Schedule for: 19w5076 - Hamiltonian PDEs: KAM, Reducibility, Normal Forms and Applications

Abstract: Modulation, or amplitude, equations are simplified partial differential equations which approximately describe the evolution of solutions of more complicated PDEs in certain asymptotic regimes. The Korteweg-de Vries, nonlinear Schroedinger, and Ginzburg-Landau equations are all examples of modulation equations. In this talk I will describe how such equations serve as a sort of normal form for large classes of nonlinear partial differential equations and describe a general method of proving that they give accurate approximations to the solutions of the original, more complicated equations.

The Gross-Pitaevskii (GP) equation is named after Eugene P. Gross and Lev P. Pitaevskii who introduced it independently in 1961 for the purpose of describing vortices in superfluids. At that time the only known superfluid was liquid helium, but since the first experimental realization of Bose-Einstein condensates in trapped gases of alkali atoms in 1995 the GP equation has become a basic tool for theoretical investigations of the manifold quantum phenomena exhibited by dilute, ultracold Bose gases. Mathematically, the GP equation is a special case of a non-linear Schrödinger equation and the mathematics and physics literature about it is vast. In the talk the focus will be on the following topics: 1) The interpretation of the equation on the basis of many-body quantum mechanics. 2) Vortices in rapidly rotating trapped superfluids at strong coupling and the determination of critical speeds that mark phase transitions from single vortices to a vortex lattice pattern and finally to a giant vortex.

Landau-Lifshitz equations are the basic dynamical equations in a micromagnetic description of a ferromagnet. They are naturally viewed as geometric evolution PDE of dispersive, Hamiltonian type (``Schrodinger maps") , which scale critically with respect to the physical energy in two space dimensions. We describe here recent results on the stability of important topological soliton solutions known as ``chiral magnetic skyrmions". This is joint work with Li Wang.

Considering general classes of nonlinear Schr\"odinger equations on the circle with nontrivial cubic part and without external parameters, I will present the construction a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant $M$ and a sufficiently small parameter $\varepsilon$, for generic initial data of size $\varepsilon$, the flow is conjugated to an integrable flow up to an arbitrary small remainder of order $\varepsilon^{M+1}$. This implies that for such initial data $u(0)$ we control the Sobolev norm of the solution $u(t)$ for time of order $\varepsilon^{-M}$. Furthermore this property is locally stable: if $v(0)$ is sufficiently close to $u(0)$ (of order $\varepsilon^{3/2}$) then the solution $v(t)$ is also controled for time of order $\varepsilon^{-M}$. This is a joint work with Erwan Faou and Benoît Grébert.

In the talk I will present some recent results on upper and lower bounds for the growth of Sobolev norms in linear time dependent Schrödinger equations. After discussing the known upper bounds on the Sobolev norms of the solutions, I will explain a strategy to construct time dependent perturbations which provoke transfer of energy from low to high modes, leading to unbounded solutions.

Steady states in Hamiltonian PDEs are often constrained minimizers of energy subject to fixed mass and momentum. I will discuss two examples when the minimizers are degenerate so that spectral stability of minimizers does not imply their nonlinear stability due to lack of coercivity of the second variation of energy. For the example related to the conformally invariant cubic wave equation on three-sphere, we prove that integrability of the normal form equations results in nonlinear stability of steady states. For the other example related to the nonlinear Schrodinger equation on a star graph, we prove that the lack of momentum conservation results in the irreversible drift of steady states and their nonlinear instability.

It is well known that the phase space of a finite dimensional integrable system is filled by invariant tori; KAM theory ensures that most of them persist under small perturbation. In PDEs the situation is much more complicated, and the only quite complete theory that we know, allows to deal with Hamiltonian PDEs in one space dimension and only allows to show persistence of finite dimensional tori. A few eamples in higher dimensions are also known. Also some preliminary results on the persistence of infinite dimensional tori are known. In this talk I will present some of the above results and discuss some open problems related to the possible application to Gross Pitaevskii equation.

We study the issue of collisions in finite time for vortex filaments in 3D incompressible fluids, according to a model introduced by Klein, Majda and Damodaran. We also mention another more precise model for the dynamics of one filament introduced by Zhakarov. This is joint work with Valeria Banica and Erwan Faou.

We consider the NLS equation with focusing nonlinearities in a presence of an external field. We investigate the soliton motions that correspond to a free soliton escaping the well created by the potential. We exhibit the dynamical system driving the exiting trajectory and construct the associated nonlinear dynamics for untrapped motions.

We consider the defocusing cubic nonlinear Schrödinger equation (NLS) on the two-dimensional torus. This equation admits a special family of elliptic invariant quasiperiodic tori called finite-gap solutions. These solutions 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. In my talk, I will also say some words about the ongoing work concerning Sobolev instability of more general 2D quasi-periodic solutions. The subject of this talk is partly motivated by the problem of infinite energy cascade for 2D NLS, and it is a joint work with M. Guardia, Z. Hani, A. Maspero and M. Procesi.

