Mathematical Research Letters

Volume 26 (2019)

Number 1

Randomized final-data problem for systems of nonlinear Schrödinger equations and the Gross–Pitaevskii equation

Pages: 253 – 279

DOI: https://dx.doi.org/10.4310/MRL.2019.v26.n1.a12

Authors

Kenji Nakanishi (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan)

Takuto Yamamoto (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan)

Abstract

We consider the final-data problem for systems of nonlinear Schrödinger equations (NLS) with $L^2$ subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in $L^2 (\mathbb{R}^d)$, extending the result of J. Murphy [29] to powers equal to or lower than the Strauss exponent. In particular, systems with quadratic nonlinearity can be treated in three space dimensions, and by the same argument, the Gross–Pitaevskii equation in the energy space. The extension is by use of the Strichartz estimate with a time weight.

Received 15 May 2018

Accepted 31 October 2018

Published 7 June 2019