We are concerned with the global behavior of the solutions of the focusing mass supercritical nonlinear Schr{ö}dinger equation under partial harmonic confinement. We establish a necessary and sufficient condition on the initial data below the ground states to determine the global behavior (blow-up/scattering) of the solution. Our proof of scattering is based on the varia-tional characterization of the ground states, localized virial estimates, linear profile decomposition and nonlinear profiles.
We prove the existence of orbitally stable ground states to NLS with a partial confinement together with qualitative and symmetry properties. This result is obtained for nonlinearities which are $L^2$-supercritical, in particular we cover the physically relevant cubic case. The equation that we consider is the limit case of the cigar-shaped model in BEC.
We give a new method to prove the existence, non-existence, multiplicity, orbital stability/instability of standing waves for NLS with partial confinement without the subcritical hypothesis, even in the reduction equation. Using this method, we give an affirmative answer for an open problem proposed by [7, Remark 1.10] where the authors conjectured the existence of more than a normalized solution. We also establish uniqueness results of the ground state solutions depending on the bifurcation parameters. We explain that when the effect of partial confinement is strong, a dimension reduction app…
We prove polynomial upper bounds on the growth of solutions to 2d cubic NLS where the Laplacian is confined by the harmonic potential. Due to better bilinear effects our bounds improve on those available for the $2d$ cubic NLS in the periodic setting: our growth rate for a Sobolev norm of order s=2k, $k\in \mathbb{N}$, is $t^{2(s-1)/3+\varepsilon}$. In the appendix we provide an direct proof, based on integration by parts, of bilinear estimates associated with the harmonic oscillator.
We consider the nonlinear Schrodinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.
In this paper we study radial solutions of certain two-dimensional nonlinear Schrödinger equation with harmonic potential, which is supercritical with respect to the initial data. By combining the nonlinear smoothing effect of Schr ödinger equation with Lp estimates of Laguerre functions, we are able to prove an almost sure global well-posedness result and the invariance of the Gibbs measure. We also discuss an application to the NLS equation without harmonic potential.
We study nonlinear Schrödinger $i\partial_tu-Hu=F(u)$ (NLSH) equation associated to harmonic oscillator $H=-Δ+|x|^2$ in modulation spaces $M^{p,q}.$ When $F(u)= (|x|^{-γ}\ast |u|^2)u, $ we prove global well-posedness for (NLSH) in modulation spaces $M^{p,p}(\mathbb R^d)$ $ (1\leq p < 2d/(d+γ), 0<γ< \min \{ 2, d/2\}).$ When $F(u)= (K\ast |u|^{2k})u$ with $K\in \mathcal{F}L^q $ (Fourier-Lebesgue spaces) or $M^{\infty,1}$ (Sjöstrand's class) or $M^{1, \infty},$ some local and global well-posedness for (NLSH) are obtained in some modulation spaces. When $F$ is real entire and $F(0)=0$, we prove lo…
The uniqueness of the positive ground state solutions of fractional Shrodinger equations with a harmonic potential has not been covered by the breakthrough method developed in [1, 2]. It has remained an open question for years. [3] and [5] were quite recently able to prove the existence of the ground state solutions and to derive important properties but they failed to address the uniqueness. [3] left it as an open question, and [5] run numerical simulations that were in favor of the uniqueness. Very recently, the authors of this note developed a general and unified method to prove the uniquen…
We study the continuous resonant (CR) equation which was derived by Faou-Germain-Hani as the large-box limit of the cubic nonlinear Schrödinger equation in the small nonlinearity (or small data) regime. We first show that the system arises in another natural way, as it also corresponds to the resonant cubic Hermite-Schrödinger equation (NLS with harmonic trapping). We then establish that the basis of special Hermite functions is well suited to its analysis, and uncover more of the striking structure of the equation. We study in particular the dynamics on a few invariant subspaces: eigenspaces …
We show that in the presence of a rough external potential, a 3d cubic defocusing NLS is globally well-posed in H^s for s>5/6. The proof is based on the approach of Colliander-Keel-Staffilani-Takaoka-Tao, called the I-method, but in order to deal with a rough potential, we modify harmonic analysis tools by spectral theory.
