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

TL;DR

This paper investigates the 1D focusing nonlinear Schrödinger equation with point nonlinearity, establishing scattering criteria, analyzing dynamics at thresholds, and proving blow-up conditions for large energies.

Contribution

It introduces new scattering criteria and analyzes solution dynamics at the mass-energy threshold for the 1D NLS with point nonlinearity.

Findings

Established scattering criteria based on Kenig-Merle's method.

Proved energy scattering both below and above the threshold.

Derived blow-up conditions for solutions with large initial energy.

Abstract

In this paper, we study the nonlinear Schr\"odinger equation with focusing point nonlinearity in dimension one. First, we establish a scattering criterion for the equation based on Kenig-Merle's compactness-rigidity argument. Then we prove the energy scattering below and above the mass-energy threshold. We also describe the dynamics of solutions with data at the ground state threshold. Finally, we prove a blow-up criteria for the equation with initial data with arbitrarily large energy.