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

Abstract

In this paper we study the one-dimensional logarithmic Schrödinger equation perturbed by an attractive ${\delta }^{\prime }$-interaction \[ i\partial_{t}u+\partial^{2}_{x}u+ γδ^{\prime}(x)u+u\, \mbox{Log}\left|u\right|^{2}=0, \quad (x,t)\in\mathbb{R}\times\mathbb{R}, \] where $\gamma >0$. We establish the existence and uniqueness of the solutions of the associated Cauchy problem in a suitable functional framework. In the attractive ${\delta }^{\prime }$-interaction case, the set of the ground state is completely determined. More precisely: if $0<\gamma \le 2$, then there is a single ground state and it is an odd function; if $\gamma >2$, then there exist two non-symmetric ground states. Finally, we show that the ground states are orbitally stable via a variational approach.

21 pages, 1 figure

References in corpus (1)

• Orbital stability of Gausson solutions to logarithmic Schrödinger equations

Cited by in corpus (2)

• On the dynamical nature of nonlinear coupling of logarithmic quantum wave equation, Everett-Hirschman entropy and temperature
• Orbital stability of standing waves for supercritical NLS with potential on graphs