Final state problem for the defocusing long-range nonlinear Schrödinger equations in one space dimension
A talk in the Oberseminar Analysis series by
Haruya Mizutani from Osaka
| Abstract: | I will discuss recent progress on the final state problem (or equivalently, the existence of modified wave operators) for the defocusing NLS in one space dimension with cubic and a range of subcritical power-type nonlinearities. In this setting, the asymptotic behavior of global solutions cannot be approximated by a free solution, and nonlinear effects must be taken into account not only in the scattering data, but also in the dynamics of the leading term of solutions. Given a prescribed asymptotic profile obtained by modifying the free solution with a nonlinear phase correction, we construct a unique global solution that scatters to this profile. The proof relies on a modified energy estimate for the linearized equation around the prescribed asymptotic profile. In the subcritical case, we also use certain algebraic structures of the nonlinearity that emerge after linearization. This talk is based on joint work with Masaki Kawamoto (Okayama University). Zoom: https://uni-bielefeld.zoom-x.de/j/64951709096?pwd=EfJynCkou8bBw77CsNYMuh88IrKazJ.1 Within the CRC this talk is associated to the project(s): A1 |