Scattering and Blow-Up in Both Time Directions Above the Ground State for the Focusing Nonlinear Schr\"{o}dinger Equation

TL;DR

This paper introduces new solutions for the focusing nonlinear Schrödinger equation that exhibit scattering and blow-up behavior in both time directions, even with arbitrarily large mass-energy, expanding understanding of solution dynamics above the ground state.

Contribution

It provides novel scattering and blow-up solutions with large mass-energy not derived from Galilean transformations, and establishes new criteria for blow-up and scattering in both time directions.

Findings

Constructed solutions with arbitrarily large mass-energy that scatter in both time directions.

Established new criteria for blow-up in both time directions.

Provided improved conditions for scattering and blow-up in one time direction.

Abstract

In this article we obtain new scattering and blow-up solutions for intercritical focusing nonlinear Schr\"{o}dinger equations (NLS) above the ground state mass-energy threshold. The main focus of this article is the establishment of some solutions with arbitrarily large mass-energy which scatter in both time directions. In particular, large mass-energy which does not arise from the Galilean transformation. We additionally obtain new criteria for blow-up in both time directions, as well as improved sufficient conditions for scattering and blow-up in one time direction.