Modulational and numerical solutions for the steady discrete Sine–Gordon equation in two space dimensions

2009 ◽  
Vol 238 (14) ◽  
pp. 1229-1240 ◽  
Author(s):  
L.A. Cisneros ◽  
J. Ize ◽  
A.A. Minzoni
Keyword(s):  
Gordon Equation ◽  
Numerical Solutions ◽  
Sine Gordon Equation ◽  
Two Space Dimensions

Numerical solutions of a damped Sine-Gordon equation in two space variables

1995 ◽  
Vol 29 (4) ◽  
pp. 347-369 ◽  
Author(s):  
K. Djidjeli ◽  
W. G. Price ◽  
E. H. Twizell
Keyword(s):  
Gordon Equation ◽  
Numerical Solutions ◽  
Sine Gordon Equation

Polynomial conserved densities for the sine-Gordon equations

1977 ◽  
Vol 352 (1671) ◽  
pp. 481-503 ◽  
Keyword(s):  
Inverse Scattering ◽  
Wave Equations ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Soliton Solutions ◽  
Inverse Scattering Method ◽  
Bäcklund Transformations ◽  
Scattering Method ◽  
Sine Gordon Equation ◽  
Backlund Transformations

Like a number of other nonlinear dispersive wave equations the sine–Gordonequation z , xt = sin z has both multi-soliton solutions and an infinity of conserved densities which are polynomials in z , x , z , xx , etc. We prove that the generalized sine–Gordon equation z , xt = F ( z ) has an infinity of such polynomial conserved densities if, and only if, F ( z ) = A e αz + B e – αz for complex valued A, B and α ≠ 0. If F ( z ) does not take the form A e αz + B e βz there is no p. c. d. of rank greater than two. If α ≠ – β there is only a finite number of p. c. ds. If α = – β then if A and B are non-zero all p. c. ds are of even rank; if either A or B vanishes the p. c. ds are of both even and odd ranks. We exhibit the first eleven p. c. ds in each case when α = – β and the first eight when α ≠ – β . Neither the odd rank p. c. ds in the case α = – β , nor the particular limited set of p. c. ds in the case when α ≠ – β have been reported before. We connect the existence of an infinity of p. c. ds with solutions of the equations through an inverse scattering method, with Bäcklund transformations and, via Noether’s theorem, with infinitesimal Bäcklund transformations. All equations with Bäcklund transformations have an infinity of p. c. ds but not all such p. c. ds can be generated from the Bäcklund transformations. We deduce that multiple sine–Gordon equations like z , xt = sin z + ½ sin ½ z , which have applications in the theory of short optical pulse propagation, do not have an infinity of p. c. ds. For these equations we find essentially three conservation laws: one and only one of these is a p. c. d. and this is of rank two. We conclude that the multiple sine–Gordons will not be soluble by present formulations of the inverse scattering method despite numerical solutions which show soliton like behaviour. Results and conclusions are wholly consistent with the theorem that the generalized sine–Gordon equation has auto-Bäcklund transformations if, and only if Ḟ ( z ) – α 2 F ( z ) = 0.

scholarly journals Solutions to Benjamin-Bona-Mahony Equation in Two Space Dimensions by Various Methods

2018 ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Gordon Equation ◽  
Expansion Method ◽  
Algebraic Method ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Hyperbolic Tangent ◽  
Real Solutions ◽  
Two Space Dimensions ◽  
Bbm Equation

Four methods in two different families have been constructed to derive the exact solutions to Benjamin-Bona-Mahony equation in two space dimensions. Simply defined hyperbolic tangent, hyperbolic secant and hyperbolic cosecant ansatzes and the expansion method based on the Sine-Gordon equation in two dimensions are directly substituted into the governing ODE reduced from the two dimensional BBM equation. Classical algebraic method is used to find the relations among the target parameters representing the nonzero coefficients in the predicted solutions and the wave transform parameters. Some complex and real solutions have been constructed in explicit forms.

Exact and numerical solutions to the perturbed sine-Gordon equation

1984 ◽  
Vol 11 (3) ◽  
pp. 349-358 ◽  
Author(s):  
O.A. Levring ◽  
M.R. Samuelsen ◽  
O.H. Olsen
Keyword(s):  
Gordon Equation ◽  
Numerical Solutions ◽  
Sine Gordon Equation

Chebyshev Split-step Scheme for the Sine-Gordon Equation in 2+1 Dimensions

2013 ◽  
Vol 14 (1) ◽  
pp. 69-75
Author(s):  
Pablo Suarez ◽  
Stephen Johnson ◽  
Anjan Biswas
Keyword(s):  
Numerical Solution ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Time Integration ◽  
One Dimension ◽  
Fractional Step ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Spectral Convergence ◽  
Chebyshev Spectral Method

Abstract This article studies the numerical solution of the two-dimensional sine-Gordon equation (SGE) using a split-step Chebyshev Spectral Method. In our method we split the 2D SGE by considering one dimension at a time, first along x and then along y. In each fractional step we solve a 1D SGE. Time integration is handled by a finite difference scheme. The numerical solution is then compared with many of the known numerical solutions found throughout the literature. Our method is simple to implement and second order accurate in time and has spectral convergence. Our method is both fast and accurate.

scholarly journals Numerical Solution of Sine-Gordon Equation with the Local Kriging Meshless Method

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Pengfei Guo ◽  
Ariunkhishig Boldbaatar ◽  
Dutao Yi ◽  
Pengxiang Dai
Keyword(s):  
Meshless Method ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Interpolation Method ◽  
Time Discretization ◽  
Delta Function ◽  
Kriging Interpolation ◽  
Sine Gordon Equation ◽  
Field Function ◽  
Kriging Interpolation Method

This paper develops a local Kriging meshless solution to the nonlinear 2 + 1-dimensional sine-Gordon equation. The meshless shape function is constructed by Kriging interpolation method to have Kronecker delta function property for the two-dimensional field function, which leads to convenient implementation of imposing essential boundary conditions. Based on the local Petrov–Galerkin formulation and the center difference method for time discretization, a system of nonlinear discrete equations is obtained. The numerical examples are presented and the numerical solutions are found to be in good agreement with the results in the literature to validate the ability of the present meshless method to handle the 2 + 1-dimensional sine-Gordon equation related problems.

scholarly journals Solving Two Dimensional Coupled Burger's Equations and Sine-Gordon Equation Using El-Zaki Transform-Variational Iteration Method

2021 ◽  
Vol 24 (2) ◽  
pp. 41-47
Author(s):  
Marwa H. Al-Tai ◽  
◽  
Ali Al-Fayadh ◽  
Keyword(s):  
Differential Equation ◽  
Iteration Method ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Variational Iteration Method ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Partial Differential ◽  
Burger's Equations ◽  
Variation Iteration Method

In this paper, the combined form of the Elzaki transform and variation iteration method is implemented efficiently in finding the analytical and numerical solutions of the two-dimensional nonlinear coupled Burger's partial differential equations and sine-Gordon partial differential equation. The obtained solutions were compared to the exact solutions and other existing methods. Illustrative examples show the efficiency and the power of the used method.

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