Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

Takahisa Inui; Nobu Kishimoto; Kuranosuke Nishimura

doi:10.3934/dcds.2019275

Article Contents

2019, Volume 39, Issue 11: 6299-6353. Doi: 10.3934/dcds.2019275

This issue Previous Article Thermodynamic formalism for topological Markov chains on standard Borel spaces Next Article Oblique projection based stabilizing feedback for nonautonomous coupled parabolic-ode systems

Scattering for a mass critical NLS system below the ground state with and without mass-resonance condition

1.
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka, 560-0043, Japan
2.
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502, Japan
3.
Department of Mathematics, Graduate School of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

^* Corresponding author: Kuranosuke Nishimura
^* Corresponding author: Kuranosuke Nishimura

Received: November 2018

Published online: November 2019

The first author is partially supported by JSPS Grant-in-Aid for Early-Career Scientists JP18K13444. The second author is supported in part by Grant-in-Aid for Young Scientists (B) JP24740086 and JP16K17626.

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Abstract

We consider a mass-critical system of nonlinear Schrödinger equations

$ \left\{ \begin{split} i\partial_t u + \;\; \Delta u & = \bar{u}v,\\ i\partial_t v +\kappa \Delta v & = u^2, \end{split} \right. \qquad (t,x)\in \mathbb{R}\times \mathbb{R}^4, $

where $ (u,v) $ is a $ \mathbb{C}^2 $-valued unknown function and $ \kappa >0 $ is a constant. If $ \kappa = 1/2 $, we say the equation satisfies mass-resonance condition. We are interested in the scattering problem of this equation under the condition $ M(u,v)<M(\phi ,\psi) $, where $ M(u,v) $ denotes the mass and $ (\phi ,\psi) $ is a ground state. In the mass-resonance case, we prove scattering by the argument of Dodson [5]. Scattering is also obtained without mass-resonance condition under the restriction that $ (u,v) $ is radially symmetric.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 35P25.

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References

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