A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation

Abstract

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 timescales, they exhibit a strong form of transverse instability in Sobolev spaces H s (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.1.1. The dynamical system and its quasiperiodic objects. We start by describing the dynamical system and its quasiperiodic invariant objects at the center of our analysis. Consider the periodic cubicNow, we describe the invariant objects around which we will study and construct our long-time nonlinear dynamics. Of course, such a task requires a very precise understanding of the linearized dynamics around such objects. For this reason, we take the simplest non-trivial family of invariant quasiperiodic tori admitted by (2D-NLS), namely those inherited from its completely integrable 1D counterpartThis is a subsystem of (2D-NLS) if we consider solutions that depend only on the first spatial variable. It is well known that equation (1D-NLS) is integrable and its phase space is foliated by tori of finite or infinite dimension with periodic, quasiperiodic, or almost periodic dynamics. The quasiperiodic orbits are usually called finite-gap solutions. Such tori are Lyapunov stable (for all time!) as solutions of (1D-NLS) (as will be clear once we exhibit its integrable structure) and some of them are linearly stable as solutions of (2D-NLS), but we will be interested in their long-time nonlinear stability (or lack of it) as invariant objects for the 2D equation (2D-NLS). In fact, we shall show that they are nonlinearly unstable as solutions of (2D-NLS), and in a strong sense, in certain topologies and after very long times. Such instability is transversal in the sense that one drifts along the purely 2-dimensional directions: solutions which are initially very close to 1-dimensional become strongly 2-dimensional after some long time scales 2 .1.2. Energy Cascade, Sobolev norm growth, and Lyapunov instability. In addition to studying long-time dynamics close to invariant objects for NLS, another purpose of this work is to make progress on a fundamental problem in nonlinear wave theory, which is the transfer of energy between characteristically different scales for a nonlinear dispersive PDE. This is called the energy cascade phenomenon....

Long time dynamics of Schrödinger and wave equations on flat tori

“…The second group analyzes phenomenon of growth of Sobolev norms [DG17, Den17, SW18], see also [PTV17] for 2 and 3 dimensional arbitrary compact manifolds. It would be interesting to see if the techniques developed in this paper could allow to extend these results for flat tori, as well as those in [MP18,GHHMP18] concerning stability and instability of finite gap solutions of NLS on the standard torus.…”

On a Damped Szegö Equation (With an Appendix in Collaboration With Christian Klein)

“…In the last decade, a number of papers have tried to display growth of Sobolev norms of high regularity for solutions of globally wellposed nonlinear Hamiltonian partial differential equations. This question, raised by Bourgain in [1], [2] for the defocusing nonlinear Schrödinger equation on the torus, led to several contributions constructing solutions with a small initial Sobolev norm of high regularity and a big Sobolev norm at some later time, see [4], [5], [8], [13], [16], [19], [15], [14], [12]. The actual existence of unbounded trajectories was proved in [21], [17], [18], [7], [23], [24], [3], [22], [9].…”

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

“…Before closing this introduction let us mention that the construction of unbounded orbits in nonlinear Schrödinger equations is an extremely difficult and challeging problem. A first breakthrough was achieved in [10], which constructs solutions of the cubic nonlinear Schrödinger equation on T 2 whose Sobolev norms become arbitrary large (see also [21,17,23,19,18] for generalizations of this result). At the moment, existence of unbounded orbits has only been proved by Gérard and Grellier [16] for the cubic Szegő equation on T, and by Hani, Pausader, Tzvetkov and Visciglia [22] for the cubic NLS on R × T 2 .…”