Nikolay Tzvetkov, Nicola Visciglia
We study upper bounds and the asymptotic behavior of high order moments for solutions to a family of linear and nonlinear Schroedinger equations.
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We prove the sharp L^1-L^{\infty} time-decay estimate for the 2D-Schroedinger equation with a general family of scaling critical electromagnetic potentials.
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
We prove long time Anderson localization for nonlinear random Schroedinger equation in $\ell^2$ by making a Birkoff normal form type transform to creat an energy barrier where there is essentially no mode propagation. One of the new features is that this transform is in a small neighborhood enabling us to treat "rough" data, where there are no moment conditions. The formulation of the present result is inspired by the RAGE theorem.
We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.
We consider the 3d cubic focusing nonlinear Schroedinger equation (NLS) i\partial_t u + Δu + |u|^2 u=0, which appears as a model in condensed matter theory and plasma physics. We construct a family of axially symmetric solutions, corresponding to an open set in H^1_{axial}(R^3) of initial data, that blow-up in finite time with singular set a circle in xy plane. Our construction is modeled on Raphaël's construction \cite{R} of a family of solutions to the 2d quintic focusing NLS, i\partial_t u + Δu + |u|^4 u=0, that blow-up on a circle.
We consider a class of nonlinear Schroedinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time d…
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroedinger Equation $- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$ on a manifold or in the Euclidean space. Here V represents the potential, p is an exponent greater than 1 and $ε$ a small parameter corresponding to the Planck constant. As $ε$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase in highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In the first part of this wor…
The main objective of this paper is understanding the propagation laws obeyed by high-frequency limits of Wigner distributions associated to solutions to the Schroedinger equation on the standard d-dimensional torus T^{d}. From the point of view of semiclassical analysis, our setting corresponds to performing the semiclassical limit at times of order 1/h, as the characteristic wave-length h of the initial data tends to zero. It turns out that, in spite that for fixed h every Wigner distribution satisfies a Liouville equation, their limits are no longer uniquely determined by those of the Wigne…
For nonlinear Schroedinger equations with a power nonlinearity, a new approach to derive the conservation law of the momentum and the pseudo conformal conservation law is obtained. Since this approach does not contain approximating procedure, the argument is simplified to derive these conservation laws.
We consider the Schrödinger Map equation in $2+1$ dimensions, with values into $§^2$. This admits a lowest energy steady state $Q$, namely the stereographic projection, which extends to a two dimensional family of steady states by scaling and rotation. We prove that $Q$ is unstable in the energy space $\dot H^1$. However, in the process of proving this we also show that within the equivariant class $Q$ is stable in a stronger topology $X \subset \dot H^1$.
We prove that, for any real $λ$, the system $-Δu +λu = u^3-βuv^2$, $ -Δv+λv =v^3-βvu^2$, $ u,v\in H^1_0(Ω),$ where $Ω$ is a bounded smooth domain of $R^3$, admits a bounded family of positive solutions $(u_β, v_β)$ as $β\to +\infty$. An upper bound on the number of nodal sets of the weak limits of difference $u_β-v_β$ is also provided. Moreover, for any sufficiently large fixed value of $β>0$ the system admits infinitely many positive solutions.
Existence and stability of traveling waves for discrete nonlinear Schroedinger equations over long times
研究离散非线性薛定谔方程中长时 traveling waves 的存在性与稳定性，采用调制方法证明其长期稳定性。
We prove existence of a special class of solutions to the (elliptic) Nonlinear Schroeodinger Equation $- ε^2 Δψ+ V(x) ψ= |ψ|^{p-1} ψ$, on a manifold or in the Euclidean space. Here V represents the potential, p an exponent greater than 1 and $ε$ a small parameter corresponding to the Planck constant. As $ε$ tends to zero (namely in the semiclassical limit) we prove existence of complex-valued solutions which concentrate along closed curves, and whose phase is highly oscillatory. Physically, these solutions carry quantum-mechanical momentum along the limit curves. In this first part we provide …
We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger operators and the Dirichlet Laplacian, on which the operators of interest are modeled. In addition we produce inequalities that are new even in the model case. These include a family of differential inequalities for generalized Riesz means and theorems stating that arithmetic means of lambda_k^p for p <= 3 are universally bounded from above by multiples of the …
We shall study the following initial value problem: \begin{equation}{\bf i}\partial_t u - Δu + V(x) u=0, \hbox{} (t, x) \in {\mathbf R} \times {\mathbf R}^n, \end{equation} $$u(0)=f,$$ where $V(x)$ is a real short--range potential, whose radial derivative satisfies some supplementary assumptions. More precisely we shall present a family of identities satisfied by the solutions to the previous Cauchy problem. As a by--product of these identities we deduce some uniqueness results and a lower bound for the so called local smoothing which becomes an identity in a precise asymptotic sense.
Scattering of radial $H^1$ solutions to the 3D focusing cubic nonlinear Schrödinger equation below a mass-energy threshold $M[u]E[u] < M[Q]E[Q]$ and satisfying an initial mass-gradient bound $\|u_0\|_{L^2} \|\nabla u_0 \|_{L^2} < \|Q\|_{L^2} \|\nabla Q\|_{L^2}$, where $Q$ is the ground state, was established in Holmer-Roudenko (2007). In this note, we extend the result in Holmer-Roudenko (2007) to non-radial $H^1$ data. For this, we prove a non-radial profile decomposition involving a spatial translation parameter. Then, in the spirit of Kenig-Merle (2006), we control via momentum conservation…
We consider a nonlinear Schroedinger equation with a finite bands periodic potential in R . We assume the existence of an orbitally stable family of ground states. We prove that under appropriate hypotheses the ground states are asymptotically stable.
We consider wavepackets composed of two modulated carrier Bloch waves with opposite group velocities in the one dimensional periodic Nonlinear Schroedinger/Gross-Pitaevskii equation. These can be approximated by first order coupled mode equations (CMEs) for the two slowly varying envelopes. Under a suitably selected periodic perturbation of the periodic structure the CMEs possess a spectral gap of the corresponding spatial operator and allow families of exponentially localized solitary waves parametrized by velocity. This leads to a family of approximate solitary waves in the periodic nonlinea…
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