Collision of two solitons for $1d$ Nonlinear Schrodinger Equation with the same mass

Abstract

We study the global dynamics of the collision of two solitons having the same mass for one-dimensional Nonlinear Schr\"odinger models with multi-power nonlinearity. For any natural number k, it is verified that if the incoming speed v between the two solitary waves is small enough, then, after the collision, the two solitons will move away with an outcoming speed v_{f}=v+O(v^{k}) and the remainder of the solution will also have energy and weighted norms of order O(v^{k}). This is applied to the one-dimensional models with polynomial odd nonlinearity having a stable soliton such as the cubic NLS and the cubic-quintic NLS.