scholarly journals Localized structures on librational and rotational travelling waves in the sine-Gordon equation

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200490
Author(s):  
Dmitry E. Pelinovsky ◽  
Robert E. White
Keyword(s):  
Exact Solution ◽  
Classical Limit ◽  
Gordon Equation ◽  
Rogue Wave ◽  
Modulational Instability ◽  
Travelling Waves ◽  
Sine Gordon Equation ◽  
Localized Structures ◽  
Modulational Stability ◽  
Rotational Waves

We derive exact solutions to the sine-Gordon equation describing localized structures on the background of librational and rotational travelling waves. In the case of librational waves, the exact solution represents a localized spike in space-time coordinates (a rogue wave) that decays to the periodic background algebraically fast. In the case of rotational waves, the exact solution represents a kink propagating on the periodic background and decaying algebraically in the transverse direction to its propagation. These solutions model the universal patterns in the dynamics of fluxon condensates in the semi-classical limit. The different dynamics are related to modulational instability of the librational waves and modulational stability of the rotational waves.

Separation Transformation and New Exact Solutions for the (1+N)-Dimensional Triple Sine-Gordon Equation

2011 ◽  
Vol 66 (1-2) ◽  
pp. 19-23 ◽  
Author(s):  
Yifang Liu ◽  
Jiuping Chen ◽  
Weifeng Hu ◽  
Li-Li Zhu
Keyword(s):  
Exact Solution ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Gordon Equation ◽  
Transformation Method ◽  
Nonlinear Wave Equations ◽  
High Dimensional ◽  
Sine Gordon Equation ◽  
New Exact Solutions ◽  
Localized Structures

The separation transformation method is extended to the (1+N)-dimensional triple Sine-Gordon equation and a special type of implicitly exact solution for this equation is obtained. The exact solution contains an arbitrary function which may lead to abundant localized structures of the high dimensional nonlinear wave equations. The separation transformation method in this paper can also be applied to other kinds of high-dimensional nonlinear wave equations

scholarly journals Travelling waves in a singularly perturbed sine-Gordon equation

2003 ◽  
Vol 180 (1-2) ◽  
pp. 40-70 ◽  
Author(s):  
Gianne Derks ◽  
Arjen Doelman ◽  
Stephan A van Gils ◽  
Timco Visser
Keyword(s):  
Gordon Equation ◽  
Travelling Waves ◽  
Singularly Perturbed ◽  
Sine Gordon Equation

scholarly journals Solutions of Two-Dimensional Nonlinear Sine-Gordon Equation via Triple Laplace Transform Coupled with Iterative Method

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Alemayehu Tamirie Deresse ◽  
Yesuf Obsie Mussa ◽  
Ademe Kebede Gizaw
Keyword(s):  
Exact Solution ◽  
Iterative Method ◽  
Laplace Transform ◽  
Gordon Equation ◽  
Type Systems ◽  
Mathematical Physics ◽  
Test Problems ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
The Given

This article presents triple Laplace transform coupled with iterative method to obtain the exact solution of two-dimensional nonlinear sine-Gordon equation (NLSGE) subject to the appropriate initial and boundary conditions. The noise term in this equation is vanished by successive iterative method. The proposed technique has the advantage of producing exact solution, and it is easily applied to the given problems analytically. Four test problems from mathematical physics are taken to show the accuracy, convergence, and the efficiency of the proposed method. Furthermore, the results indicate that the introduced method is promising for solving other type systems of NLPDEs.

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