Modified scattering for inhomogeneous nonlinear Schr\"odinger equations with and without inverse-square potential

We consider the final state problem for the inhomogeneous nonlinear Schr\"odinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free evolution, we construct a unique global solution which conve...

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Bibliographic Details
Main Authors Aoki, Kazuki, Inui, Takahisa, Miyazaki, Hayato, Mizutani, Haruya, Uriya, Kota
Format Journal Article
LanguageEnglish
Published 23.01.2021
Subjects
Mathematics - Analysis of PDEs
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Summary:We consider the final state problem for the inhomogeneous nonlinear Schr\"odinger equation with a critical long-range nonlinearity. Given a prescribed asymptotic profile, which has a logarithmic phase correction compared with the free evolution, we construct a unique global solution which converges to the profile. As a consequence, the existence of modified wave operators for localized small scattering data is obtained. We also study the same problem for the case with the critical inverse-square potential under the radial symmetry. In particular, we construct the modified wave operators for the long-range nonlinear Schr\"odinger equation with the critical inverse-square potential in three space dimensions, under the radial symmetry.
DOI:10.48550/arxiv.2101.09423