Ali Mezher | Searcharxiv

arXiv2026

Trapped bosons in mean field QED, nonlinear resonance cascades and dynamical BEC formation

In this paper, we study a system of bosons trapped in a confining potential, interacting with a quantized field of coherent photons in the mean field description of non-relativistic Quantum Electrodynamics (QED) obtained by [N. Leopold and P. Pickl , 2017]. We derive the effective nonlinear cascade equations governing the emission and absorption of coherent photons by the boson subsystem in a combined weak-coupling and macroscopic time limit. We demonstrate that solutions to this nonlinear cascade determine a monotone decreasing energy flow in the boson subsystem, and thereby describe the dynamical formation of a Bose-Einstein condensate (BEC) with full ground state occupation, under conservation of the total boson $L^2$ mass. We note that this process is crucially different from thermal relaxation to the ground state, and fundamentally depends on the nonlinear nature of the cascade dynamics.

math-ph↗

arXiv2024

On the modified Zakharov--Kuznetsov equation on the cylinder: A trilinear Bourgain-space estimate and its application to local well-posedness

We study the modified Zakharov--Kuznetsov equation on $\mathbb R \times \mathbb T$. We establish a trilinear Bourgain-space estimate on the full time line for the derivative cubic term, with input exponent $\frac12+\delta$ and output exponent $-\frac12+3\delta$, and use it to recover local well-posedness in $H^s(\mathbb R \times \mathbb T)$ for $s>1$. After time localization, this output exponent yields a factor $T^{2\delta}$ in the contraction argument. Lower Sobolev regularity thresholds are already known; the purpose here is to give a self-contained dyadic proof of this different modulation balance.

math.AP↗