UT AUSTIN - Nov 21-23

Light filamentation is a highly nonlinear process that results from light-matter interactions. It is important area in nonlinear optics with much ongoing theoretical and experimental research efforts. While there has been much progress in both fronts, there remain many challenges to produce long-lived filaments. Modeling requires good understanding of light matter interactions with a particular objective of determining conditions that limits spatio-temporal instabilities thus producing quasi-stable dynamics.  While typical studies concentrate on optical filaments at a single frequency, mostly in the infrared (IR) and the ultraviolet (UV) wavelengths, under suitable conditions nonlinear wave-mixing can lead to multicolored filaments. In this talk I will present new results that consider the co-existence of UV/IR filaments under a variety of spatial configurations.

It is widely believed in the mathematics and physics community that extrema of dispersion relations of generic periodic Schr\"odinger operators are non-degenerate. (Here by non-degeneracy we mean extrema having non-degenerate Hessian.) Unfortunately, the conjecture remains unproven. The most advanced result, obtained by F. Klopp and J. Ralston, proves that each band edge of a generic periodic Schr\"odinger operator is an extremal value of just a single band function. In this talk, we will present a study of the conjecture for a class of periodic differential operators on graphs.

A Stokes wave is a fully nonlinear wave that travels over the surface of deep water. We solve Euler equations with free surface in the framework of conformal variables via Newton Conjugate Gradient method and find Stokes waves in regimes dominated by nonlinearity. By investigating Stokes waves with increasing steepness we observe peculiar oscillations occur as we approach Stokes limiting wave. Finally by analysing Pade approximation of Stokes waves we infer that analytic structure associated with those waves has branch cut nature.

In this talk, we consider general soliton solution to a vector nonlinear Schr"{o}dinger (vNLS) equation of all possible combinations of nonlinearities including all-focusing, all-defocusing and mixed types. By using the KP-hierarchy reduction method based on the Sato-theory, we construct an unified formula for general soliton solutions including bright, dark and bright-dark ones expressed in term of Gram-type determinants for a vNLS equation. The condition for the reality of this type of soliton solution with all possible combinations of nonlinearities is also elucidated.

In three dimensions the quantum statistics of indistinguishable particles are either Bose-Einstein or Fermi-Dirac.   Constraining the particles to a plane a new form of statistics appears, anyon statistics.  Restricting the dimension of the space further to a quasi-one-dimensional quantum graph opens new forms of statistics determined by the connectivity of the graph. We develop a full characterization of abelian quantum statistics.  For two connected graphs the statistics are independent of the particle number.  On three connected non-planar graphs particles are either bosons or fermions while in three connected planar graphs they are anyons. Graphs with more general connectivity can exhibit interesting mixtures of these behaviors.  For example, a graph can be constructed where particles behave as bosons, fermions and anyons depending on the region of the graph they inhabit.   This is work with Jon Keating, Jonathan Robbins and Adam Sawicki, arXiv:0809.3476.

In this talk, I will discuss about some well-posedness and ill-posedness results on the fractional nonlinear Schr\"odinger equations in Sobolev spaces. An important feature of the equation is a loss of regularity in the linear Strichartz estimates. We overcome this problem by simple L^\infty bounds, and prove local well-posedness of the equation. Another interesting feature is that there is no exact Galilean invariance, which is a common tool to prove ill-posedness of this type of equations. However, we could prove that non-dispersing solutions, obtained by the method of Christ-Colliander-Tao, satisfies "almost" invariance. This allows us to obtain ill-posedness of the equation below L^2. This is a joint work with Yannick Sire.

For the nonautonomous linear equation x' = A(t)x that may exhibit stable, unstable and central behaviors in different directions. In this talk, we give a complete characterization of nonuniform $(\mu,\nu)$ trichotomies in terms of strict Lyapunov functions. In particular, we present an inverse theorem giving explicitly Lyapunov functions for each given trichotomy. The main novelty of this work is that we consider a very general type of nonuniform exponential trichotomy, which admits different growth rates in the uniform and the nonuniform parts.

We investigate quantum dynamics with the underlying Hamiltonian being a Jacobi or a block Jacobi matrix with the diagonal and the off-diagonal terms modulated by a periodic or a limit-periodic sequence. In particular, we investigate the transport exponents. In the periodic case we demonstrate ballistic transport, while in the limit-periodic case we discuss various phenomena such as quasi-ballistic transport and weak dynamical localization. We also present applications to some quantum many body problems. In particular, we establish for the anisotropic XY chain on $\mathbb{Z}$ with periodic parameters an explicit strictly positive lower bound for the Lieb-Robinson velocity. This is joint work with David Damanik and William Yessen.

