Interacting localized waves for the regularized long wave equation via a Galerkin spectral method

We develop a Fourier–Galerkin spectral technique for computing the solutions of type of interacting localized waves. To this end, a special complete orthonormal system of functions in L2(−∞,∞) is used and a time-stepping algorithm implementing the spectral method is developed. The rate of convergence of the coefficients is shown to be exponential. We consider the regularized long wave equation (RLWE) which is not fully integrable. We demonstrate the stability of the algorithm and find numerically the threshold for the existence of such interactions. We also calculate the phase shifts of the interactions and compare them to the finite-difference solution.