Logarithmic Bose–Einstein condensates with harmonic potential

Abstract

In this paper, by using a compactness method, we study the Cauchy problem of the logarithmic Schrödinger equation with harmonic potential. We then address the existence of ground states solutions as minimizers of the action on the Nehari manifold. Finally, we explicitly compute ground states (Gausson-type solution) and we show their orbital stability.2010 Mathematics Subject Classification. 35Q55; 35Q40.

Finite dimensional global attractor for a damped fractional anisotropic Schrödinger type equation with harmonic potential

“…On the contrary, to the best of our knowledge, the literature for fractional Schrödinger equations is still expending and rather young. If α = 2, (1.3) becomes the standard Gross-Pitaevskii equation which has been extensively studied as a fundamental equation in modern mathematical physics specially for Bose-Einstein condensates (see [10,4,34] for instance). In the special case when V = 0 we refer the reader; for well-posedness results and existence of traveling waves for the resulting conservative fractional NLS; to [5,25,26,21,13,45,7] and the references therein.…”

“…Lemma 2.3. Let α ∈ (1, 2) and q ∈ [2,4]. Then there exists a reel constant C α > 0 that depends on α and q such that for all…”

Optimal bilinear control of the logarithmic Schrödinger equation

“…The LNSE was introduced as a dimensionless model in quantum physics by Bialynicki-Birula and Mycielski 1 and has received a lot of attentions in the past decades; see Martino et al 2 and Zloshchastiev 3 in physics and previous studies [4][5][6][7][8][9][10] in mathematics. However, to the best of our knowledge, the study on optimal control concerned with LNSE is still lacking in mathematics literatures.…”

Orbital stability of standing waves for supercritical NLS with potential on graphs

“…(1.1) is known as a model to describe the Bose-Einstein condensate under a magnetic trap. We refer the readers to [4,12,21,23] for more information. If L Ω = 0, the model equation (1.1) describes the Bose Einstein condensate with rotation, which appears in a variety of physical settings such as the description of nonlinear waves and propagation of a laser beam in the optical fiber [11,22].…”

“…In particular, we infer that the standing wave e ωn 2 it φ ωn (x) in Corollary 1.10 is strongly unstable in Σ (see the proof of Corollary 1.10 and Lemma 4.1 below). Now, we focus on the stability of standing waves in the mass supercritical regimen p > 1+ 4 N . The more common approach to construct orbitally stable standing waves to (1.1) is to consider the following constrained minimization problems…”