Piotr Bizoń

Piotr Bizoń

Dynamics in spatially confined Hamiltonian systems

Understanding of propagation of nonlinear waves in spatially confined systems is much more challenging than in spatially unbounded systems because the waves cannot escape to infinity and keep self-interacting for all times, inducing complicated long-time behavior. The central physical question is: how the energy injected to the system gets distributed over the degrees of freedom during the evolution? In particular, can energy migrate to arbitrarily small spatial scales? Such turbulent cascades of energy can be measured by the growth of high Sobolev norms of solutions and the key mathematical question is whether these norms can become unbounded in finite time (strong turbulence) or infinite time (weak turbulence). During the past decade this question has been intensively studied for the nonlinear Schrödinger equation (NLS) on a two-torus, starting from the pioneering work by Colliander et al. [1] who proved that there exist solutions for which high Sobolev norms can grow from arbitrarily small to arbitrarily large in finite time. Later, the existence of weakly turbulent solutions has been established for some artificially `designed' Hamiltonian systems (most notably, for the cubic Szegő equation [2] ) and, somewhat paradoxically, for the NLS on $R\times T^2$ [3] . The characteristic property of any confined Hamiltonian system is that the associated linearized system has a purely discrete spectrum of frequencies, hence it is natural to expand solutions in the basis of linear eigenstates. This transforms the original partial differential equation into an infinite dimensional dynamical system with discrete degrees of freedom (usually referred to as `modes'). The nonlinearity generates new frequencies that may lead to resonances between the modes. For small amplitude solutions, these resonant interactions can be shown to dominate the evolution for a long (but finite) time, therefore by dropping all nonresonant terms from the Hamiltonian one obtains a good long-time approximation, called the resonant system. It is believed that understanding the dynamics of resonant systems will open the door to understanding the dynamics of the corresponding PDEs. Thus, our strategy is to analyze the resonant systems for some physically relevant nonlinear dispersive equations: the Gross-Pitaevskii equation (a mean field model of the Bose-Einstein condensate), the Einstein equation with negative cosmological constant (describing the dynamics of asymptotically anti-de Sitter spacetimes which play a prominent role in the gauge/gravity duality), semilinear wave equations on compact manifolds, surface water wave equations, and others. Note that for complicated equations or domains, the very derivation of the resonant system may be highly nontrivial [4] . Having understood the properties of the resonant system, the next step is to export these properties to the original PDE. Some preliminary results of our studies are described in [5,6,7] . In [5] and [6] , we have found finite-dimensional invariant subspaces for two resonant systems of completely different physical origins. Intriguing analogies between these systems suggest that we have stumbled upon a novel rich mathematical structure, which deserves further studies.

  1. J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181, 39 (2010) ↩
  2. P. Gérard, S. Grellier, The cubic Szegő equation, Ann. Scient. éc. Norm. Sup. 43, 761 (2010) ↩
  3. Z. Hani, B. Pausader, N. Tzvetkov, N. Visciglia, Modified scattering for the nonlinear Schrödinger equation on product space and applications, Forum of Math. Pi, 3, e4 (2015)↩
  4. B. Craps, O. Evnin, J. Vanhoof, Renormalization, averaging, conservation laws and AdS (in)stability, JHEP 1501, 108 (2015) ↩
  5. P. Bizoń, B. Craps, O. Evnin, D. Hunik, V. Luyten, M. Maliborski, Conformal flow on $S^3$ and weak field integrability in AdS$_4$, Comm. Math. Phys. 353, 1179 (2017) ↩
  6. A. Biasi, P. Bizoń, B. Craps, O. Evnin, Exact LLL solutions for BEC vortex precession, arXiv:1705.00867, submitted to Phys. Rev. A ↩
  7. P. Bizoń, D. Hunik-Kostyra, D. Pelinovsky, Ground state of the conformal flow on $S^3$, arXiv:1706.07726, submitted to Comm. Pure Appl. Math. ↩