Texas Analysis and Mathematical Physics Symposium

RICE UNIVERSITY - Oct 25-27 - 2013

Contributed Talks

Linear stability and instability of self-interacting spinor field, Andrew Comech (Texas A&M University)

We consider the linear stability of solitary waves of the nonlinear Dirac equation and classical Dirac-Maxwell system. We prove that in the nonrelativistic limit the stability is described by the Vakhitov-Kolokolov stability criterion. In particular, this shows that Dirac-Maxwell solitary waves are linearly stable in the nonrelativistic limit (\omega near -m).

We also obtain a new stability criterion which proves linear instability of Dirac-Maxwell solitary waves away from the nonrelativistic limit and estimate that the transition to instability happens at \omega near 0.3 m.

The stability results are obtained together with N. Boussaid [arXiv:1211.3336] and D. Stuart [arXiv:1210.7261].

The instability results are obtained together with G. Berkolaiko and A. Sukhtayev [arXiv:1306.5150]

Pseudo-integrable billiards, measurable foliations, and arithmetic dynamics, Vladimir Dragovic (University of Texas at Dallas)

We introduce a class of nonconvex billiards with a boundary composed of arcs of confocal conics which contain reflex angles. We present their basic dynamical, topological, and arithmetic properties. We study their periodic orbits and establish a local Poncelet porism. Some of the key properties are consequences of the existence of a significant measurable foliation. This research is done jointly with M. Radnovic.

[1] V. Dragovic, M. Radnovic, Bicentennial of the Great Poncelet Theorem (1813-2013): Current Advances, to appear Bulletin of the AMS
[2] V. Dragovic, M. Radnovic, Ellipsoidal billiards in pseudo-Euclidean spaces and relativistic quadrics. Adv. Math. 231 (2012), no. 3-4, 1173–1201
[3] V. Dragovic, M. Radnovic, Poncelet porisms and beyond. Integrable billiards, hyperelliptic Jacobians and pencils of quadrics. Frontiers in Mathematics. BirkhaÃàuser/Springer Basel AG, Basel, 2011
[4] V. Dragovic, M. Radnovic, Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: beyond Poncelet porisms. Adv. Math. 219 (2008), no. 5, 1577–1607

Integrable discretizations and self-adaptive moving mesh methods for a class of nonlinear wave equations, Baofeng Feng (University of Texas-Pan American)

Recently, much attention has been paid to a class of nonlinear wave equations which are derived from physics context such as water waves and nonlinear optics. These equations include the Camassa-Holm equation, the Degasperis-Procesi equation and their short-wave limits (the Hunter-Saxton and the reduced Ostrovsky equations), the short pulse and coupled short equations etc. In this talk, I will first construct integrable semi-discretizations of these equations based on Hirota’s bilinear method and reductions from the Kadomtsev-Petviashvili hierarchy. Then I will show how these integrable semi-discretizations can be successfully used as a self-adaptive moving mesh method for the numerical simulation of these PDEs. Various numerical experiments including loop, breather and loop-breather interaction reveal very good results when compared with exact solutions.

This is a joint work with Dr. Ohta at Kobe University and Dr. Maruno at the University of Texas-Pan American.

Approximate Solutions to the Korteweg-de Vries-Burgers Equation, Zhaosheng Feng (University of Texas-Pan American)

n this talk, we mainly compute the approximate solutions to the Korteweg-de Vries-Burgers equation. We provide a connection between the Abel equation of the first kind, an ordinary differential equation that is cubic in the unknown function, and the Korteweg-de Vries-Burgers equation, a partial differential equation that describes the propagation of waves on liquid-filled elastic tubes. We present an integral form of the Abel equation with the initial condition. By virtue of the integral form and the Banach Contraction Mapping Principle we derive the asymptotic expansion of bounded solutions in the Banach space, and use the asymptotic formula to construct approximate solutions to the Korteweg-de Vries- Burgers equation.

