Stable and Unstable Dynamical Systems

Sergei Kuksin

Long-time behaviour of solutions for equations with randomness: a panorama

In this talk, I will begin by discussing the problem of long-time behaviour of solutions to equations with both discrete and continuous time, incorporating randomness. I will then present results on this problem obtained jointly with A. Shirikyan in our recent paper published in GAFA (2025), along with subsequent developments.

Raphaël Krikorian

On the approximation of real analytic Hamiltonians by locally integrable ones.

Let \(H\) be a real analytic Hamiltonian with an elliptic Diophantine fixed point and satisfying a Kolmogorov non-degeneracy condition. Then, it admits arbitrarily small real analytic perturbations that are integrable on (small depending on the perturbation) neighborhoods of the origin. The proof is based on deformations of compatible complex and symplectic structures and on a KAM scheme where at each step nonlinear Cousin problems are solved.

Shulamit Terracina

Time Quasi-Periodic Three-dimensional Traveling Gravity Water Waves

Starting with the pioneering computations of Stokes in 1847, traveling wave solutions have played a central role in the study of water waves, both as physically relevant objects and as fundamental building blocks for understanding long-time dynamics. In this talk I will discuss recent results on the existence of time quasi-periodic traveling waves for the three-dimensional gravity water waves equations in finite depth with periodic boundary conditions. The solutions we construct evolve globally in time and propagate with an arbitrary number of rationally independent velocities; their profile is well approximated by finite superpositions of Stokes waves traveling at different speeds.

From a mathematical point of view, the problem is highly delicate due to the intricate resonance structure of the equations. Our result is the first KAM (Kolmogorov-Arnold-Moser) result for an autonomous, dispersive, quasi-linear PDE in dimension greater than one and it is the first example of global solutions, which do not reduce to steady ones in any moving reference frame, for 3D water waves equations on compact domains.

Wei-Min Wang

Quasi-periodic solutions to the nonlinear Klein-Gordon equations in two dimensions

We construct time quasi-periodic solutions to the nonlinear Klein-Gordon equation with polynomial nonlinear terms on the two dimensional torus: \[ \partial_t^2 u −\Delta u+u+u^{p+1} =0. \] To our knowledge, this is the first such result for the Klein-Gordon equation above dimension one. The analysis explores in an essential way the submodule structure on a dual Fourier lattice. We shall discuss the main ideas of the approach.

Emanuele Haus

Normal form and dynamics of some Kirchhoff-type equations

The Kirchhoff equation for a vibrating string or plate is a quasilinear model from elasticity theory for which several fundamental questions remain open, most notably the existence of global solutions for general initial data in Sobolev spaces. In this talk, based on works in collaboration with P. Baldi, F. Giuliani, M. Guardia and S. Marrocco, I am going to review a series of recent results regarding long-time existence and dynamics of Kirchhoff-type equations for initial data of small amplitude.

Tommaso Barbieri

On the bifurcation of 3d gravity-capillary Stokes waves

In this talk, I will discuss a novel existence result for 3d Stokes waves. The result is achieved through a refined variational Lyapunov-Schmidt reduction and Morse-Conley theoretic argument for a functional invariant under the action of a torus.

Eugene Wayne

Breathers in nonlinear wave equations and "hard" implicit function theorems.

Breathers are temporally periodic, spatially localized, solutions of nonlinear wave equations or other, spatially extended, infinite dimensional dynamical systems. They are rare in PDEs defined on the whole line, because localized oscillations in such systems tend to be destroyed by resonances with the dispersive modes in the system. While breathers have been shown to exist some nonlinear wave equations with spatially periodic coefficients, these examples require rather special conditions on the coefficients. I'll explain why Nash-Moser, or "hard", implicit function theorems are a natural tool to try to expand the class of equations for which breathers exist.

Jean-Marc Delort

Blowing up solutions for one dimensional Klein-Gordon equations.

Consider a quasi-linear (or semi-linear) cubic Klein-Gordon equation in one space dimension with small, smooth and decaying initial data. It is known that when the nonlinearity satisfies a convenient ``null condition'', the solution is global. It is conjectured that, if this null condition is not satisfied, generic initial data should give rise to solutions blowing up at a time of magnitude similar to \(\pm e^{S_*/\epsilon^2}\), where \(S_*>0\) and \(0<\epsilon\ll 1 \) is the size of the initial data. We consider in this talk equation \[(\partial_t^2-\partial_x^2 +1)u = (\partial_t u)^3,\] where the cubic semi-linear nonlinearity does not satisfy the null condition. We construct a blowing up solution at time \(T_*(\epsilon) = e^{S_*/\epsilon^2}\), which at time \(T_0(\epsilon) = \epsilon^{-2}T_*(\epsilon)^{1-b}\) (\(b>0\) small), satisfies smallness and decay conditions, compatible with those allowing one to prove global existence for the similar problem when the nonlinearity satisfies the null condition. Moreover, we have an asymptotic description of $u$ close to the unique blowing up point \((T_*(\epsilon),0)\).