In this talk I will present a reducibility result holding for time quasi-periodic unbounded perturbations of a non-resonant transport equation on the d−dimensional torus $T^d$ . Such a result is obtained combining pseudo-differential calculus, used to conjugate the initial system to a new one with a smoothing perturbation, and a KAM scheme, implemented to actually obtain reducibility. This makes possible to exhibit an example of reducibility for a higher dimensional Hamiltonian PDE, in a case where, in addition, the unperturbed problem has a spectrum which is dense on the real axis.

A new compound method to compute periodic orbits in symplectic maps without the use of symmetries is presented. The results corroborate classical results from renormalization theory in examples not considered in the literature. Moreover the findings suggest a natural procedure to compute critical self-similar geometrical coefficients for these new cases.

In 1995, Francois Lalonde and Leonid Polterovich conjectured a characterization of Hamiltonian diffeomorphisms on a symplectic manifold in terms of the symplectic diffeomorphisms and the Hofer metric. They were successful in proving such characterization for symplectic manifolds of dimension two and their carterisian products. In this talk we will describe the above conjecture and a new class of manifolds for which the conjecture is true. This is joint work with C. Campos.

In this talk we describe, within the Ginzburg-Landau (GL) theory, the response of a type-II superconducting wire with non-smooth cross section to an external time-independent magnetic field parallel to it. We focus on the fields whose intensity varies in the so called surface superconductivity regime, i.e., superconductivity is confined near the boundary of the sample. A natural question is then how does the ground state of the GL functional depend on the geometry of the boundary? In a preliminary result we show that the presence of corners does not affect the leading order of the energy density. To describe their contribution we then prove a refined energy asymptotics, however this is not enough to have an explicit description. A deeper understanding of the problem in the case of flat angles has led us to conjecture the explicit expression for corner contribution to the energy. Joint work with Michele Correggi.

The quantum vortices of a superfluid are described as nodal lines of a solution to the time-dependent Gross-Pitaevskii equation. Experiments and extensive numerical computations show that quantum vortices cross, each of them breaking into two parts and exchanging part of itself for part of the other. This phenomenon is known as quantum vortex reconnection, and usually leads to a change of topology of the quantum vortices. In this talk I will show that, given any initial and final congurations of quantum vortices (which do not need to be topologically equivalent) and any conceivable way of reconnecting them (that is, of transforming one into the other), there is a Schwartz initial datum whose associated solution is smooth and realizes this specific vortex reconnection scenario. Key for the proof of this result is a new global approximation theorem for the linear Schrodinger equation, and the construction of pseudo-Seifert surfaces in spacetime. This is based on joint work with Alberto Enciso.

In this talk we discuss families of homographic standing waves appearing from n vortex filaments rotating uniformly at a central configuration. The solution of the filaments are time periodic with periodic boundary conditions, i.e. this is a small divisor problem for a Hamiltonian partial differential equation which requires techniques related to KAM theory. In this case the Nash-Moser method gives rise to a family of solutions over a Cantor set of parameters. On the other hand, we show that when the relation between temporal and spatial periods is fixed at certain rational numbers, the contraction mapping theorem gives existence of an infinite number of families of standing waves that bifurcate from these configurations.

The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components...). From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of Grébert, Paturel and Thomann (2013). In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions. The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions. This is a work in collaboration with Benoît Grébert (Université de Nantes).

A full theory for hinged beams and degenerate plates with multiple intermediate piers is developed. The analysis starts with the variational setting and the study of the linear stationary problem in one dimension. Well-posedness results are provided and the possible loss of regularity, due to the presence of the piers, is underlined. A complete spectral theorem is then proved, explicitly determining the eigenvalues on varying of the position of the piers and exhibiting the fundamental modes of oscillation. The obtained eigenfunctions are used to tackle the study of the nonlinear evolution problem in presence of different nonlinearities, focusing on the appearance of instabilities and determining the position of the piers that maximizes the stability of the structure. Then degenerate plate models are introduced, where the structure is composed by a central beam moving vertically and by cross-sections free to rotate around the beam. The torsional instability of the structure is investigated, taking into account the impact of different nonlinear terms aiming at modeling the action of cables and hangers in a suspension bridge. Again, the optimal position of the piers in terms of stability is discussed. The stability analysis is carried out both by means of analytical tools, such as Floquet theory, and numerical experiments. Several open problems and possible future developments are presented. This is joint work with Maurizio Garrione (Politecnico di Milano).

Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Exchanging the roles of time and position, breathers can be interpreted as homoclinic solutions to a steady solution. In this talk, I will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. Their distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied. This is a joint work with O. Gomide, T. Seara and C. Zeng.

