Simple high-order boundary conditions for computing rogue waves in the nonlinear Schrödinger equation

This paper proposes some simple and accurate artificial boundary conditions in order to develop numerical methods to compute the rational solution of the nonlinear Schrödinger equation (NLSE) modeling the dynamics of rogue waves. To overcome the nonzero background condition and the algebraically slow decaying ratio at spatial infinity, we design the boundary conditions by means of a far field asymptotic expansion formulation of the rational solution and some transformations for resting the oscillation in time. Compared to the existing periodic boundary condition, the proposed boundary conditions can achieve fourth-order accuracy at most, i.e., O(L−4) with the interval length L, which means that for getting the presupposed accuracy, a smaller computational interval can be chosen. Meanwhile, their forms are simple, so there is hardly any additional computational cost to implement the numerical methods. Then we compare the proposed boundary conditions and numerical methods in terms of efficiency and accuracy, and identify the most efficient and accurate one for computing the rational solution of NLSE and numerically study their stability and interaction.