A fourth-order split-step pseudospectral scheme for the Kuramoto–Tsuzuki equation

The numerics of the Kuramoto–Tsuzuki equation is dealt with in this paper. We propose a split-step Fourier pseudospectral discretization for solving the problem, which is split into one linear subproblem and one nonlinear subproblem. The nonlinear subproblem is integrated exactly via solving the equations for the amplitude and phase angle of the unknown complex-valued function respectively. The linear subproblem is first approximated by Fourier pseudospectral discretization to the spatial derivative, and then integrated exactly in phase space via solving the equations for the Fourier coefficients analytically. We apply a fourth-order splitting integration in time advances, and therefore the overall error in space discretization is of spectral order and the overall error in time discretization is of fourth order which merely comes from the splitting. The scheme is fully explicit, easy to implement and quite efficient thanks to FFT. Moreover, it is time reversible and gauge invariant which are two properties in the continuous problem. Extensive numerical results are reported, which are geared towards testing the convergence and demonstrating the efficiency and accuracy.