Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction

Alex H. Ardila

doi:10.3934/eect.2017009

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2017, Volume 6, Issue 2: 155-175. Doi: 10.3934/eect.2017009

This issue Previous Article Next Article Asymptotic for the perturbed heavy ball system with vanishing damping term

Stability of ground states for logarithmic Schrödinger equation with a $δ^{\prime}$-interaction

Alex H. Ardila

Department of Mathematics, IME-USP, Cidade Universitária, CEP 05508-090, São Paulo, SP, Brazil

Received: July 2016

Revised: February 2017

Published: May 2017

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Abstract

In this paper we study the one-dimensional logarithmic Schrödin-\break ger equation perturbed by an attractive $δ^{\prime}$-interaction

$i{\partial _t}u + \partial _x^2u + {\rm{ }}{\gamma ^\prime }(x)u + u{\mkern 1mu} {\rm{Log|}}u|2 = 0,(x,t) \in \mathbb{R} \times \mathbb{R} ,$

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 $δ^{\prime}$-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.

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Mathematics Subject Classification: 35Q51, 35Q55, 37K40, 34B37.

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Figure 1. The graph of the curve $\mathcal{I}_{1}\cup \mathcal{I}_{2}$.

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