Modified scattering for the nonlinear Schrödinger equation in an external field.

Abstract: In this talk, we consider the cubic nonlinear Schrödinger equation with an external potential. We prove the existence of modified scattering for this model, that is, linear scattering modulated by a phase. Our approach is based on the spectral theorem for the perturbed linear Schrödinger operator and a factorization technique, which allows us to control the resonant nonlinear term. This approach requires a detailed and subtle study of the low-energy properties of the scattering data. Therefore, we study the cases of generic and exceptional potentials separately. The exceptional case is more delicate: we need some parity assumptions in order to control the small-energy behavior of the scattering coefficients and of the wave functions.

Posted on Oct 21, 2020 in Differential Equations, Seminars