scholarly journals Isospectral flow in loop algebras and quasiperiodic solutions of the sine‐Gordon equation

1993 ◽  
Vol 34 (8) ◽  
pp. 3518-3526 ◽  
Author(s):  
J. Harnad ◽  
M.‐A. Wisse
Keyword(s):  
Gordon Equation ◽  
Sine Gordon Equation ◽  
Quasiperiodic Solutions ◽  
Isospectral Flow ◽  
Loop Algebras

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.

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 The validity of modulation equations for extended systems with cubic nonlinearities

1992 ◽  
Vol 122 (1-2) ◽  
pp. 85-91 ◽  
Author(s):  
Pius Kirrmann ◽  
Guido Schneider ◽  
Alexander Mielke
Keyword(s):  
Gordon Equation ◽  
Landau Equation ◽  
Hyperbolic Problems ◽  
Sine Gordon Equation ◽  
Ginzburg Landau ◽  
Modulation Equation ◽  
Modulation Equations ◽  
Quadratic Interaction ◽  
Long Time ◽  
Dissipative Case

SynopsisModulation equations play an essential role in the understanding of complicated systems near the threshold of instability. Here we show that the modulation equation dominates the dynamics of the full problem locally, at least over a long time-scale. For systems with no quadratic interaction term, we develop a method which is much simpler than previous ones. It involves a careful bookkeeping of errors and an estimate of Gronwall type.As an example for the dissipative case, we find that the Ginzburg–Landau equation is the modulation equation for the Swift–Hohenberg problem. Moreover, the method also enables us to handle hyperbolic problems: the nonlinear Schrodinger equation is shown to describe the modulation of wave packets in the Sine–Gordon equation.

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