PERIODIC WAVE SOLUTIONS FOR POCHHAMMER–CHREE EQUATION WITH FIVE ORDER NONLINEAR TERM AND THEIR RELATIONSHIP WITH SOLITARY WAVE SOLUTIONS

2010 ◽  
Vol 24 (19) ◽  
pp. 3769-3783 ◽  
Author(s):  
WEIGUO ZHANG ◽  
YAN ZHAO ◽  
GANG LIU ◽  
TONGKE NING
Keyword(s):  
Solitary Wave ◽  
Nonlinear Term ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Solitary Wave Solutions ◽  
Systematic Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Fifth Order ◽  
The Given

In this paper, periodic wave solutions for Pochhammer–Chree equation (PC-equation) with fifth order nonlinear term and their relationship with solitary wave solutions are studied. By designing innovative structure of solution, sixteen bounded periodic wave solutions in fractional form of Jacobi elliptic function (JEF) for PC-equation are given. Furthermore, global phase figure in the plane of the traveling solution for the PC-equation are obtained through dynamic systematic method, we indicate the region in the phase where the given sixteen solutions for PC-equation belong to. We find that two couples of these solutions change into two bell profile solitary wave solutions as k → 1 and four solutions change into four periodic wave solutions in fractional form of cosine function as k → 0. Finally, four figures are shown to describe the evolvement from periodic wave solutions to bell profile solitary wave solutions as k → 1.

scholarly journals Extended Jacobi Elliptic Function Expansion Method to the ZK-MEW Equation

2011 ◽  
Vol 2011 ◽  
pp. 1-11 ◽  
Author(s):  
Weimin Zhang
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Expansion Method ◽  
Periodic Wave ◽  
Jacobi Elliptic Function ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Function Expansion ◽  
Jacobi Elliptic Function Expansion

The extended Jacobi elliptic function expansion method is applied for Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. With the aid of symbolic computation, we construct some new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.

scholarly journals Periodic Wave Solutions and Their Limits for the Generalized KP-BBM Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Ming Song ◽  
Zhengrong Liu
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Blow Up ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave ◽  
Bbm Equation

We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limits for the generalized KP-BBM equation. A number of explicit periodic wave solutions are obtained. These solutions contain smooth periodic wave solutions and periodic blow-up solutions. Their limits contain periodic wave solutions, kink wave solutions, unbounded wave solutions, blow-up wave solutions, and solitary wave solutions.

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.

TRAVELING WAVES FOR AN INTEGRABLE HIGHER ORDER KDV TYPE WAVE EQUATIONS

2006 ◽  
Vol 16 (08) ◽  
pp. 2235-2260 ◽  
Author(s):  
JIBIN LI ◽  
JIANHONG WU ◽  
HUAIPING ZHU
Keyword(s):  
Solitary Wave ◽  
Wave Equations ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Higher Order ◽  
Solitary Wave Solutions ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Kink Wave ◽  
Planar Dynamical Systems

Using the method of planar dynamical systems to a higher order wave equations of KdV type, the existence of smooth and nonsmooth solitary wave, kink wave and uncountably infinite many periodic wave solutions is proved. In different regions of the parametric space, the sufficient conditions to guarantee the existence of the above solutions are given. In some spatial conditions, the exact explicit parametric representations of solitary wave solutions are determined.

scholarly journals Time Evolution of Folded (2+1)-Dimensional Solitary Waves

2009 ◽  
Vol 64 (5-6) ◽  
pp. 309-314 ◽  
Author(s):  
Song-Hua Ma ◽  
Yi-Pin Lu ◽  
Jian-Ping Fang ◽  
Zhi-Jie Lv
Keyword(s):  
Solitary Wave ◽  
Elliptic Function ◽  
Water Wave ◽  
Periodic Wave ◽  
Wave System ◽  
Solitary Wave Solutions ◽  
Variable Separation ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Variable Separation Approach

Abstract With an extended mapping approach and a linear variable separation approach, a series of solutions (including theWeierstrass elliptic function solutions, solitary wave solutions, periodic wave solutions and rational function solutions) of the (2+1)-dimensional modified dispersive water-wave system (MDWW) is derived. Based on the derived solutions and using some multi-valued functions, we find a few new folded solitary wave excitations.

scholarly journals Solitary Wave and Periodic Wave Solutions for a Class of Singular p-Laplacian Systems with Impulsive Effeects

2018 ◽  
Vol 23 (1) ◽  
pp. 17-32 ◽  
Author(s):  
Fanchao Kong ◽  
Zhiguo Luo ◽  
Hongjun Qiu
Keyword(s):  
Solitary Wave ◽  
Mountain Pass Theorem ◽  
Periodic Wave ◽  
Impulsive Effects ◽  
Approximation Technique ◽  
Solitary Wave Solutions ◽  
Sufficient Criterion ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Innovative Method

This work deals with the existence of periodic wave solutions and nonexistence of solitary wave solutions for a class of second-order singular p-Laplacian systems with impulsive effects. A sufficient criterion for the solutions of the considered system is provided via an innovative method of the mountain pass theorem and an approximation technique. Some corresponding results in the literature can be enriched and extended.

Exact Solutions and Bifurcations of a Modulated Equation in a Discrete Nonlinear Electrical Transmission Line (I)

2015 ◽  
Vol 25 (01) ◽  
pp. 1550016 ◽  
Author(s):  
Jibin Li ◽  
Lin Jiang
Keyword(s):  
Solitary Wave ◽  
Transmission Line ◽  
Dynamical Behavior ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Electrical Transmission ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Electrical Transmission Line ◽  
Planar Dynamical Systems

In this paper, we consider a model which is a modulated equation in a discrete nonlinear electrical transmission line. By investigating the dynamical behavior and bifurcations of solutions of the planar dynamical systems, we derive all explicit exact parametric representations of solutions (including smooth solitary wave solutions, smooth periodic wave solutions, peakons, compactons, periodic cusp wave solutions, etc.) under different parameter conditions.

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 Solitary and Periodic Wave Solutions of Sasa–Satsuma Equation and Their Relationship with Hamilton Energy

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17 ◽  
Author(s):  
Weiguo Zhang ◽  
Xingqian Ling ◽  
Bei-Bei Wang ◽  
Shaowei Li
Keyword(s):  
Solitary Wave ◽  
Integral Method ◽  
First Integral ◽  
Periodic Wave ◽  
Evolutionary Relationships ◽  
Solitary Wave Solutions ◽  
First Integral Method ◽  
Periodic Wave Solutions ◽  
Hamilton Energy ◽  
Wave Solutions

In this paper, we study the exact solitary wave solutions and periodic wave solutions of the S-S equation and give the relationships between solutions and the Hamilton energy of their amplitudes. First, on the basis of the theory of dynamical system, we make qualitative analysis on the amplitudes of solutions. Then, by using undetermined hypothesis method, the first integral method, and the appropriate transformation, two bell-shaped solitary wave solutions and six exact periodic wave solutions are obtained. Furthermore, we discuss the evolutionary relationships between these solutions and find that the appearance of these solutions for the S-S equation is essentially determined by the value which the Hamilton energy takes. Finally, we give some diagrams which show the changing process from the periodic wave solutions to the solitary wave solutions when the Hamilton energy changes.

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