Numerical solutions of some stochastic hyperbolic wave equations including sine-Gordon equation

Wave Motion ◽  
2016 ◽  
Vol 65 ◽  
pp. 130-146 ◽  
Author(s):  
Henry C. Tuckwell
Keyword(s):  
Wave Equations ◽  
Gordon Equation ◽  
Numerical Solutions ◽  
Sine Gordon Equation ◽  
Hyperbolic Wave

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.

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

scholarly journals Interaction of Solitons for Sine-Gordon-Type Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Georgii A. Omel’yanov ◽  
Israel Segundo-Caballero
Keyword(s):  
Numerical Simulation ◽  
Asymptotic Analysis ◽  
Small Parameter ◽  
Wave Equations ◽  
Gordon Equation ◽  
Sine Gordon Equation ◽  
Semilinear Wave Equations ◽  
The Subject ◽  
Interaction Of Solitons

The subject of our consideration is a family of semilinear wave equations with a small parameter and nonlinearities which provide the existence of kink-type solutions (solitons). Using asymptotic analysis and numerical simulation, we demonstrate that solitons of the same type (kinks or antikinks) interact in the same manner as for the sine-Gordon equation. However, solitons of the different type preserve the shape after the interaction only in the case of two or three waves, and, moreover, under some additional conditions.

Separation Transformation and New Exact Solutions for the (1+N)-Dimensional Triple Sine-Gordon Equation

2011 ◽  
Vol 66 (1-2) ◽  
pp. 19-23 ◽  
Author(s):  
Yifang Liu ◽  
Jiuping Chen ◽  
Weifeng Hu ◽  
Li-Li Zhu
Keyword(s):  
Exact Solution ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Gordon Equation ◽  
Transformation Method ◽  
Nonlinear Wave Equations ◽  
High Dimensional ◽  
Sine Gordon Equation ◽  
New Exact Solutions ◽  
Localized Structures

The separation transformation method is extended to the (1+N)-dimensional triple Sine-Gordon equation and a special type of implicitly exact solution for this equation is obtained. The exact solution contains an arbitrary function which may lead to abundant localized structures of the high dimensional nonlinear wave equations. The separation transformation method in this paper can also be applied to other kinds of high-dimensional nonlinear wave equations

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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