Bifurcations and Exact Traveling Wave Solutions of the Celebrated Green–Naghdi Equations

2017 ◽  
Vol 27 (07) ◽  
pp. 1750114 ◽  
Author(s):  
Zhenshu Wen
Keyword(s):  
Traveling Wave ◽  
Bifurcation Theory ◽  
Blow Up ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

In this paper, we study the bifurcations and exact traveling wave solutions of the celebrated Green–Naghdi equations by using the qualitative theory of differential equations and the bifurcation theory of dynamical systems. We obtain all possible phase portraits of bifurcations of the system under various conditions about the parameters associated with the planar dynamical system. Then we show the existence of traveling wave solutions including solitary wave solutions, blow-up solutions, periodic wave solutions and periodic blow-up solutions, and give their exact explicit expressions. These results can help to understand the dynamical behavior of the traveling wave solutions of the system.

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.

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.

EXACT TRAVELING WAVE SOLUTIONS OF A HIGHER-DIMENSIONAL NONLINEAR EVOLUTION EQUATION

2010 ◽  
Vol 24 (10) ◽  
pp. 1011-1021 ◽  
Author(s):  
JONU LEE ◽  
RATHINASAMY SAKTHIVEL ◽  
LUWAI WAZZAN
Keyword(s):  
Traveling Wave ◽  
Function Method ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions ◽  
Higher Dimensional

The exact traveling wave solutions of (4 + 1)-dimensional nonlinear Fokas equation is obtained by using three distinct methods with symbolic computation. The modified tanh–coth method is implemented to obtain single soliton solutions whereas the extended Jacobi elliptic function method is applied to derive doubly periodic wave solutions for this higher-dimensional integrable equation. The Exp-function method gives generalized wave solutions with some free parameters. It is shown that soliton solutions and triangular solutions can be established as the limits of the Jacobi doubly periodic wave solutions.

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.

scholarly journals Bifurcations of Traveling Wave Solutions for the Coupled Higgs Field Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Shengqiang Tang ◽  
Shu Xia
Keyword(s):  
Field Equation ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Sufficient Conditions ◽  
Higgs Field ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions

By using the bifurcation theory of dynamical systems, we study the coupled Higgs field equation and the existence of new solitary wave solutions, and uncountably infinite many periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All exact explicit parametric representations of the above waves are determined.

ON A CLASS OF SINGULAR NONLINEAR TRAVELING WAVE EQUATIONS (II): AN EXAMPLE OF GCKdV EQUATIONS

2009 ◽  
Vol 19 (06) ◽  
pp. 1995-2007 ◽  
Author(s):  
JIBIN LI ◽  
YI ZHANG ◽  
XIAOHUA ZHAO
Keyword(s):  
Traveling Wave ◽  
Wave Equations ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solutions ◽  
Kdv Equations ◽  
Wave Solutions ◽  
Kink Wave ◽  
Coupled Kdv Equations

By using the method of dynamical systems, we continuously study the dynamical behavior for the first class of singular nonlinear traveling wave systems. As an example, the traveling wave solutions for a generalized coupled KdV equations are discussed. Exact explicit parametric representations of solitary wave solutions, periodic wave solutions and kink wave solutions are given.

Bifurcations and Exact Traveling Wave Solutions of a Modified Nonlinear Schrödinger Equation

2016 ◽  
Vol 26 (06) ◽  
pp. 1650106 ◽  
Author(s):  
KitIan Kou ◽  
Jibin Li
Keyword(s):  
Dynamical Systems ◽  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
One Dimensional ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Planar Dynamical Systems

In this paper, we consider two singular nonlinear planar dynamical systems created from the studies of one-dimensional bright and dark spatial solitons for one-dimensional beams in a nonlocal Kerr-like media. On the basis of the investigation of the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we obtain all possible explicit exact parametric representations of solutions (including solitary wave solutions, periodic wave solutions, peakon and periodic peakons, compacton solutions, etc.) under different parameter conditions.

scholarly journals Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type forK(m,n)Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-23 ◽  
Author(s):  
Xianbin Wu ◽  
Weiguo Rui ◽  
Xiaochun Hong
Keyword(s):  
Exact Solutions ◽  
Traveling Wave ◽  
Soliton Solutions ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Exact Traveling Wave Solutions ◽  
Wave Solutions

By using the integral bifurcation method, we study the nonlinearK(m,n)equation for all possible values ofmandn. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions ofK(n,n),K(2n−1,n),K(3n−2,n),K(4n−3,n), andK(m,1)equations are chosen to illustrate with the concrete features.

SOME TRAVELING WAVE SOLITONS OF THE GREEN–NAGHDI SYSTEM

2011 ◽  
Vol 21 (02) ◽  
pp. 575-585 ◽  
Author(s):  
SHENGFU DENG ◽  
BOLING GUO ◽  
TINGCHUN WANG
Keyword(s):  
Qualitative Analysis ◽  
Traveling Wave ◽  
Autonomous Systems ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Equivalent System ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Soliton Wave Solutions

The traveling wave solutions for an equivalent system of the Green–Naghdi system are considered. The qualitative analysis methods of planar autonomous systems yield their phase portraits. The exact expressions of smooth soliton wave solutions, cusp soliton wave solutions, smooth periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also given. These reveal some new properties of the Green–Naghdi system.

On the Exact Traveling Wave Solutions of (2 + 1)-Dimensional Higher Order Broer–Kaup Equation

2014 ◽  
Vol 24 (01) ◽  
pp. 1450007 ◽  
Author(s):  
Jibin Li
Keyword(s):  
Traveling Wave ◽  
Dynamical Behavior ◽  
Traveling Wave Solutions ◽  
Periodic Wave ◽  
Higher Order ◽  
Solution Component ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave ◽  
Wave Profiles

In this paper, we study the dynamical behavior and exact parametric representations of all traveling wave solutions for (2 + 1)-dimensional higher order Broer–Kaup equation. By using the method of dynamical systems, under different parametric conditions, for the solution component U, exact monotonic and nonmonotonic kink wave solutions, two-peak wave solutions, periodic wave solutions, as well as unbounded traveling wave solutions are obtained. Exact wave profiles of traveling wave solutions for all solution components U, V, W, P are shown.

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