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

Abstract

We consider a mass-critical system of nonlinear Schrödinger equations (i∂i∂ t t uv+ + κ ∆∆ u v == ū u2 v,, (t, x) ∈ R × R4, where (u, v) is a C2-valued unknown function and κ > 0 is a constant. If κ = 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(φ, ψ), where M(u, v) denotes the mass and (φ, ψ) 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.