Propagation of coherent states in the logarithmic Schrodinger equation
Rémi Carles, Fangyuan Dong
TL;DR
This work analyzes the semiclassical limit of the logarithmic Schrödinger equation with a smooth at-most-quadratic potential $V$ and coherent-state initial data. It develops a sharp, regime-dependent description of the wave function: in the subcritical case $\alpha>1$ the solution is effectively governed by the linear dynamics up to Ehrenfest time, with quantitative proximity to the quadratic-envelope approximation; at the critical case $\alpha=1$ a nonlinear envelope equation with a log-nonlinearity governs the amplitude, yielding precise $L^2$-error estimates, especially for Gaussian envelopes where the envelope remains Gaussian under quadratic potentials; and when $V$ decouples coordinates and the initial data is a sum of two Gaussians, a nonlinear superposition principle holds at leading order, with explicit error decay $\mathcal{O}(\varepsilon^{\gamma})$ for any $\gamma<\tfrac{1}{2}$ on bounded times (and controlled growth in 1D under energy separation $E_1\neq E_2$). The analysis relies on a coherent-state ansatz, a detailed envelope equation, energy-type estimates, and separation lemmas for Gaussian interactions, yielding insights into how logarithmic nonlinearity interacts with semiclassical concentration and classical Hamiltonian flow. Overall, the results clarify the interplay between nonlinearity strength, Gaussian structure, and the underlying classical dynamics, with implications for logarithmic models in quantum mechanics and Bose–Einstein condensates.
Abstract
We consider the logarithmic Schr{ö}dinger equation in a semiclassical scaling, in the presence of a smooth, at most quadratic, external potential. For initial data under the form of a single coherent state, we identify the notion of criticality as far as the nonlinear coupling constant is concerned, in the semiclassical limit. In the critical case, we prove a general error estimate, and improve it in the case of initial Gaussian profiles. In this critical case, when the initial datum is the sum of two Gaussian coherent states with different centers in phase space, we prove a nonlinear superposition principle.