Related papers: Weakly nonlinear Schr\"odinger equation with rando…
The nonlinear Schr\"odinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that…
We consider the cubic nonlinear Schr\"odinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness…
We study the kinetic, weak coupling limit of the dynamics governed by a discrete random Schr\"odinger operator on $\mathbb{Z}^3$. For sequences of $\ell^2\left(\mathbb{Z}^3\right)$-bounded initial states and convergent initial Wigner…
We consider random Schr\"odinger equations on $\bR^d$ or $\bZ^d$ for $d\ge 3$ with uncorrelated, identically distributed random potential. Denote by $\lambda$ the coupling constant and $\psi_t$ the solution with initial data $\psi_0$.…
We analyze the weak-coupling limit of the random Schr\"odinger equation with low frequency initial data and a slowly decorrelating random potential. For the probing signal with a sufficiently long wavelength, we prove a homogenization…
The propagation of an initially Gaussian wave packet of width $\Delta_0$ in a cubic non-linear Schrodinger equation with a negative coupling constant for the nonlinear term is considered . It is predicted analytically and verified…
We consider the initial value problem for cubic derivative nonlinear Schr\"odinger equation in one space dimension. Under a suitable weakly dissipative condition on the nonlinearity, we show that the small data solution has a logarithmic…
We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrodinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the…
We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schroedinger equations in which either the nonlinear coupling constant or, equivalently, the…
We consider the linear Schr\"odinger equation under periodic boundary condition, driven by a random force and damped by a quasilinear damping: $$ \frac{d}{dt}u+i\big(-\Delta+V(x)\big) u=\nu \Big(\Delta u-\gr |u|^{2p}u-i\gi |u|^{2q}u \Big)…
We study a non-linear Schroedinger equation with a Hartree-type nonlinearity and a localized random time-dependent external potential. Sharp dispersive estimates for the linear Schroedinger equation with a random time-dependent potential…
We consider a nonlinear semi-classical Schrodinger equation for which it is known that quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. If the initial data is an energy bounded sequence, we…
We consider the free linear Schroedinger equation on a torus $\mathbb T^d$, perturbed by a Hamiltonian nonlinearity, driven by a random force and damped by a linear damping: $$u_t -i\Delta u +i\nu \rho |u|^{2q_*}u = - \nu f(-\Delta) u +…
The dynamics of an initially localized wavepacket is studied for the generalized nonlinear Schroedinger Equation with a random potential, where the nonlinearity term is |\psi|^p*\psi and "p" is arbitrary. Mainly short times for which the…
We consider the long time limit theorems for the solutions of a discrete wave equation with a weak stochastic forcing. The multiplicative noise conserves the energy and the momentum. We obtain a time-inhomogeneous Ornstein-Uhlenbeck…
We derive a new kinetic and a porous medium equations from the nonlinear Schr\"odinger equation with random potentials. The kinetic equation has a very similar form with the 4-wave turbulence kinetic equation in the wave turbulence theory.…
We consider the Schr\"odinger equation with a time-independent weakly random potential of a strength $\epsilon\ll 1$, with Gaussian statistics. We prove that when the initial condition varies on a scale much larger than the correlation…
We study the cubic weakly nonlinear Schr\"odinger equation with randomized spatially quasi-periodic initial data in higher dimensions. Under a polynomial decay assumption in Fourier space, we establish a {\em Large Deviations Principle} for…
We consider the small time semi-classical limit for nonlinear Schrodinger equations with defocusing, smooth, nonlinearity. For a super-cubic nonlinearity, the limiting system is not directly hyperbolic, due to the presence of vacuum. To…
We study the time evolution of a quantum particle in a Gaussian random environment. We show that in the weak coupling limit the Wigner distribution of the wave function converges to a solution of a linear Boltzmann equation globally in…