Scattering for the 𝐿² supercritical point NLS

Abstract

We consider the 1D nonlinear Schrödinger equation with focusing point nonlinearity. "Point" means that the pure-power nonlinearity has an inhomogeneous potential and the potential is the delta function supported at the origin. This equation is used to model a Kerr-type medium with a narrow strip in the optic fibre. There are several mathematical studies on this equation and the local/global existence of solution, blow-up occurrence and blow-up profile have been investigated. In this paper we focus on the asymptotic behavior of the global solution, i.e, we show that the global solution scatters as t → ±∞ in the L 2 supercritical case. The main argument we use is due to Kenig-Merle, but it is required to make use of an appropriate function space (not Strichartz space) according to the smoothing properties of the associated integral equation.

Blow-up and scattering for the 1D NLS with point nonlinearity above the mass-energy threshold

“…The remainder of the proof is similar to that of [1,Proposition 3.1]. This completes the proof of proposition.…”

“…Using the same argument developed in the proof of Lemma 4.2 in [1], we can show that given ψ ∈ H 1 (R), there exists…”

“…As a first consequence of the Theorem 1.1, we have the energy scattering below the ground state threshold, which was originally proved by Adami-Fukuizumi-Holmer [1] based on the ideas of [9,17].…”

“…We call Q the ground state. Recently, Adami-Fukuizumi-Holmer [1] determined the long time dynamics to (1.1) with data below the ground state threshold. Our purpose in this paper is to discuss the the global behavior of the solutions to (1.1) at and above the mass and energy ground states threshold.…”

On stability and instability of standing waves for 2d-nonlinear Schrödinger equations with point interaction

“…As related topics, we mention the NLS with the concentrated nonlinearity. See [3,17,42] for N = 1, [1,2] for N = 2, and [5] for N = 3.…”