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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.

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A three-time level finite-difference scheme based on a fourth order in time and second order in space approximation has been proposed for the numerical solution of the nonlinear two-dimensional sine-Gordon equation. The method, which is analysed for local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor-corrector scheme using as predictor a secondorder explicit scheme is proposed. The procedure of the corrector has been modified by considering as known the already evaluated corrected values instead of the predictor ones. This modified scheme has been tested on the line and circular ring soliton and the numerical experiments have proved that there is an improvement in the accuracy over the standard predictor-corrector implementation.

A predictor-corrector (P-C) scheme based on the use of rational approximants of second-order to the matrixexponential term in a three-time level reccurence relation is applied to the nonlinear Klein-Gordon equation. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. Both the predictor and the corrector scheme are analyzed for local truncation error and stability. The proposed method is applied to problems possessing periodic, kinks and single, double-soliton waves. The accuracy as well as the long time behavior of the proposed scheme is discussed.