scholarly journals The stability spectrum for elliptic solutions to the sine-Gordon equation

2017 ◽  
Vol 360 ◽  
pp. 17-35 ◽  
Author(s):  
Bernard Deconinck ◽  
Peter McGill ◽  
Benjamin L. Segal
Keyword(s):  
Gordon Equation ◽  
Sine Gordon Equation ◽  
Stability Spectrum ◽  
Elliptic Solutions ◽  
The Stability

Solitions and separable elliptic solutions of the sine-Gordon equation

1979 ◽  
Vol 3 (4) ◽  
pp. 265-269 ◽  
Author(s):  
Alan C. Bryan ◽  
Christopher R. Haines ◽  
Allan E. G. Stuart
Keyword(s):  
Gordon Equation ◽  
Sine Gordon Equation ◽  
Elliptic Solutions

scholarly journals Salient features of dressed elliptic string solutions on $$\mathbb {R}\times \hbox {S}^2$$

2019 ◽  
Vol 79 (10) ◽  
Author(s):  
Dimitrios Katsinis ◽  
Ioannis Mitsoulas ◽  
Georgios Pastras
Keyword(s):  
Equation Of State ◽  
Topological Charge ◽  
Gordon Equation ◽  
Qualitative Difference ◽  
Dispersion Relations ◽  
Sine Gordon Equation ◽  
Periodic Disturbances ◽  
Elliptic Solutions ◽  
Salient Features ◽  
Specific Equation

Abstract We study several physical aspects of the dressed elliptic strings propagating on $$\mathbb {R} \times \mathrm {S}^2$$R×S2 and of their counterparts in the Pohlmeyer reduced theory, i.e. the sine-Gordon equation. The solutions are divided into two wide classes; kinks which propagate on top of elliptic backgrounds and non-localised periodic disturbances of the latter. The former class of solutions obey a specific equation of state that is in principle experimentally verifiable in systems which realize the sine-Gordon equation. Among both of these classes, there appears to be a particular class of interest the closed dressed strings. They in turn form four distinct subclasses of solutions. One of those realizes instabilities of the seed elliptic solutions. The existence of such solutions depends on whether a superluminal kink with a specific velocity can propagate on the corresponding elliptic sine-Gordon solution. Unlike the elliptic strings, the dressed ones exhibit interactions among their spikes. These interactions preserve an appropriately defined turning number, which can be associated to the topological charge of the sine-Gordon counterpart. Finally, the dispersion relations of the dressed strings are studied. A qualitative difference between the two wide classes of dressed strings is discovered.

scholarly journals Duality of positive and negative integrable hierarchies via relativistically invariant fields

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
S. Y. Lou ◽  
X. B. Hu ◽  
Q. P. Liu
Keyword(s):  
Integrable Hierarchies ◽  
Gordon Equation ◽  
Formal Series ◽  
Two Dimensional ◽  
Relativistic Invariance ◽  
Sine Gordon Equation ◽  
Recursion Operators ◽  
Symmetry Approach ◽  
Duality Relations ◽  
Duality Problem

Abstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

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