Global dynamics below the ground state for the quadratic Schödinger system in 5d

Scattering for quadratic-type Schrödinger systems in dimension five without mass-resonance

“…Other results concerning global well-posedness and blow-up for (1.2) have also appeared in the current literature. Indeed, the dichotomy global existence versus blow-up in finite time in H 1 (R 5 ) was discussed in [9] and [18] (see also [21]). In [5] the author studied the stability of ground states (for 1 ≤ n ≤ 3) as well as the characterization of minimal mass blow-up solutions (for n = 4).…”

“…First note that (1.2) satisfies the mass-resonance condition if and only if κ = 1/2. In dimension n = 5, under similar assumption as in Theorem A, in [9] the author established the scattering of radially symmetric solutions in H 1 (R 5 ) in the case κ = 1/2; the mass-resonance assumption was dropped in [10]. In both cases, the authors used the concentration-compactness and rigidity method introduced in [14].…”

Blow-up of the radially symmetric solutions for the quadratic nonlinear Schrödinger system without mass-resonance

“…Hayashi, Li, and Ozawa [8] investigated the small data scattering. Recently, scattering below the ground state was also obtained by Hamano [7] when d = 5 and the authors [11] when d = 4, where scattering means that the solution of nonlinear system (NLS) approches to a free solution to the Schrödinger equations as time goes to infinity.…”

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

“…Recently, under the massresonance condition κ = 1/2, scattering below the ground state was obtained by the first author [7], where scattering means that the solution of nonlinear system (NLS) approaches to a free solution to the Schrödinger equations as time goes to infinity. The first author also proved the blow-up or grow-up result below the ground state in [7]. In the mass-resonance case, (NLS) has the Galilean invariance.…”

“…That is, (e ix•ξ e −it|ξ| 2 u(t, x − 2tξ), e 2ix•ξ e −2it|ξ| 2 v(t, x − 2tξ)) for any ξ ∈ R 5 is solution to (NLS) if (u, v) is a solution. This invariance plays an important role in the argument of the first author [7]. Roughly speaking, (NLS) with the mass-resonance condition is similar to the single nonlinear Schrödinger equation, whose global dynamics were investigated by many researchers (see e.g.…”