Global dynamics below the ground state for the focusing Schrödinger equation with a potential

Abstract

In this paper we consider the nonlinear Schrödinger system (NLS) with quadratic interaction in five dimensions. We determine the global behavior of the solutions to the system with data below the ground state. Our proof of the scattering result is based on an argument by . In particular, the new part of this paper is to deal with asymmetric interaction. A blowing-up or growing-up result is proved by combining the argument by Du-Wu-Zhang in [6] and a variational characterization of minimizers. Moreover, we show a blowing-up result if the data has finite variance or is radial.

A sharp scattering threshold level for mass-subcritical nonlinear Schrödinger system

“…Recall that mass M is scaling invariant in four dimensions, and the product of two quantities M E is a scaling invariant in five dimensions. In the previous results [8,12], these quantities play a crucial role in the criteria there.…”

“…This subject is recently extensively studied based on a concentration compactness/rigidity type argument after Kenig and Merle [15]. As for the Schrödinger system (NLS), the first author treated five dimensions [8] and Inui, Kishimoto, and Nishimura treated four dimensions [12]. In these results, a sharp condition for scattering is given in terms of conserved quantities.…”

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

“…The mass-critical case d = 4 is quite different from theḢ 1 2 -critical case d = 5. Hamano [9] gave the threshold for scattering or blow-up below the ground state inḢ 1 2 -critical case d = 5 and H 1 setting under the mass-resonance condition. To prove scattering, Hamano used the argument of Kenig-Merle [12] which is organized by stability, profile decomposition, construction of critical element, and rigidity of it.…”

The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

“…where K is the class of Kato potential, The energy scattering for (5.1) with α = 2 was studied by Hong [23] using the concentration-compactness-rigidity argument of Kenig-Merle [25]. Recently, Hamano-Ikeda [21] adapted the Dodson-Murphy's method to extend the result in [23] to the whole range of the intercritical case. The energy scattering for (5.2) was first established by Farah-Guzman [15] with 0 < b < 1 2 , α = 2 and N = 3.…”