A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS

•We apply the modulation theory to study a moving soliton with radiation in the NLS equation.•The full modulation equations are derived and solved accurately by proposed numerical method.•The modulation equations approach captures the soliton and radiation separately.•The approach gives correct long time simulation of the soliton on a fixed domain.•The multichannel dynamics are explored numerically by the method.

Based on our previous work for solving the nonlinear Schrödinger equation with multichannel dynamics that is given by a localized standing wave and radiation, in this work we deal with the multichannel solution which consists of a moving soliton and radiation. We apply the modulation theory to give a system of ODEs coupled to the radiation term for describing the solution, which is valid for all times. The modulation equations are solved accurately by the proposed numerical method. The soliton and radiation are captured separately in the computation, and they are solved on the translated domain that is moving with them. Thus for a fixed finite physical domain in the lab frame, the multichannel solution can pass through the boundary naturally, which cannot be done by imposing any existing boundary conditions. We comment on the differences of this method from the collective coordinates.