Soliton-potential interactions for nonlinear Schr\"odinger equation in ${\mathbb{R}}^3$

In this work we mainly consider the dynamics and scattering of a narrow soliton of NLS equation with a potential in ${\mathbb{R}}^3$, where the asymptotic state of the system can be far from the initial state in parameter space. Specifically, if we let a narrow soliton state with initial velocity ${\upsilon }_0$ to interact with an extra potential $V(x)$, then the velocity ${\upsilon }_{+}$ of outgoing solitary wave in infinite time will in general be very different from ${\upsilon }_0$. In contrast to our present work, previous works proved that the soliton is asymptotically stable under the assumption that ${\upsilon }_{+}$ stays close to ${\upsilon }_0$ in a certain manner.