Constructing Quasi-Periodic Wave Solutions of Differential-Difference Equation by Hirota Bilinear Method

2016 ◽  
Vol 71 (12) ◽  
pp. 1159-1165
Author(s):  
Qi Wang
Keyword(s):  
Difference Equation ◽  
Wave Solution ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Periodic Wave Solution ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Differential Difference Equation ◽  
Hirota Bilinear

AbstractIn the present paper, based on the Riemann theta function, the Hirota bilinear method is extended to directly construct a kind of quasi-periodic wave solution of a new integrable differential-difference equation. The asymptotic property of the quasi-periodic wave solution is analyzed in detail. It will be shown that quasi-periodic wave solution converge to the soliton solutions under certain conditions and small amplitude limit.

Construction of Quasi-Periodic Wave Solutions for Differential- Difference Equation

2012 ◽  
Vol 67 (1-2) ◽  
pp. 21-28 ◽  
Author(s):  
Y. C. Hon ◽  
Qi Wang
Keyword(s):  
Difference Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Constructive Method ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Equation Analysis ◽  
Differential Difference Equation ◽  
Generalized Differential

Based on the use of the Hirota bilinear method and the Riemann theta function, we develop in this paper a constructive method for obtaining explicit quasi-periodic wave solutions of a new integrable generalized differential-difference equation. Analysis on the asymptotic property of the quasiperiodic wave solutions is given, and it is shown that the quasi-periodic wave solutions converge to the soliton solutions under certain conditions.

EXACT SOLITON SOLUTIONS AND QUASI-PERIODIC WAVE SOLUTIONS TO THE FORCED VARIABLE-COEFFICIENT KdV EQUATION

2012 ◽  
Vol 26 (19) ◽  
pp. 1250072 ◽  
Author(s):  
YI ZHANG ◽  
ZHILONG CHENG
Keyword(s):  
Variable Coefficient ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Time Dependent ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Hirota Bilinear ◽  
Exact Soliton Solutions

In this paper, the time-dependent variable-coefficient KdV equation with a forcing term is considered. Based on the Hirota bilinear method, the bilinear form of this equation is obtained, and the multi-soliton solutions are studied. Then the periodic wave solutions are obtained by using Riemann theta function, and it is also shown that classical soliton solutions can be reduced from the periodic wave solutions.

Interaction among a lump, periodic waves, and kink solutions to the KP-BBM equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Junjie Li ◽  
Jalil Manafian ◽  
Nguyen Thi Hang ◽  
Dinh Tran Ngoc Huy ◽  
Alla Davidyants
Keyword(s):  
Soliton Solutions ◽  
Periodic Wave ◽  
Contour Plot ◽  
Periodic Waves ◽  
Hirota Bilinear Method ◽  
Algebraic Equations ◽  
Bilinear Method ◽  
Kink Wave ◽  
Hirota Bilinear ◽  
Bbm Equation

Abstract The Hirota bilinear method is prepared for searching the diverse soliton solutions to the (2+1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony (KP-BBM) equation. Also, the Hirota bilinear method is used to find the lump and interaction with two stripe soliton solutions. Interaction among lumps, periodic waves, and one-kink soliton solutions are investigated. Also, the solitary wave, periodic wave, and cross-kink wave solutions are examined for the KP-BBM equation. The graphs for various parameters are plotted to contain a 3D plot, contour plot, density plot, and 2D plot. We construct the exact lump and interaction among other types of solutions, by solving the underdetermined nonlinear system of algebraic equations with the associated parameters. Finally, analysis and graphical simulation are presented to show the dynamical characteristics of our solutions, and the interaction behaviors are revealed. The existing conditions are employed to discuss the available got solutions.

The quasi-periodic solutions for the supersymmetric variable-coefficient KdV equation

2020 ◽  
Vol 34 (16) ◽  
pp. 2050171
Author(s):  
Chao Dong ◽  
Shu-Fang Deng
Keyword(s):  
Variable Coefficient ◽  
Kdv Equation ◽  
Soliton Solutions ◽  
Periodic Wave ◽  
Singularity Analysis ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Painlevé Property ◽  
Hirota Bilinear

The supersymmetric variable-coefficient KdV equation is presented and it admits Painlevé property by the standard singularity analysis. Based on Hirota bilinear method and Riemann theta function, one and two quasi-periodic wave solutions for the supersymmetric variable-coefficient KdV equation are studied. In addition, we give the asymptotic relations between quasi-periodic wave solutions and soliton solutions.

Y-shaped soliton solutions for the (2+1)-dimensional bidirectional Sawada–Kotera equation

2021 ◽  
Author(s):  
Shuxin Yang ◽  
Zhao Zhang ◽  
Biao Li
Keyword(s):  
Soliton Solutions ◽  
Complex Interaction ◽  
High Order ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Dynamic Behaviors ◽  
Interaction Solutions ◽  
Localized Waves ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.

