Traveling wave solutions of the nonlinear Gilson–Pickering equation in crystal lattice theory

This paper focuses on obtaining the traveling wave solutions of the nonlinear Gilson–Pickering equation (GPE), which describes the prorogation of waves in crystal lattice theory and plasma physics. The solution of the GPE is approximated via the finite difference technique and the localized meshless radial basis function generated finite difference. The association of the technique results in a meshless approach that does not require linearizing the nonlinear terms. At the first step, the PDE is converted to a system of nonlinear ODEs with the help of the radial kernels. In the second step, a high-order ODE solver is adopted to discretize the nonlinear ODE system. The global collocation techniques pose a considerable computationl burden due to the calculation of the dense algebraic system. The proposed method approximates differential operators over the local support domain, leading to sparse differentiation matrices and decreasing the computational burden. Numerical results and comparisons are provided to confirm the efficiency and accuracy of the method.