Long-time simulations of rogue wave solutions in the nonlinear Schrödinger equation

Abstract

. Although several short-time simulations have been reported nicely reproducing rogue wave solutions in the nonlinear Schr¨odinger equation, rogue wave solutions are linearly unstable as shown by theoretical studies. In the present work, we perform long-time simulations for two kinds of rogue wave solutions, namely, the Akhmediev breather and Peregrine soliton. Numerical evidences demonstrate that spurious oscillations that emerge in the central domain in both simulations arise from round-off error and evolve under the mechanism of modulational instability. For the periodic approximation of the Peregrine soliton, the modulational instability also gives rise to additional oscillations on the boundary. We obtain a fitting formula to forecast the time when the boundary oscillations occur. Our simulation results show that a clean and faithful long-time reproduction of rogue wave solutions would be difficult because of the modulational instability.