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In this paper, the finite point method is discussed for solving the initial‐boundary value problem associated with the sine‐Gordon equation in two‐dimensional domains arising in the Josephson junctions problem. The resulting nonlinear system is solved using an appropriate predictor‐corrector scheme. The proposed scheme is simple and efficient. The collisional properties for cases involving the most known from the bibliography, line, and ring solitons are studied in numerical results. Also the birth of a single Josephson vortex in a Josephson transmission line at a T‐shaped junction is studied.

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In this paper, the finite point method (FPM) is presented for solving the 2D, nonlinear, elliptic p-Laplace or p-harmonic equation. The FPM is a truly meshfree technique based on the combination of the moving least squares approximation on a cloud of points with the point collocation method to discretize the governing equation. The lack of dependence on a mesh or integration procedure is an important feature, which makes the FPM simple, efficient and applicable to solve nonlinear problems. Applications are demonstrated through illustrative examples.

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