An improved numerical scheme for the sine-Gordon equation in 2+1 dimensions

2008 ◽  
Vol 75 (7) ◽  
pp. 787-799 ◽  
Author(s):  
A. G. Bratsos
Keyword(s):  
Numerical Scheme ◽  
Gordon Equation ◽  
Sine Gordon Equation

Chebyshev Wavelet Quasilinearization Scheme for Coupled Nonlinear Sine-Gordon Equations

2016 ◽  
Vol 12 (1) ◽  
Author(s):  
K. Harish Kumar ◽  
V. Antony Vijesh
Keyword(s):  
Basis Function ◽  
Numerical Scheme ◽  
Function Method ◽  
Gordon Equation ◽  
Basic Function ◽  
Sine Gordon Equation ◽  
One Dimensional ◽  
Improve Accuracy ◽  
Chebyshev Wavelet ◽  
Grid Points

Radial basis function (RBF) has been found useful for solving coupled sine-Gordon equation with initial and boundary conditions. Though this approach produces moderate accuracy in a larger domain, it requires more grid points. In the present study, we develop an alternative numerical scheme for solving one-dimensional coupled sine-Gordon equation to improve accuracy and to reduce grid points. To achieve these objectives, we make use of a wavelet scheme and solve coupled sine-Gordon equation. Based on the numerical results from the wavelet-based scheme, we conclude that our proposed method is more efficient than the radial basic function method in terms of accuracy.

scholarly journals The Spectral Method for Solving Sine-Gordon Equation Using a New Orthogonal Polynomial

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Zoleikha Soori ◽  
Azim Aminataei
Keyword(s):  
Orthogonal Polynomial ◽  
Spectral Method ◽  
Numerical Scheme ◽  
Gordon Equation ◽  
Weighting Function ◽  
Sine Gordon Equation ◽  
One Dimensional

We have presented a numerical scheme for solving one-dimensional nonlinear sine-Gordon equation. We apply the spectral method with a basis of a new orthogonal polynomial which is orthogonal over the interval with weighting function one. The results show the accuracy and efficiency of the proposed method.

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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