A deep learning improved numerical method for the simulation of rogue waves of nonlinear Schrödinger equation

Abstract Modulation instability (MI) is a pervasive phenomenon in nonlinear science. It is inevitable for simulating rogue wave or breather solutions of the focusing nonlinear Schrodinger equation (NLSE) and other application problems with MI involved. Due to MI, the small perturbation on the boundary can lead to large and non-negligible errors for the simulation of initial-boundary problems. To deal with this challenging problem, we propose a method to modify the boundary problem through a deep learning algorithm so that the long time simulation for the rogue wave or breather solutions to the NLSE can be performed with a superior numerical errors. We impose different types of rogue wave and breather solutions for the focusing NLSE as initial data to test the proposed method. It turns out that the proposed method gives rise to the better numerical results in compared with the ones obtained by traditional methods, which paves a way to simulate other physical problems with MI.