Stability of ground states for logarithmic Schrödinger equation with a δ^′-interaction

2016/08/31 by Alex Hernandez Ardila · 2 citations
Mathematics · #math.AP #msc:76B25 #msc:35Q51 #msc:35Q55 #msc:35J60 #msc:37K40 #msc:34B37

paper · pdf · doi:10.3934/eect.2017009

published as Evolution Equations and Control Theory (EECT), Pages: 155 - 175, Volume 6, Issue 2, June 2017 · 21 pages, 1 figure

arxiv created 2017/10/31 · arxiv updated 2017/11/02

Abstract

In this paper we study the one-dimensional logarithmic Schrödinger equation perturbed by an attractive δ^′-interaction i∂_tu+∂²_xu+ γδ^′(x)u+u Log|u|²=0, (x,t)∈ℝ×ℝ, where γ>0. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive δ^′-interaction case, the set of the ground state is completely determined. More precisely: if 0<γ≤ 2, then there is a single ground state and it is an odd function; if γ>2, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.

Cited by

• Logarithmic NLS equation on star graphs: existence and stability of standing waves
• Existence and stability of standing waves for nonlinear fractional Schrödinger equation with logarithmic nonlinearity

Related

• Stability of standing waves for logarithmic Schrödinger equation with attractive delta potential
• A logarithmic Schrödinger equation with asymptotic conditions on the potential
• Semiclassical states for fractional logarithmic Schrödinger equations
• Existence and asymptotics of normalized solutions for logarithmic Schrödinger system
• Sharp interaction estimates and their application: existence of normalized ground states to coupled Schrödinger systems with potentials