations for the Generalized HD Type Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-15
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Solitary Wave ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Sufficient Conditions ◽  
Traveling Wave Solutions ◽  
Type Equation ◽  
Periodic Wave ◽  
Point Of View ◽  
Bifurcation Method ◽  
Wave Solutions

The generalized HD type equation is studied by using the bifurcation method of dynamical systems. From a dynamic point of view, the existence of different kinds of traveling waves which include periodic loop soliton, periodic cusp wave, smooth periodic wave, loop soliton, cuspon, smooth solitary wave, and kink-like wave is proved and the sufficient conditions to guarantee the existence of the above solutions in different regions of the parametric space are given. Also, all possible exact parametric representations of the bounded waves are presented and their relations are stated.

scholarly journals Traveling Wave Solutions for the Generalized Zakharov Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Traveling Wave ◽  
Blow Up ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We use the bifurcation method of dynamical systems to study the traveling wave solutions for the generalized Zakharov equations. A number of traveling wave solutions are obtained. Those solutions contain explicit periodic wave solutions, periodic blow-up wave solutions, unbounded wave solutions, kink profile solitary wave solutions, and solitary wave solutions. Relations of the traveling wave solutions are given. Some previous results are extended.

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