Logarithmic Schrödinger equation with quadratic potential*

Abstract

We analyze dynamical properties of the logarithmic Schrödinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected.

Logarithmic Schrödinger Equations in Infinite Dimensions

“…where ε ∈ R, W X ∈ C 2 (R X ) has uniformly in X bounded derivatives of order n = 0, 1, 2 and W X ≡ 0 if diam(X) > R. Suppose C − ≤ C j ≤ C + for some 0 < C − ≤ C + , then for sufficiently small |ε| the measures E Λ and E • Λ both satisfy a uniform LSI by Remark 3.2. Moreover, by [46,Proposition 1.3], the Cauchy problem (LSE • Λ ) associated to the potential {J X } admits a unique solution for all λ ∈ R and localised initial condition f ∈ H 1 (µ). Therefore, all conditions in Theorem 6.1 are satisfied and there exists a solution of the infinite-volume LSE associated to the perturbed quadratic potential on R Z .…”

History of Mathematics in American Mathematical Textbooks of the Late 19th and Early 20th Centuries

“…Moreover, according to [13], no solution is dispersive for λ < 0, while for λ > 0, solutions have a dispersive behavior (with a nonstandard rate of dispersion). It is noteworthy that there has been a growing recent interest in the LogSE with a potential V (x), where (∆ + V )u is in place of ∆u in (1.1) (see, e.g., [3,12,14,18,30,33]).…”

“…In Bao et al [6], they further introduced the first-order Lie-Trotter splitting and Fourier spectral method for the RLogSE (1.3), where the conservation of mass is preserved and the constraints for discrete parameters in [5] could be relaxed. This regularized splitting method has been recently extended in Carles and Su [15] to numerically solve the LogSE with a harmonic potential studied in their work [12]. The notation of regularization was also imposed at the energy level in Bao et al [7] that resulted in a regularization different from (1.3) in their first work [5].…”

On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation

“…Also, this choice gives us the opportunity to observe the tensorization property (see e.g. [6], [9]) of the logarithmic Schrödinger equation, where, when f vanishes, the tensor product of two solutions to the one dimensional logarithmic Schrödinger equation is a solution to the two dimensional one. Different types of boundary condition can be imposed on ∂Ω, however we have shown our preference to the homogeneous Dirichlet boundary conditions since they can be easily included in a finite difference method.…”

“…For mathematical results related to the problem above, we refer the reader to [11] and [16] on the existence and uniqueness of a solution to the Cauchy problem, to [10], [2] and [8] on the asymptotic behaviour of solutions to the Cauchy problem, to [13] on the existence and multiplicity of standing wave solutions to the Cauchy problem, to [18] on existence and uniqueness of a solution over a general domain, to [9] on the asymptotic behaviour of the solution to the Cauchy problem under the presence of a quadratic potential and to [5] for existence and uniqueness of a solution to a stochastic version of the logarithmic Schrödinger equation. More references could be found in the bibliography of the aforementioned works.…”