Energy scattering of a modified Davey–Stewartson system in three dimensions

Abstract

We obtain the global well-posedness and the scattering theory of the solution for the modified Davey–Stewartson system $\{\begin{array}{rl}& {\displaystyle i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=|u{|}^{4}u+u{v}_{x_1},(t,x)\in \mathbb{R}\times {\mathbb{R}}^3,}\\ & {\displaystyle -\mathrm{\Delta }v=(|u{|}^2{)}_{x_1}}\end{array}$ in the energy space $H^1({\mathbb{R}}^3)$ in this paper. Since the interaction Morawetz estimate fails and the nonlinearity is non-local, we employ the concentration-compactness argument introduced by Kenig and Merle (Invent Math. 2006;166:645–675) to establish the scattering result.

Keywords:

• Davey–Stewartson system
• scatter
• global well-posedness
• profile decomposition

2020 Mathematics Subject Classifications:

• Primary: 35Q55
• Secondary: 47J35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Natural Science Foundation of China [grant number 12101604].