MULTI-SOLITON SOLUTIONS AND THEIR INTERACTIONS FOR THE (2+1)-DIMENSIONAL SAWADA-KOTERA MODEL WITH TRUNCATED PAINLEVÉ EXPANSION, HIROTA BILINEAR METHOD AND SYMBOLIC COMPUTATION

2009 ◽  
Vol 23 (25) ◽  
pp. 5003-5015 ◽  
Author(s):  
XING LÜ ◽  
TAO GENG ◽  
CHENG ZHANG ◽  
HONG-WU ZHU ◽  
XIANG-HUA MENG ◽  
...  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Symbolic Computation ◽  
Bilinear Form ◽  
Soliton Solutions ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Truncated Painlevé Expansion ◽  
Hirota Bilinear ◽  
Bilinear Equations

In this paper, the (2+1)-dimensional Sawada-Kotera equation is studied by the truncated Painlevé expansion and Hirota bilinear method. Firstly, based on the truncation of the Painlevé series we obtain two distinct transformations which can transform the (2+1)-dimensional Sawada-Kotera equation into two bilinear equations of different forms (which are shown to be equivalent). Then employing Hirota bilinear method, we derive the analytic one-, two- and three-soliton solutions for the bilinear equations via symbolic computation. A formula which denotes the N-soliton solution is given simultaneously. At last, the evolutions and interactions of the multi-soliton solutions are graphically discussed as well. It is worthy to be noted that the truncated Painlevé expansion provides a useful dependent variable transformation which transforms a partial differential equation into its bilinear form and by means of the bilinear form, further study of the original partial differential equation can be conducted.

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.

Multiple-soliton solutions for coupled KdV and coupled KP systems

2009 ◽  
Vol 87 (12) ◽  
pp. 1227-1232 ◽  
Author(s):  
Abdul-Majid Wazwaz
Keyword(s):  
Soliton Solutions ◽  
Resonance Phenomenon ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Multiple Soliton Solutions ◽  
Two Systems ◽  
Kp Equations ◽  
Hirota Bilinear ◽  
Singular Soliton ◽  
Multiple Singular Soliton Solutions

In this work we study two systems of coupled KdV and coupled KP equations. The Hirota bilinear method is applied to show that these two systems are completely integrable. Multiple-soliton solutions and multiple singular-soliton solutions are derived for each system. The resonance phenomenon is examined as well.

RIEMANN THETA FUNCTIONS SOLUTIONS WITH RATIONAL CHARACTERISTICS FOR (2+1)-DIMENSIONAL SINH-GORDON EQUATION

2010 ◽  
Vol 24 (06) ◽  
pp. 575-584
Author(s):  
YANG FENG ◽  
HONG-QING ZHANG
Keyword(s):  
Gordon Equation ◽  
Theta Functions ◽  
Periodic Wave ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Wave Numbers ◽  
Riemann Theta Functions ◽  
The Waves ◽  
Hirota Bilinear

In this letter, we use the Riemann theta functions with rational characteristics and the Hirota bilinear method to construct quasi-periodic wave solutions for (2+1)-dimensional sinh-Gordon equation. This method not only conveniently obtains quasi-periodic solutions of nonlinear equations, but also directly gets the explicit expressions of frequencies, wave numbers, phase and amplitudes for the waves.

scholarly journals Solitary wave solutions and integrability for generalized nonlocal complex modified Korteweg-de Vries (cmKdV) equations

2021 ◽  
Vol 6 (10) ◽  
pp. 11046-11075
Author(s):  
Wen-Xin Zhang ◽  
◽  
Yaqing Liu
Keyword(s):  
Soliton Solutions ◽  
Solitary Wave Solutions ◽  
Hirota Bilinear Method ◽  
Reverse Time ◽  
Local Equation ◽  
Bilinear Method ◽  
Dynamical Behaviors ◽  
Wave Solutions ◽  
Nonlocal Equations ◽  
Hirota Bilinear

<abstract><p>In this paper, the reverse space cmKdV equation, the reverse time cmKdV equation and the reverse space-time cmKdV equation are constructed and each of three types diverse soliton solutions is derived based on the Hirota bilinear method. The Lax integrability of three types of nonlocal equations is studied from local equation by using variable transformations. Based on exact solution formulae of one- and two-soliton solutions of three types of nonlocal cmKdV equation, some figures are used to describe the soliton solutions. According to the dynamical behaviors, it can be found that these solutions possess novel properties which are different from the ones of classical cmKdV equation.</p></abstract>

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