Existence, Uniqueness, and Energy of Approximate Fourier Solutions of Modified Stochastic Sine-Gordon Equation with Power-Law Nonlinearity in 1D

2021 ◽  
Vol 7 (4) ◽  
Author(s):  
Henri Schurz ◽  
Abdallah M. Talafha
Keyword(s):  
Power Law ◽  
Gordon Equation ◽  
Sine Gordon Equation

scholarly journals Static Kinks in Chains of Interacting Atoms

2020 ◽  
Vol 5 (2) ◽  
pp. 35
Author(s):  
Haggai Landa ◽  
Cecilia Cormick ◽  
Giovanna Morigi
Keyword(s):  
Power Law ◽  
Gordon Equation ◽  
Ion Trap ◽  
Finite Size ◽  
Coulomb Repulsion ◽  
Periodic Lattice ◽  
Sine Gordon Equation ◽  
Integral Term ◽  
Power Law Exponent ◽  
A Chain

We theoretically analyse the equation of topological solitons in a chain of particles interacting via a repulsive power-law potential and confined by a periodic lattice. Starting from the discrete model, we perform a gradient expansion and obtain the kink equation in the continuum limit for a power-law exponent n ≥ 1 . The power-law interaction modifies the sine-Gordon equation, giving rise to a rescaling of the coefficient multiplying the second derivative (the kink width) and to an additional integral term. We argue that the integral term does not affect the local properties of the kink, but it governs the behaviour at the asymptotics. The kink behaviour at the center is dominated by a sine-Gordon equation and its width tends to increase with the power law exponent. When the interaction is the Coulomb repulsion, in particular, the kink width depends logarithmically on the chain size. We define an appropriate thermodynamic limit and compare our results with existing studies performed for infinite chains. Our formalism allows one to systematically take into account the finite-size effects and also slowly varying external potentials, such as for instance the curvature in an ion trap.

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.

scholarly journals Physical problem in interpreting classical energy of soliton of sine Gordon equation

2021 ◽  
Vol 1869 (1) ◽  
pp. 012191
Author(s):  
T B Prayitno ◽  
E Budi ◽  
S Sunaryo ◽  
R Fahdiran
Keyword(s):  
Gordon Equation ◽  
Physical Problem ◽  
Sine Gordon Equation ◽  
Classical Energy

On the method for solving the complex sine-Gordon equation

1983 ◽  
Vol 38 (2) ◽  
pp. 60-64 ◽  
Author(s):  
Hang Nian-Ning
Keyword(s):  
Gordon Equation ◽  
Sine Gordon Equation

Numerical modeling of long Josephson junctions in the frame of double sine-gordon equation

2011 ◽  
Vol 3 (3) ◽  
pp. 389-398 ◽  
Author(s):  
P. Kh. Atanasova ◽  
T. L. Boyadjiev ◽  
Yu. M. Shukrinov ◽  
E. V. Zemlyanaya
Keyword(s):  
Numerical Modeling ◽  
Josephson Junctions ◽  
Gordon Equation ◽  
Sine Gordon Equation

scholarly journals Application of DRBEM for 2D sine-Gordon equation

2021 ◽  
Vol 15 (1) ◽  
pp. 225-238
Author(s):  
Nagehan Alsoy-Akgün
Keyword(s):  
Gordon Equation ◽  
Sine Gordon Equation
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