Duohui Xiang

Almost global solutions of Kirchhoff equation

We concerned the original Kirchhoff equation (with Dirichlet boundary condition), and proved almost global existence and stability of solutions for almost any small initial data. In Sobolev spaces, we obtained polynomial bounds on the stability times, and in Gevrey and analytic spaces, we obtain sub-exponential bounds. To achieve these, we build rational normal form for infinite dimensional reversible vector fields without external parameters. We emphasize that for vector fields, the homological equation and the definition of rational normal form are significantly different from those for Hamiltonian functions. This is a joint work with Prof. Jianjun Liu.

Antonio Ponno

24 years of FPUT along the Bambusian way

A review of the Fermi-Pasta-Ulam-Tsingou problem is presented. In particular, the focus will be on the general picture we got on it during the last two decades under the significant scientific influence of Professor Dario Bambusi, who taught many of us the philosophy and the techniques of Hamiltonian perturbation theory and founded an Italian school of Hamiltonian PDEs.

Evelyne Miot

On the dynamics of point vortices and vortex filaments in fluid models

This talk will review several results obtained in collaboration with Martin Donati, Lars Eric Hientzsch and Christophe Lacave. We study the asymptotic evolution of sharply concentrated vorticities in some models for incompressible inviscid fluids: the lake equation on the one hand, and the three-dimensional Euler equations with helical or axial symmetry without swirl on the other hand. Under quite general concentration assumptions on the initial data, we prove that the vorticity remains concentrated in some strong sense to point vortices (in the lake equations), to helical vortex filaments (for the Euler equation with helical symmetry) for all times, or to vortex rings (for the axisymmetric Euler equation) on some finite interval of time. We derive moreover the asymptotic motion law of vortices .

Patrick Gérard

The half-wave maps flow on the energy space

The half-wave maps equation posed on the one dimensional torus with target the two-dimensional sphere admits a flow on the critical energy space with almost periodic trajectories. I will discuss the main ingredients of the proof of this fact, which relies on an explicit representation formula of the solutions derived from a Lax pair. This talk is based on a joint work with Enno Lenzmann.

Laurent Thomann

On the stability of the Abrikosov lattice in the Lowest Landau Level

We study the Lowest Landau Level equation set on simply and doubly-periodic domains (in other words, rectangles and strips with appropriate boundary conditions). To begin with, we study well-posedness and establish the existence of stationary solutions. Then we investigate the linear stability of the lattice solution and prove it is stable for the (hexagonal) Abrikosov lattice, but unstable for rectangular lattices. We also mention some recent results on the LLL equation posed on the complex space. This is a joint work with Pierre Germain and Valentin Schwinte.

Maria Teresa Rotolo

Sobolev instability for perturbations of periodic transport equations

In this talk, we consider linear, time-dependent perturbations of periodic transport equations on the two-dimensional torus. We show that, for a generic class of perturbations, there exists a large set of initial data whose Sobolev norms grow exponentially fast in time. In higher dimensions, this remains true under some dynamical assumptions on the resonant part of the perturbation. The proof relies on a normal form procedure combined with the study of dynamical properties of Morse-Smale transport systems.

During the talk, we describe these results, and we outline the main tools from microlocal analysis and hyperbolic dynamics needed. This is a joint work with Gabriel Rivière.

Xiaoping Yuan

Full-Dimensional Invariant Tori for the Derivative Nonlinear Schrödinger Equation

This talk addresses the existence of full-dimensional invariant tori for the derivative nonlinear Schrödinger equation (DNLS) under periodic boundary conditions: \[ i u_t - u_{xx} + V(x) *u + f(|u|^2) \partial_x u = 0 \] where V is a random convolution potential. A core technical challenge lies in solving small-divisor equations with large variable coefficients of the form \[ i \omega \cdot \partial_\phi u + \lambda u + \lambda^\alpha b(\phi) u = R(\phi), \] where \(\lambda > 0\) is a large parameter, \(0 < \alpha \leq 1\), and \(b(\phi), R(\phi)\) are analytic functions on the complex torus \(\overline{\mathbb{T}}_s^n\) with \(s > 0\). Let \(\varepsilon > 0\) be a small perturbation parameter, \(\varepsilon_m \sim \varepsilon^{2^m}\), and \(K_m \sim -\log \varepsilon_m\). Classical results show that:

• For \(0 < \alpha < 1\), Kuksin's lemma yields the estimate \[ \|u\|_{s/2} \lesssim K_m^{C_1}, \] where \(C_1 > 0\) is a constant.
• For the critical case \(\alpha = 1\), Liu--Yuan's lemma gives \[ \|u\|_{s/2} \lesssim \varepsilon_m^{-C_2}, \] with a sufficiently small constant \(C_2 > 0\).

