Georgia Institute of Technology College of Sciences

Let X_{N,n} be an iid product of N real Gaussian matrices of size n x n. In this talk, I will explain some recent joint work with G. Paouris 
(arXiv:2005.08899) about a non-asymptotic analysis of the singular values of X_{N,n} . I will begin by giving some intuition and motivation for studying such matrix products. Then, I will explain two new results. The first gives a rate of convergence for the global distribution of singular values of X_{N,n} to the so-called Triangle Law in the limit where N,n tend to infinity. The second is a kind of quantitative version of the multiplicative ergodic theorem, giving estimates at finite but large N on the distance between the joint distribution of all Lyapunov exponents of X_{N,n} and appropriately normalized independent Gaussians in the near-ergodic regime (N >> n).

Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work of O. Gomide, M. Guardia, T. Seara, and C. Zeng. 

In this talk we present the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution for a flat bottom, with the exception of a sequence of velocities $c_{k}$.  Furthermore, we prove that at least two steady solutions for a near-flat bottom persist close to a non-degenerate $S^1$-orbit of steady waves for a flat bottom. As a consequence, we obtain the persistence of at least two steady waves close to a non-degenerate $S^1$-orbit of Stokes waves bifurcating from the velocities $c_{k}$ for a flat bottom. This is a joint work with W. Craig.

Consider the three body problem with masses $m_0,m_1,m_2>0$. Take units such that $m_0+m_1+m_2 = 1$. In 1922 Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of them are what is known as oscillatory motions, that is, solutions of the three body problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy
\[
\liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad 
\limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty.
\] At the time of Chazy, all types of final motions were known, except the oscillatory ones. We prove that, if all three masses $m_0,m_1,m_2>0$ are not equal to $1/3$, such motions exist. In fact, we prove more, since our result is based on the construction of a hyperbolic invariant set whose dynamics is conjugated to the Bernouilli shift of infinite symbols, we prove (if all masses are not all three equals to $1/3$) 1) the existence of chaotic motions and positive topological entropy for the three body problem, 2) the existence of periodic orbits of arbitrarily large period in the 3BP. Reversing time, Chazy's classification describes ``starting'' motions and then, the question if starting and final motions need to coincide or may be different arises.  We also prove that one can construct solutions of the three body problem whose starting and final motions are of different type.