Scattering for Nonlinear Schrödinger Equations with a potential

In this talk, I'll discuss the asymptotics of the cubic nonlinear Schrödinger equation with potential in dimension 1 for small, localized initial data. In the case when the potential is equal to 0, it has been known for some time that solutions exhibit modified scattering. Due to additional complications introduced by the potential, the case with V nonzero has not been addressed until recently.

Here, we present a method to obtain asymptotics for this problem.  The main ingredients are  (1) a new linear identity, which allows us to relate certain vector field-like quantities for the problem with a potential to those for the problem with no potential, and (2) an adaptation of the method of testing with wave packets introduced by Ifrim and Tataru. Compared to previous results, this method can handle potentials with slower decay at infinity.