Cyclicity in Dirichlet-type spaces and extremal polynomials: Functions on the disk vs. functions on the bidisk, Constanze Liaw (Baylor University)

A function f is said to be cyclic in a space of analytic functions H, if any function in H can approximated in H-norm by the sequence {p_n f}. On the disc, a function is cyclic in the Hardy space H^2 if and only if it is outer. On the bidisk, the situation is much more complicated. In fact in the Dirichlet space of the bidisk (i.e. the space of analytic functions on the bidisk whose Fourier coefficients satisfy a certain decay condition), we can find a simple polynomial with no zeros on the bidisk that is not cyclic. This exemplifies the contrast between cyclicity in one versus two complex variables, since in one complex variable all analytically continuable functions with no zeros on the unit disk are cyclic.

Regularity and Global Existence of Solutions to a System of Cross-Diffusion Equations, Tuoc Van Phan (University of Tennessee, Knoxville)

In this talk, we show how to use the perturbation technique and maximal functions to derive the $L^p$-gradient estimate of bounded solutions of a class of nonlinear diffusion equations. We then apply the results to show the global time existence of smooth solutions of a system of cross-diffusion equations in bounded spacial domains of any dimensions.

The talk is based on the joint work with Luan Hoang (Texas Tech University) and Truyen Nguyen (Akron University).

Capacities in nonlinear PDEs with power nonlinearities, Nguyen Cong Phuc

We give a brief survey on the use of Sobolev capacities in the study of fully nonlinear and quasilinear equations of Lane-Emden type, and the stationary Navier-Stokes equations with strongly singular external forces. Topics discussed include complete characterizations of existence and removable singular sets for Lane-Emden type equations, as well as the stability result for stationary Navier-Stokes equations. This talk is based on joint work with Igor E. Verbitsky and Tuoc Van Phan.

Eigenvalues of the magnetic Schroedinger operator on quantum graphs and nodal count of eigenfunctions, Tracy Weyand (Texas A&M University)

We consider the eigenvalues of the magnetic Schroedinger operator on a quantum graph as functions of the magnetic potential. A simple gauge transformation shows that these eigenvalues are functions only of the magnetic flux through the cycles of the graph. We relate the magnetic eigenvalues in question to eigenvalues of a parameter-dependent family of spanning trees. This allows us to establish a simple relation between the Morse index of the magnetic eigenvalue and the number of zeros of the corresponding non-magnetic eigenfunction.

New developments of LOBPCG for large-scale nonlinear eigenvalue problems, Fei Xue (University of Louisiana at Lafayette)

In this talk, we discuss several new observations and applications of the locally optimal block preconditioned conjugate gradient (LOBPCG) method. We present an interpretation of LOBPCG as an optimal variant of the nonlinear PCG methods for optimization, and we show consequently that LOBPCG can be adapted to compute "extreme eigenvalues" of nonlinear eigenproblems that can be characterized by the variational principle (minimax theorem). In addition, we show that the convergence rate of LOBPCG can be greatly accelerated by the use of variable and indefinite preconditioners. This observation is particularly important for the fast computation of many eigenvalues using the single-vector algorithm (LOPCG) with deflation.

The Aluthge and Mean transforms of bounded linear operators, Jasang Yoon (The University of Texas-Pan American)

The Aluthge transform of a bounded linear operator on a complex Hilbert space was first studied in Aluthge [1990] and has received much attention in recent years. One reason is because there exists a connection with the invariant subspace problem. However, in the view of the practical use, it is so hard to find the Aluthge transform of the given operator because it involves the positive square roots of positive operators, and it is quite difficult to find the positive square roots in general. In this talk we first introduce the mean transform of bounded operators and study whether this transform preserves subnormality, k-hyponormality of operators. In contrast to Aluthge transform, the mean transform involves the sums of two operators, so it is easy to get the mean transforms if we know the polar decompositions of the operators. Thus, the mean transform may be useful in the practical use. In this talk we also explore how the mean transform of weighted shifts behave, in comparison with the Aluthge transform. We also extend the notion of Aluthge transform of single operators to a commuting pair of operators.

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