Erwan Faou

Quasi-singularities and power laws solutions for PDEs with forcing

I this talk I will address the problem of singularity and quasi-singularity formation in partial differential equations with smooth forcing and smooth coefficients. In simple models like the nonlinear Schroedinger equation and the shallow water equation, I will show how a smooth forcing can yield to solutions exhibiting quasi singularities. While the forcing terms considered are smooth and well localized in Fourier, these solutions have Fourier spectra decaying at power laws rate depending on the algebraic structure of the nonlinearity of the equation. I will then study the stability of such solutions, and show some numerical examples. These are joint works with R. Carles (CNRS), L. Martaud and G. Beck (INRIA).

Antonio Milosh Radakovic

The instability of three-dimensional water waves

Since the pioneering works of Benjamin–Feir, Whitham and many others in the 1960s, a central question in fluid dynamics has concerned the instability of steadily propagating water waves (the so-called Stokes waves), whose leading destabilizing mechanism is now known as the Benjamin–Feir instability. In the last decade, this problem has seen a strong resurgence from the analytical viewpoint, following the works of Nguyen–Strauss and Berti–Maspero–Ventura. The latter provided a detailed description of the instability spectrum under two-dimensional perturbations, showing that it consists of a “figure eight” and a sequence of progressively smaller “isolas”. The first natural extension of this line of research is to characterize what happens in three dimensions. In this setting, instabilities may arise only near specific resonant frequencies known as McLean curves. In this novel work, in collaboration with Massimiliano Berti and Alberto Maspero, we provide a complete description of the instabilities arising near the principal resonance, unifying and generalizing both the figure-eight and first-isola instabilities. We also establish a criterion for the emergence of instability isolas near any McLean resonance.

Annalaura Stingo

Trivial resonances for a system of Klein-Gordon equations and statistical applications

In the derivation of the kinetic equation from the cubic NLS, a key feature is the invariance of the Schrödinger equation under the action of U(1), which allows the quasi-resonances of the equation to drive the effective dynamics of the correlations. In this talk, I will give an example of equation that does not enjoy such type of invariance and show that the exact resonances always take precedence over quasi-resonances. As a result, the effective dynamics is not of kinetic type but still nonlinear. I will present the problem, the ideas behind the derivation of the effective dynamics and some elements of the proof. This is based on a recent work in collaboration with de Suzzoni (Université Evry Paris-Saclay) and Touati (CNRS and Université de Bordeaux).

Carlos Villegas

On the spectral invariants for the Dirichlet to Neumann map in the unit ball. An inverse spectral result.

In this talk we consider the Dirichlet to Neumann map (D-N) for the unit sphere in \(\mathbb{R}^3\). When we are sufficiently far from the origin, the spectrum of such an operator consists of eigenvalue clusters around the natural numbers. The distribution of the corresponding scaled eigenvalue shifts has an asymptotic expansion when the label of the cluster goes to infinity. The asymptotic expansion consists of distributions called spectral invariants. By using the averaging method, asymptotics of the Berezin symbol of the D-N map and a suitable symbol calculus, we compute the first terms of such an expansion in terms of the Radon transform (averages along geodesis of the unit sphere) of derivatives of the function that encodes the conductivity properties of the media in the unit ball. On base of the first two computed spectral invariants, we obtain an inverse spectral result.

Massimiliano Berti

3d instabilities of Stokes water waves

The stability/instability of traveling periodic Stokes waves—the first global-in-time solutions ever discovered for nonlinear quasi-linear dispersive PDEs—is a central, long-standing question in fluid mechanics. In 1967, Benjamin and Feir proposed a famous heuristic mechanism suggesting instability under longitudinal long-wave perturbations, complemented around 1980 by McLean's numerical identification of additional transverse instability. In this talk I will present a mathematically comphensive description of the Fourier-Bloch-Floquet spectral bands for the linearized water wave operator at small-amplitude Stokes waves under 3d longitudinal and transverse wave perturbations. We achieve this by exploiting the Hamiltonian and reversible strucure of the water waves equations, and rigorously combining spectral and perturbation theory, jointly with dynamical systems techniques. Joint work with Antonio Radakovic and Alberto Maspero..