We study the existence and stability of standing waves associated to the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose-Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, …
We study analytically the existence and uniqueness of the ground state of the nonlinear Schrödinger equation (NLSE) with a general power nonlinearity described by the power index $σ\ge0$. For the NLSE under a box or a harmonic potential, we can derive explicitly the approximations of the ground states and their corresponding energy and chemical potential in weak or strong interaction regimes with a fixed nonlinearity $σ$. Besides, we study the case where the nonlinearity $σ\to\infty$ with a fixed interaction strength. In particular, a bifurcation in the ground states is observed. Numerical res…
We study two resonant Hamiltonian systems on the phase space $L^2(\mathbb{R} \rightarrow \mathbb{C})$: the quintic one-dimensional continuous resonant equation, and a cubic resonant system that has appeared in the literature as a modified scattering limit for an NLS equation with cigar shaped trap. We prove that these systems approximate the dynamics of the quintic and cubic one-dimensional NLS with harmonic trapping in the small data regime on long times scales. We then pursue a thorough study of the dynamics of the resonant systems themselves. Our central finding is that these resonant equat…
We consider the cubic nonlinear Schrödinger equation with harmonic trapping on $\mathbb{R}^D$ ($1\leq D\leq 5$). In the case when all but one directions are trapped (a.k.a "cigar-shaped" trap), following the approach of Hani-Pausader-Tzvetkov-Visciglia, we prove modified scattering and construct modified wave operators for small initial and final data respectively. The asymptotic behavior turns out to be a rather vigorous departure from linear scattering and is dictated by the resonant system of the NLS equation with full trapping on $\mathbb{R}^{D-1}$. In the physical dimension $D=3$, this sy…
We consider a 2D time-dependent quantum system of $N$-bosons with harmonic external confining and \emph{attractive} interparticle interaction in the Gross-Pitaevskii scaling. We derive stability of matter type estimates showing that the $k$-th power of the energy controls the $H^{1}$ Sobolev norm of the solution over $k$-particles. This estimate is new and more difficult for attractive interactions than repulsive interactions. For the proof, we use a version of the finite-dimensional quantum di Finetti theorem from [49]. A high particle-number averaging effect is at play in the proof, which is…
We consider the final-state problem for the nonlinear Schrödinger equations (NLS) with a suitable time-decaying harmonic oscillator. In this equation, the power of nonlinearity $|u|^ρu $ is included in the long-range class if $0 < ρ\leq 2/(n(1- λ)) $ with $0 \leq λ<1/2$, which is determined by the harmonic potential and a coefficient of Laplacian. In this paper, we find the final state for this system and obtain the decay estimate for asymptotics.
Consider the global wellposedness problem for nonlinear Schrödinger equation \[ i\partial_t u = [-\tfrac{1}{2} Δ+ V(x)] u \pm |u|^{4/(d-2)} u, \ u(0) \in Σ(\mathbf{R}^d), \] where $Σ$ is the weighted Sobolev space $\dot{H}^1 \cap |x|^{-1} L^2$. The case $V(x) = \tfrac{1}{2}|x|^2$ was recently treated by the author. This note generalizes the results to a class of "approximately quadratic" potentials. We closely follow the previous concentration compactness arguments for the harmonic oscillator. A key technical difference is that in the absence of a concrete formula for the linear propagator, we…
We consider the 3d cubic nonlinear Schrödinger equation (NLS) with a strong 2d harmonic potential. The model is physically relevant to observe the lower-dimensional dynamics of the Bose-Einstein condensate, but its ground state cannot be constructed by the standard method due to its supercritical nature. In Bellazzini-Boussaïd-Jeanjean-Visciglia \cite{BBJV}, the authors constructed a proper ground state introducing a constrained energy minimization problem. In this paper, we further investigate the properties of the ground state. First, we show that as the partial confinement is increased, the…
We consider the focusing mass supercritical nonlinear Schrödinger equation with rotation \begin{equation*} iu_{t}=-\frac{1}{2}Δu+\frac{1}{2}V(x)u-|u|^{p-1}u+L_Ωu,\quad (x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and $V(x)$ is an anisotropic harmonic potential. Here $L_Ω$ is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stabl…
We interpret the lens transformation (a variant of the pseudoconformal transformation) as a pseudoconformal compactification of spacetime, which converts the nonlinear Schrödinger equation (NLS) without potential with a nonlinear Schrödinger equation with attractive harmonic potential. We then discuss how several existing results about NLS can be placed in this compactified setting, thus offering a new perspective to view this theory.
The Hyperbolic Nonlinear Schrodinger equation (HypNLS) arises as a model for the dynamics of three-dimensional narrowband deep water gravity waves. In this study, the Petviashvili method is exploited to numerically compute bi-periodic time-harmonic solutions of the HypNLS equation. In physical space they represent non-localized standing waves. Non-trivial spatial patterns are revealed and an attempt is made to describe them using symbolic dynamics and the language of substitutions. Finally, the dynamics of a slightly perturbed standing wave is numerically investigated by means a highly acccura…
Dynamics of focusing nonlinear Schrödinger equation with partial harmonic confinement in higher dimensions
Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所）；National Key Laboratory of Computational Physics（计算物理国家重点实验室）；Graduate School of China Academy of Engineering Physics（中国工程物理研究院研究生院）
We study the instability of standing wave solutions for nonlinear Schrödinger equations with a one-dimensional harmonic potential in dimension $N\ge 2$. We prove that if the nonlinearity is $L^2$-critical or supercritical in dimension $N-1$, then any ground states are strongly unstable by blowup.
We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schrödinger operators. Such refinements begin to characterize the near-optimizers of the Strichartz estimate, and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the spacetime Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials …
In this article, we first present the construction of Gibbs measures associated to nonlinear Schrödinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is globally well-posed for rough initial conditions in a statistical set (the support of the measures). Finally, we prove that the Gibbs measures are indeed invariant by the flow of the equation. As a byproduct of our analysis, we give a global well-posedness and scattering result for the $L^2$ critical and super-critical NLS (without harmonic potential).
We propose an approach to nonlinear evolution equations with large and decaying external potentials that addresses the question of controlling globally-in-time the nonlinear interactions of localized waves in this setting. This problem arises when studying localized perturbations around (possibly non-decaying) special solutions of evolution PDEs, and trying to control the projection onto the continuous spectrum of the nonlinear radiative interactions. One of our main tools is the Fourier transform adapted to the Schrödinger operator $H=-Δ+V$, which we employ at a nonlinear level. As a first st…
