An explicit numerical scheme for the Sine-Gordon equation in 2+1 dimensions

Abstract

The paper presents an explicit finite-difference method for the numerical solution of the Sine-Gordon equation in two space variables, as it arises, for example, in rectangular large-area Josephson junction. The dispersive nonlinear partial differential equation of the system allows for soliton-type solutions, an ubiquitous phenomenon in a large-variety of physical problems.The method, which is based on fourth order rational approximants of the matrix-exponential term in a three-time level recurrence relation, after the application of finite-difference approximations, it leads finally to a second order initial value problem. Because of the existing sinus term this problem becomes nonlinear. To avoid solving the arising nonlinear system a new method based on a predictor-corrector scheme is applied. Both the nonlinear method and the predictor-corrector are analyzed for local truncation error, stability and convergence. Numerical solutions for cases involving the most known from the bibliography ring and line solitons are given.

A modified predictor–corrector scheme for the two-dimensional sine-Gordon equation

“…1b). All graphs are in agreement with those published in [2,11,12] and [6]. It is known in the bibliography -see [12,19] etc.…”

“…The results obtained are also compared with those published in [2,6,12]. In all experiments t 0 = 0 and h x = h y = h = 0.2 were used.…”

“…Also in [17] who studied the nonlinear wave propagation in a planar wave guide consisting of two rectangular regions joined by a bent of constant curvature using as a model the kink solutions of the SG equation, where this can be considered as a part of analogous works examining the effect of the curvature to analogous nonlinear physical phenomena -see [3,13,20,24,30,33] etc. Finally, by [12] where the method arises from a two-step one-parameter leapfrog scheme, which is a generalization to that used by [11] and [6].…”

“…Numerical solutions for the undamped SG equation have been given among others by [19] using two difference schemes, [34] who studied SG as an asymptotic reduction of the two level dissipation less Maxwell-Bloch system, [11] using a generalized leapfrog method and [2] with finite-elements where both methods using appropriate initial conditions have been applied successfully with the latter one giving slightly better results, [32] with a split cosine scheme, [6] using a three time-level fourth order explicit finite-difference scheme. Numerical approaches to the damped SG equation can be found in [31] who considers dimensionless loss factors and unit less normalized bias.…”

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

“…As it was expected-see [11][12][13]15]-the presence of the dissipative term delays the propagation of the solitons. …”