I will present a method for constructing quasi-periodic response solutions (i.e. quasi-periodic solutions with the same frequency as the forcing) for over-damped systems. Our method applies to non-linear wave equations subject to very strong damping and quasi-periodic external forcing and to the varactor equation in electronic engineering. The strong damping leads to very few small divisors which allows to prove the existence by using a contraction mapping argument requiring very weak non-resonance conditions on the frequency. This is joint work with A. Celletti, L. Corsi, and R. de la Llave.

The problem of persistence of invariant tori in infinite dimension is a challenging problem in the study of PDEs. There is a rather well established literature on the persistence of n-dimensional invariant tori carrying a quasi-periodic Diophantine flow (for one-dimensional system) but very few on the persistence of infinite-dimensional ones. Inspired by the classical "twisted conjugacy theorem" of M. Herman for perturbations of degenerate Hamiltonians possessing a Diophantine invariant torus, we intend to present a compact and unified frame in which recover the results of Bourgain and Pöschel on the existence of almost-periodic solutions for the Nonlinear Schrödinger equation. We shall discuss the main advantages of our approach as well as new perspectives. This is a joint work with L. Biasco and M. Procesi.

The so called "time-dependent point interactions” are solvable models with singular potentials whose “strength” changes in time. They are typically useful to investigate the ionization of a bound state by the action of a time-dependent localized interaction. We prove that time-dependent point interactions can be derived from the microscopic dynamics of a quantum particle – a Frohlich polaron – interacting with a bosonic scalar quantum field – the lattice field –, in suitable field’s configurations.

Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the "wave kinetic equation". This is the wave-analog of Boltzmann's equation for particle collisions. We shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and explore some of its effective long-time behaviors at time scales that are shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah).

Using spatial dynamics and results from the KAM theory, we develop a framework to find solutions of semilinear elliptic equations on the entire space which are quasiperiodic in one variable, decaying in the other variables. These results apply to a wide class of nonhomogeneous (and some homogeneous) problems. A careful application of Birkhoff normal form allows us to obtain a nondegeneracy condition for KAM that works even for some purely quadratic nonlinearities.

I will present a renormalization problem about the convergence of normal forms in the presence of counterterms for some semiclassical systems that are close to be completely integrable. I will also explain some applications of these normal-form constructions to the study of quantum limits and semiclassical measures.

The research is concerned with the study of dispersive shock waves (DSW), also termed undular bores, on the surface of fluids, DSWs arise due to the dispersive resolution of step, or near-step, initial conditions and are a common waveform in nature. They consist of a modulated dispersive wavetrain linking distinct levels ahead and behind it. In this talk, we will present some preliminary results related to the accuracy of DSW solutions of the Hamiltonian Bidirectional Whitham compared with fully nonlinear results. All research on DSWs to date has been based on weakly nonlinear approximations of full systems of equations, for instance, the water wave equations. The research of this project is the first attempt for the study of fully nonlinear DSWs. We will use the Whitham-Boussinesq (W-B) model that I introduced in my doctoral thesis as the bridge between the water wave equations (the free-surface Euler equations) and in which the "Shock fitting method” can be applied. This "Shock fitting method", derived by G. El and his collaborators, is a general method for determining the leading and trailing edges of DSWs, based on the dispersion relation for the governing nonlinear, dispersive wave equation.

In this talk I will present a spectral result for the Schröodinger operator, perturbed with an unbounded potential of order strictly smaller than 2, on an arbitrary torus obtained as the quotient between R^d and a maximal d-dimensional lattice of R^d. I will show that it is possible to provide an asymptotic expansion of most of the eigenvalues of this operator. The proof is based on a "quantum normal form" involving pseudo differential operators and a standard "quasi-modes" argument. This is a joint work with Dario Bambusi and Beatrice Langella

The Schroedinger equation in dimension two with a nonlinearity concentrated in a point shows some peculiar features: for instance it admits standing waves also for repulsive nonlinearity and it exhibits no critical power. We report on some recent results, found in collaboration with Raffaele Carlone, Michele Correggi, and Lorenzo Tentarelli, and point out several open problems.

We shall consider the water waves equations for a bi-dimensional fluid with periodic one-dimensional free surface under the action of gravity and eventually capillary forces. We shall discuss long time existence and stability results of solutions evolving from smooth and small initial data. The main difficulties in the proof concern the quasilinear nature of the equations, the presence of small divisors arising form near-resonances and the possible presence of non-trivial three-waves or four-waves interactions.

In this talk I report on joint work with Patrick Gérard concerning the construction of global Birkhoff coordinates for the Benjamin-Ono equation. In these coordinates this equation can be solved by quadrature, meaning that it is an integrable (pseudo)differential equation in the strongest possible sense. The construction is based on the Lax pair formulation of this equation. I will present spectral properties of the Lax operator, discuss a generating function of the Benjamin-Ono hierarchy, which allows to establish various trace formulas, and introduce the Birkhoff coordinates. Furthermore, I will provide a characterization of finite gap solutions.