scholarly journals On periodic waves of the nonlinear systems

1998 ◽  
Vol 20 (4) ◽  
pp. 11-19
Author(s):  
Le Xuan Can
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Nonlinear Partial Differential Equation ◽  
Sufficient Conditions ◽  
Periodic Wave ◽  
Periodic Waves ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Electric Cables ◽  
Necessary And Sufficient

The paper is concerned with the solvability and approximate solution of the nonlinear partial differential equation describing the periodic wave propagation. Necessary and sufficient conditions for the existence of the periodic wave solutions are obtained. An approximate method for solving the equation is presented. As an illustrative example, the equation of periodic waves of the electric cables is considered.

scholarly journals Double Periodic Wave Solutions of the (2 + 1)-Dimensional Sawada-Kotera Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zhonglong Zhao ◽  
Yufeng Zhang ◽  
Tiecheng Xia
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Theta Function ◽  
Nonlinear Partial Differential Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Bilinear Forms ◽  
Periodic Wave Solutions ◽  
Wave Solutions

Based on a general Riemann theta function and Hirota’s bilinear forms, we devise a straightforward way to explicitly construct double periodic wave solution of(2+1)-dimensional nonlinear partial differential equation. The resulting theory is applied to the(2+1)-dimensional Sawada-Kotera equation, thereby yielding its double periodic wave solutions. The relations between the periodic wave solutions and soliton solutions are rigorously established by a limiting procedure.

scholarly journals New Traveling Wave Solutions of the Higher Dimensional Nonlinear Partial Differential Equation by the Exp-Function Method

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hasibun Naher ◽  
Farah Aini Abdullah ◽  
M. Ali Akbar
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solutions ◽  
Traveling Wave ◽  
Function Method ◽  
Nonlinear Partial Differential Equation ◽  
Traveling Wave Solutions ◽  
Partial Differential ◽  
Wave Solutions ◽  
Higher Dimensional

We construct new analytical solutions of the (3+1)-dimensional modified KdV-Zakharov-Kuznetsev equation by the Exp-function method. Plentiful exact traveling wave solutions with arbitrary parameters are effectively obtained by the method. The obtained results show that the Exp-function method is effective and straightforward mathematical tool for searching analytical solutions with arbitrary parameters of higher-dimensional nonlinear partial differential equation.

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 Numerical calculation of N -periodic wave solutions to coupled KdV–Toda-type equations

2021 ◽  
Vol 477 (2245) ◽  
pp. 20200752
Author(s):  
Yingnan Zhang ◽  
Xingbiao Hu ◽  
Jianqing Sun
Keyword(s):  
Toda Lattice ◽  
Kdv Equation ◽  
Direct Method ◽  
Periodic Wave ◽  
Periodic Waves ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Numerical Process ◽  
Relativistic Toda Lattice ◽  
Ito Equation

In this paper, we study the N -periodic wave solutions of coupled Korteweg–de Vries (KdV)–Toda-type equations. We present a numerical process to calculate the N -periodic waves based on the direct method of calculating periodic wave solutions proposed by Akira Nakamura. Particularly, in the case of N  = 3, we give some detailed examples to show the N -periodic wave solutions to the coupled Ramani equation, the Hirota–Satsuma coupled KdV equation, the coupled Ito equation, the Blaszak–Marciniak lattice, the semi-discrete KdV equation, the Leznov lattice and a relativistic Toda lattice.

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.

scholarly journals Peakon, Solitary and Smooth Periodic Wave Solutions for a Modification of theK(2,2)Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Qing Meng ◽  
Bin He
Keyword(s):  
Dynamical Systems ◽  
Solitary Wave ◽  
Periodic Wave ◽  
Point Of View ◽  
Phase Portraits ◽  
Periodic Waves ◽  
Solitary Wave Solutions ◽  
Bifurcation Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions

We consider a modification of theK(2,2)equationut=2uuxxx+2kuxuxx+2uuxusing the bifurcation method of dynamical systems and the method of phase portraits analysis. From dynamic point of view, some peakons, solitary, and smooth periodic waves are found and their exact parametric representations are presented. Also, the coexistence of peakon and solitary wave solutions is investigated.

scholarly journals EXACT PERIODIC WAVE SOLUTIONS OF A SINGULAR INTEGRABLE EQUATION

2011 ◽  
Vol 16 (1) ◽  
pp. 315-325
Author(s):  
Shaolong Xie ◽  
Bin Gao
Keyword(s):  
Dynamical Systems ◽  
Periodic Wave ◽  
Integrable Equation ◽  
Periodic Waves ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Fixed Parameter ◽  
Periodic Cusp Waves

In this paper, theory of dynamical systems is employed to investigate periodic waves of a singular integrable equation. These periodic waves contain smooth periodic waves, periodic cusp waves and periodic cusp loop waves. Under fixed parameter conditions, their exact parametric expressions are given.

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