Growth of Sobolev norms in quasi-integrable quantum systems

In this paper, we consider the following problem. Let iu t +Δu+V(x,t)u= 0 be a linear Schrodinger equation ( periodic boundary conditions) where V is a real, bounded, real analytic potential which is periodic in x and quasi periodic in t with diophantine frequency vector λ. Denote S(t) the corresponding flow map. Thus S(t) preserves the L 2-norm and our aim is to study its behaviour on H s (T D ), s> 0. Our main result is the growth in time is at most logarithmic; thus if φ∈H s , then More precisely, (*) is proven in 1D and 2D when V is small. We also exhibit examples showing that a growth of higher Sobolev norms may occur in this context and (*) is thus essentially best possible.

We give a detailed description in 1-D the growth of Sobolev norms for time dependent linear generalized KdV-type equations on the circle. For most initial data, the growth of Sobolev norms is polynomial in time for fixed analytic potential with admissible growth. If the initial data are given in a fixed smaller function space with more strict admissible growth conditions for $V(x,t)$ , then the growth of previous Sobolev norms is at most logarithmic in time.

Fix s > 1. Colliander et al. proved in (Invent Math 181:39–113, 2010) the existence of solutions of the cubic defocusing nonlinear Schrödinger equation in the two torus whose s-Sobolev norm undergoes arbitrarily large growth as time evolves. In this paper we generalize their result to the cubic defocusing nonlinear Schrödinger equation with a convolution potential. Moreover, we show that the speed of growth is the same as the one obtained for the cubic defocusing nonlinear Schrödinger equation in Guardia and Kaloshin (Growth of Sobolev norms in the cubic defocusing Nonlinear Schrödinger Equation. To appear in the Journal of the European Mathematical Society, 2012).

We present the recent result in [29] concerning strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap solutions of the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the H^s topology (0 < s < 1) and whose H^s norm can grow by any given factor.

1-d Quantum harmonic oscillator with time quasi-periodic quadratic perturbation: Reducibility and growth of Sobolev norms

On time dependent Schrödinger equations: Global well-posedness and growth of Sobolev norms

On growth of Sobolev norms in time dependent linear Schrödinger equations on spheres

We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s>1 . Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with s -Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is c>0 such that for any \mathcal K\gg 1 we find a solution u and a time T such that \| u(T)\|_{H^s}\geq\mathcal K \| u(0)\|_{H^s} . Moreover, the time T satisfies the polynomial bound 0 < T < \mathcal K^c .

We consider the following coupled cubic Schrödinger equationsWe prove that there exists a beating effect, i.e. an energy exchange between different modes. This construction may be transported to the linear time-dependent Schrödinger equation: we build solutions such that their Sobolev norms grow logarithmically. All of these results are stated for large but finite times.

We improve Delort’s method to show that solutions of linear Schrodinger equations with a time dependent Gevrey potential on the torus, have at most logarithmically growing Sobolev norms. In particular, it contains the result of Wang (Commun Partial Differ Equ 33:2164–2179, 2008), which deals with analytic potentials in dimension 1.

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the twodimensional torus. The equation admits a special family of elliptic invariant quasiperiodic tori called finite gap solutions. These are inherited from the integrable 1D model (cubic NLS on the circle) by considering solutions that depend only on one variable. We study the long-time stability of such invariant tori for the 2D NLS model and show that, under certain assumptions and over sufficiently long time scales, they exhibit a strong form of transverse instability in Sobolev spaces H^s(\mathbb{T}^2)(0 < s < 1) . More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the H^s topology and whose H^s norm can grow by any given factor. This work is partly motivated by the problem of infinite energy cascade for 2D NLS, and seems to be the first instance where (unstable) long-time nonlinear dynamics near (linearly stable) quasiperiodic tori is studied and constructed.

Erratum to “Growth of Sobolev norms in the cubic defocusing nonlinear Schrödinger equation”

Growth of Sobolev norms in linear Schrödinger equations as a dispersive phenomenon

In this paper we obtain a stabilization result for both linear and nonlinear Schrödinger equations under generic assumptions on the potential. Then we consider the Schrödinger equations with a potential which has a random time-dependent amplitude. We show that if the distribution of the amplitude is sufficiently non-degenerate, then any trajectory of the system is almost surely non-bounded in Sobolev spaces.

On growth of sobolev norms in linear schrödinger equations with smooth time dependent potential