Threshold odd solutions to the nonlinear Schr\"{o}dinger equation in one dimension

TL;DR

This paper investigates the behavior of odd solutions to a one-dimensional nonlinear Schrödinger equation with supercritical nonlinearity, establishing a threshold at twice the ground state action that determines whether solutions scatter or blow up.

Contribution

It proves that odd solutions with action equal to twice the ground state either scatter or blow up, extending previous results that only covered lower action levels.

Findings

Solutions with action less than twice the ground state scatter or blow up.

Solutions with action equal to twice the ground state also scatter or blow up.

The threshold at twice the ground state action is critical for solution behavior.

Abstract

We consider odd solutions to the Schr\"{o}dinger equation with the $L^2$-supercritical power type nonlinearity in one dimensional Euclidean space. It is known that the odd solution scatters or blows up if its action is less than twice as that of the ground state. In the present paper, we show that the odd solutions with the action as twice as that of the ground state scatter or blow up.