Recently John Imbrie developed a method to demonstrate localization in many body systems. We apply a similar method to a tight binding multichannel alloy model, where the potentials at each site of the lattice are matrices which may depend analytically on the random parameters, for example, these models can be realized as tight binding model in ZD with dilute randomness. Some progress has been made to understand these models by standard methods, though much of the results known for the standard (monotonic and single channel) Anderson model remain to be demonstrated. Our goal is to obtain representations of the diagonalizing basis in the case of large disorder, we achieve this by multiscale unitary transformations by Rayleigh - Schroedinger type approximations. Key difficulties which have to be overcome in such a scheme are resonant eigenstates, especially as the energies are nonmonotonic and exceed the degree of randomness. We show that the diagonalization procedure yields st retched exponential localization in expectation. This talk covers joint work with John Imbrie.

We consider the periodic defocusing cubic nonlinear Klein-Gordon equation in three dimensions in the symplectic phase space $H^{\frac{1}{2}}(\bT^3) \times H^{-\frac{1}{2}}(\bT^3)$. In this talk, we will present a local in time non-squeezing result and a global in time non-squeezing result for certain open subsets of the phase space, with no smallness condition on the size of the initial data. In this space, the global wellposedness of this equation is still open and there is no uniform control on the local time of existence of solutions. We use an almost sure global wellposedness result for this equation to define a set of full measure with respect to a suitable randomization of the initial data on which the flow is globally defined. The proof of nonsqueezing then relies on Gromov's nonsqueezing theorem and an approximation result for the flow, which uses probabilistic estimates for the nonlinear component of the flow map and deterministic critical stability theory.

Pr\"ufer variables are a standard tool in the spectral theory of dis- crete and continuous Schr\"odinger operators. Kiselev, Remling and Simon developed generalized Pr\"ufer variables to study instead perturbations of the Schr\"odinger equation. We adapt these generalized Prufer variables to the set- ting of Jacobi and Szeg\H o recursions, and present an application concerning decaying oscillatory perturbations of Jacobi and CMV operators. This project is joint work with Milivoje Lukic.

When does the spectrum of an operator determine the operator uniquely?-This question and its many versions have been studied extensively in the field of inverse spectral theory for differential operators. Several notable mathematicians have worked in this area. Among others, there are important contributions by Borg, Levinson, Hochstadt, Liebermann; and more recently by Simon, Gesztezy, del Rio and Horvath, which have further fueled these studies by relating the completeness problems of families of functions to the inverse spectral problems of the Schr ̈odinger operator. In this talk, we will discuss the role played by the Toeplitz kernel approach in answering some of these questions, as described by Makarov and Poltoratski. We will also describe some new results using this approach. This is joint work with Mishko Mitkovski.

I will describe a new method to construct finite and infinite-dimensional KAM invariant tori in some infinite-dimensional systems. This method is very flexible, relies on geometric cancelations and gives raise to efficient algorithms. I will consider first the case of coupled map lattices, which occur in statistical physics and solid state physics. If time permits, I will also deal with some kind of PDEs, like the Boussinesq system.

The Gross-Pitaevskii (GP) Hierarchy is infinite sequence of coupled PDE's that is used to rigorously derive the cubic nonlinear Schrodinger equation from the N-body linear Schrodinger equation in the limit as N goes to infinity. In this talk, we use the quantum de Finetti theorem to prove uniqueness of solutions to the GP hierarchy in R^3 in a low regularity space. In particular, we consider all subcritical regularities s > 1/2. This is a joint work with Younghun Hong and Zhihui Xie, and is an extension of the work of Chen-Hainzl-Pavlovic-Seiringer.

We consider the spatially homogeneous Boltzmann equation with non-integrable angular cross section in the case of variable hard potentials and study the behavior of exponential moments for its solution. We provide a new proof of the generation of exponential moments of order up to the rate of potentials. We also study exponential moments of order beyond the rate of potentials and show their propagation in time. For that purpose, we introduce Mittag-Leffler moments (which can be understood as a generalization of exponential moments). This is joint work with Ricardo J. Alonso, Irene M. Gamba, and Natasa Pavlovic.

We consider the spectrum of the Schr\"odinger operator acting on an infinite periodic graph $\Gamma$. Floquet-Bloch theory says that we can find the spectrum by calculating the eigenvalues of the magnetic Schr\"odinger operator acting on a fundamental domain of $\Gamma$ and then taking the union over all possible magnetic fluxes. Therefore, we consider each eigenvalue as a function of magnetic flux and analyze where the critical points occur. While most critical points occur on the boundary, counterexamples have shown that this is not always true. However, if the fundamental domain is a tree and the eigenvalue is generic, then the critical points will occur on the boundary. Similar results hold on both discrete and quantum graphs. This is based on joint work with R. Band (Technion) and G. Berkolaiko (TAMU).