Related papers: Local limit of the focusing discrete NLS
We analyze the thermodynamics of the focusing discrete nonlinear Schr\"odinger equation in dimensions $d\ge 3$ with general nonlinearity $p>1$ and under a model with two parameters, representing inverse temperature and strength of the…
We prove the existence and the invariance of a Gibbs measure associated to the defocusing sub-quintic Nonlinear Schroedinger equations on the disc of the plane $\R^2$. We also prove an estimate giving some intuition to what may happen in 3…
The paper interprets the cubic nonlinear Schr\"odinger equation as a Hamiltonian system with infinite dimensional phase space. There is a Gibbs measure which is invariant under the flow associated with the canonical equations of motion. The…
Our first purpose is to extend the results from \cite{T} on the radial defocusing NLS on the disc in $\mathbb{R}^2$ to arbitrary smooth (defocusing) nonlinearities and show the existence of a well-defined flow on the support of the Gibbs…
In this paper, we are concerned with the study of statistical equilibria for focusing nonlinear Schr\"odinger and Hartree equations on the d-dimensional torus when d=1,2,3. Due to the focusing nature of the nonlinearity in these PDEs, Gibbs…
We review some recent results concerning Gibbs measures for nonlinear Schroedinger equations (NLS), with implications for the theory of the NLS, including stability and typicality of solitary wave structures. In particular, we discuss the…
This paper concerns Gibbs measures $\nu$ for some nonlinear PDE over the $D$-torus ${\bf T}^D$. The Hamiltonian $H=\int_{{\bf T}^D} \Vert\nabla u\Vert^2 - \int_{{\bf T}^D} \vert u\vert^p$ has canonical equations with solutions in…
We study the derivation of the Gibbs measure for the nonlinear Schr\"{o}dinger equation (NLS) from many-body quantum thermal states in the high-temperature limit. In this paper, we consider the nonlocal NLS with defocusing and unbounded…
We study Gibbs measures invariant under the flow of the NLS on the unit disc of $\R^2$. For that purpose, we construct the dynamics on a phase space of limited Sobolev regularity and a wighted Wiener measure invariant by the NLS flow. The…
We consider the Gibbs measure for the focusing nonlinear Schr\"odinger equation on the one-dimensional torus $\mathbb T$, that was introduced in a seminal paper by Lebowitz, Rose and Speer (1988). We show that in the large torus limit, the…
We investigate the invariance of the Gibbs measure for the fractional Schrodinger equation of exponential type (expNLS) $i\partial_t u + (-\Delta)^{\frac{\alpha}2} u = 2\gamma\beta e^{\beta|u|^2}u$ on $d$-dimensional compact Riemannian…
We study the fluctuations of the focusing $\Phi^4$-measure on the one-dimensional torus in the infinite volume limit. This measure is an invariant Gibbs measure for the nonlinear Schr\"odinger equation. It had previously been shown by B.…
We introduce a mass conserving stochastic perturbation of the discrete nonlinear Schr\"odinger equation that models the action of a heat bath at a given temperature. We prove that the corresponding canonical Gibbs distribution is the unique…
In this work, we obtain a microscopic derivation of Gibbs measures for the focusing quintic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{T}$ from many-body quantum Gibbs states. On the quantum many-body level, the quintic…
We study the Gibbs measure associated to the periodic cubic nonlinear Schr\"odinger equation. We establish a change of variable formula for this measure under the first step of the Birkhoff normal form reduction. We also consider the case…
The soliton resolution conjecture for the focusing nonlinear Schrodinger equation (NLS) is the vaguely worded claim that a global solution of the NLS, for generic initial data, will eventually resolve into a radiation component that…
In a recent paper, in collaboration with Mathieu Lewin and Phan Th{\`a}nh Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like functionals could be derived from many-body quantum mechanics, in a mean-field limit. This…
We study the construction of the Gibbs measures for the {\it focusing} mass-critical fractional nonlinear Schr\"odinger equation on the multi-dimensional torus. We identify the sharp mass threshold for normalizability and…
We construct a Gibbs measure for the nonlinear Schrodinger equation (NLS) on the circle, conditioned on prescribed mass and momentum: d \mu_{a,b} = Z^{-1} 1_{\int_T |u|^2 = a} 1_{i \int_T u \bar{u}_x = b} exp (\pm1/p \int_T |u|^p - 1/2…
We consider the two-dimensional defocusing nonlinear Schr\"odinger equation (NLS) on the unit disc in the plane with the Gibbs initial data under radial symmetry. By using a type of random averaging operator ansatz, we build a strong…