M. Guardia, Z. Hani, E. Haus, A. Maspero and M. Procesi. (2019) A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation
. Rendiconti Lincei – Matematica e Applicazioni, 29
ABSTRACT: We present the recent result in 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 finitegap 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 Hs topology (0 < s < 1) and whose Hs norm can grow by any given factor.
M. Guardia, V. Kaloshin and J. Zhang, Asymptotic density of collision orbits in the Restricted Circular Planar 3 Body Problem
, preprint May 2018.
ABSTRACT: For the Restricted Circular Planar 3 Body Problem, we show that there exists an open set U in phase space independent of fixed measure, where the set of initial points which lead to collision is O(μ120) dense as μ→0.
M. Guardia, Z. Hani, E. Haus, A. Maspero and M. Procesi, Strong nonlinear instability and growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation
, preprint October 2018.
ABSTRACT: We consider 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 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 Hs(T2) (0<s<1). More precisely, we construct solutions of the 2D cubic NLS that start arbitrarily close to such invariant tori in the Hs topology and whose Hs 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.
O. M. L. Gomide, M. Guardia and T. M. Seara, Critical velocity in kink-defect interaction models: rigorous results
, preprint November 2018.
ABSTRACT: In this work we study a model of interaction of kinks of the sine-Gordon equation with a weak defect. We obtain rigorous results concerning the so-called critical velocity derived in by a geometric approach. More specifically, we prove that a heteroclinic orbit in the energy level 0 of a 2-dof Hamiltonian Hε is destroyed giving rise to heteroclinic connections between certain elements (at infinity) for exponentially small (in ε) energy levels. In this setting Melnikov theory does not apply because there are exponentially small phenomena.
R. Feola, F. Giuliani and M. Procesi, Reducible KAM tori for Degasperis-Procesi equation, arXiv:1812.08498
, preprint December 2018.
ABSTRACT: We develop KAM theory close to an elliptic fixed point for quasi-linear Hamiltonian perturbations of the dispersive Degasperis-Procesi equation on the circle. The overall strategy in KAM theory for quasi-linear PDEs is based on Nash-Moser nonlinear iteration, pseudo differential calculus and normal form techniques. In the present case the complicated symplectic structure, the weak dispersive effects of the linear flow and the presence of strong resonant interactions require a novel set of ideas. The main points are to exploit the integrability of the unperturbed equation, to look for special wave packet solutions and to perform a very careful algebraic analysis of the resonances. Our approach is quite general and can be applied also to other 1d integrable PDEs. We are confident for instance that the same strategy should work for the Camassa-Holm equation.