Global Dynamics below the standing waves for the focusing semilinear Schr\"{o}dinger equation with a repulsive Dirac delta potential

TL;DR

This paper investigates the global behavior of solutions to a focusing semilinear Schrödinger equation with a repulsive Dirac delta potential, establishing conditions that distinguish between global existence and blow-up.

Contribution

It provides a necessary and sufficient criterion for the global dynamics of solutions below the standing wave for the deltaNLS, extending understanding to potential-influenced cases.

Findings

Established criteria for global behavior below the standing wave.

Determined global dynamics for solutions with mass-energy between that of the standard and deltaNLS standing waves.

Analyzed radial solutions with specific mass-energy constraints.

Abstract

We consider the focusing mass supercritical semilinear Schr\"{o}dinger equation with a repulsive Dirac delta potential on the real line (deltaNLS). Our aim in the present paper is to find a necessary and sufficient condition on the data below the standing wave of NLS without a potential to determine the global behavior of the solution. Moreover, we determine the global dynamics of the radial solution whose mass-energy is larger than that of the standing wave and smaller than that of the standing wave of deltaNLS.