Scattering for the quadratic nonlinear Schrödinger system in ${\mathbb{R}}^5$ without mass-resonance condition

Abstract

We consider the quadratic nonlinear Schrödinger system (NLS system) \begin{align*}\begin{cases} i\partial_t u + Δu = v \overline{u}, \\ i\partial_t v+κΔv = u^2, \end{cases} \text{ on } I \times \mathbb{R}^5, \end{align*} where $\kappa >0$. The scattering below the standing wave solutions for NLS system was obtained by the first author when $\kappa =1/2$. The condition of $\kappa =1/2$ is called mass-resonance. In this paper, we prove scattering below the standing wave solutions when $\kappa \ne 1/2$ under the radially symmetric assumption. Our proof is based on the concentration compactness and the rigidity by Kenig--Merle. Moreover, we discuss the concentration compactness and the rigidity for non-radial solutions.

References in corpus (1)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation

Cited by in corpus (2)

• Scattering For Mass-resonance Nonlinear Schrödinger System in 5D
• Scattering theory For Quadratic Nonlinear Schrödinger System in dimension six