A locally stabilized radial basis function partition of unity technique for the sine–Gordon system in nonlinear optics

This paper develops a localized radial basis function partition of unity method (RBF-PUM) based on a stable algorithm for finding the solution of the sine–Gordon system. This system is one useful description for the propagation of the femtosecond laser optical pulse in a systems of two-level atoms. The proposed strategy approximates the unknown solution through two main steps. First, the time discretization of the problem is accomplished by a difference formulation with second-order accuracy. Second, the space discretization is obtained using the local RBF-PUM. This method authorizes us to tackle the high computational time related to global collocation techniques. However, this scheme has the disadvantage of instability when the shape parameter ɛ approaches to small value. In order to deal with this issue, we adopt RBF-QR scheme that provides the higher accuracy and stable computations for small values ɛ. Two examples are presented to show the high accuracy of the method and to compare with other techniques in the literature.