Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory

The nonlinear sine-Gordon equation (NSGE) represents the classical solitary wave model having a nonlinear sine source term in the theory of crystal dislocations. This paper focusses on an efficient and accurate localized meshless collocation method to find the approximate solution of the NSGE with Neumann boundary conditions in two space variables. The proposed method consists of two stages. At the first stage, a time stepping method with second-order accuracy is applied to discretize the temporal direction and achieve a semi-discrete approach. In this stage, we prove that the semi-discrete approach is convergent and unconditionally stable in the H¹-norm. In the second stage, a localized radial basis function partition of unity (LRBF-PU) is implemented to approximate the spatial direction and get a full-discrete scheme. A major drawback of global collocation RBF techniques is the computational burden associated with the arising dense algebraic systems. The LRBF-PU is based on the partitioning of the initial domain into a number of patches and the use of the RBF approximation on every local patch. The major advantages of this technique are its low computational burden and well-conditioned final linear system compared to global collocation techniques. Numerical results show the validity and accuracy of the proposed method for all cases associated with the NSGE containing ring and line solitons of the circular and elliptic shapes. The obtained results are also compared to those reported in the literature and found the two to be in good agreement.