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.

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

2015 ◽  
Vol 25 (03) ◽  
pp. 1550045 ◽  
Author(s):  
Jibin Li ◽  
Fengjuan Chen
Keyword(s):  
Transmission Line ◽  
Singular Systems ◽  
Dynamical Behavior ◽  
Periodic Wave ◽  
Phase Portraits ◽  
Electrical Transmission ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Electrical Transmission Line ◽  
Straight Lines

In this paper, we consider a model which is the modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems and investigating the dynamical behavior, we obtain bifurcations of the phase portraits of the system under different parameter conditions. Corresponding to some special level curves, we derive possible exact explicit parametric representations of solutions (including smooth solitary wave and periodic wave solutions, periodic cusp wave solutions) under different parameter conditions.

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.

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.

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

2016 ◽  
Vol 26 (01) ◽  
pp. 1650011 ◽  
Author(s):  
Jibin Li ◽  
Fengjuan Chen
Keyword(s):  
Transmission Line ◽  
Singular Systems ◽  
Dynamical Behavior ◽  
Phase Portraits ◽  
Solitary Wave Solutions ◽  
Electrical Transmission ◽  
Level Curves ◽  
Wave Solutions ◽  
Electrical Transmission Line ◽  
Straight Lines

In this paper, we consider a modulated equation in a discrete nonlinear electrical transmission line. This model is an integrable planar dynamical system having three singular straight lines. By using the theory of singular systems to investigate the dynamical behavior for this system, we obtain bifurcations of phase portraits under different parameter conditions. Corresponding to some special level curves, we derive exact explicit parametric representations of solutions (including smooth solitary wave solutions, peakons, compactons, periodic cusp wave solutions) under different parameter conditions.

scholarly journals New Exact Solitary Wave Solutions of a Coupled Nonlinear Wave Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
XiaoHua Liu ◽  
CaiXia He
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Undetermined Coefficient ◽  
Planar Dynamical Systems

By using the theory of planar dynamical systems to a coupled nonlinear wave equation, the existence of bell-shaped solitary wave solutions, kink-shaped solitary wave solutions, and periodic wave solutions is obtained. Under the different parametric values, various sufficient conditions to guarantee the existence of the above solutions are given. With the help of three different undetermined coefficient methods, we investigated the new exact explicit expression of all three bell-shaped solitary wave solutions and one kink solitary wave solutions with nonzero asymptotic value for a coupled nonlinear wave equation. The solutions cannot be deduced from the former references.

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.

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.

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