The University of Texas at Austin

2012/13 Focus on NLS



2012/13 Thematic Program on nonlinear Schrodinger equations and Bose gases from a multidisciplinary, integrative perspective.


With focus on research, and on the training of graduate and advanced undergraduate students.

Schedule of Events


Minicourses: Survey and Research Talks

Advanced minicourses consisting of survey and research talks by experts in their respective fields.

Speakers:

Spring 2013

Ji Oon Lee, KAIST (S. Korea). Talk on Jan 23, 2013.
      ''Rate of Convergence in Hartree Dynamics''

Christian Hainzl, Tuebingen University. Talks on March 6 and 8, 2013.
      ''From BCS to Ginzburg-Landau theory''
      ''Microscopic derivation of the Ginzburg-Landau theory''

Robert Seiringer, McGill University. Talks on March 6 and 8, 2013.
      ''A positive density analogue of the Lieb-Thirring inequalities''
      ''The Excitation Spectrum for Weakly Interacting Bosons''

Gideon Simpson, University of Minnesota *. Talks on March 20 and 22, 2013.
      '' Numerically Assisted Analysis of Nonlinear Wave Equations, I and II''
      (Part I is an expository talk)

Justin Holmer, Brown University *. Talks on March 27 and 29, 2013.
      ''Dynamics of solitons in perturbed environments''
      ''On the Klainerman-Machedon conjecture for the quantum BBGKY hierarchy with self-interaction''

Younghun Hong, Brown University *. Talk on May 1, 2013
      ''Global well-posedness of a nonlinear Schrodinger equation with a potential below the energy norm''

Fall 2012

Vedran Sohinger, University of Pennsylvania *. Two talks on Sep 17 and 19, 2012.
      ''Bounds on the growth of high Sobolev norms of solutions to nonlinear Schrodinger equations I and II''
      (Part I is an expository talk)

Andrea Nahmod, University of Massachusetts Amherst *. Two talks on Nov 12 and 14, 2012.
      ''Almost sure well posedness and randomization in nonlinear PDE, Parts I and II''
      (Part I is an expository talk)

* Joint invitations with Natasa Pavlovic.


Introductory and survey talks, held by graduate students.

Graduate Talks:

Claudia Raithel, Junior Analysis Seminar, April 5, 2013.
      ''Evolution of factored many-body wave functions in the mean field limit''

Kenneth Taliaferro, Junior Analysis Seminar, Mar 22, 2013.
      '' Well-Posedness of the Gross-Pitaevskii Hierarchy''

Kenneth Taliaferro, Junior Analysis Seminar, Nov 16, 2012.
      ''Overview of the Gross-Pitaevskii Hierarchy''


Multidisciplinary Perspective


The study of interacting Bose gases has been extraordinarily successful in recent years, both in physics (Bose-Einstein condensation, superfluidity) and mathematics. In mathematics, this topic bridges some of the most active research areas in mathematical physics and nonlinear PDE's. Due to the large number of degrees of freedom, the dynamics of a Bose gas is exceedingly complicated. Mean field limits provide a mathematically rigorous way to describe average dynamical properties of the bulk system. The nonlinear Schrodinger equation (NLS) emerges as a mean field limit for dilute Bose gases in the so-called Gross-Pitaevskii (GP) scaling limit. The study of NLS is a central and extremely successful area in the field of dispersive nonlinear PDE's.

We mainly emphasize aspects of the following disciplines:

Nonlinear PDE theory

Key concepts and results in the well-posedness theory of the Cauchy problem for NLS. Survey of recent advances.

Mathematical Physics

Derivation of NLS and Gross-Pitaevskii hierarchies from quantum many-particle systems and Quantum Field Theory. Well-posedness theory of Gross-Pitaevskii hierarchies.

Computational Simulations

Numerical study of properties of solutions of NLS beyond current grasp in PDE theory. Predictions from numerical simulations, and key problems in numerical analysis.

Applications in Physics and Engineering

Experimental observations in Bose-Einstein condensates, phenomena and questions.

Program Information


This is the first in a series of five thematic years held at the Department of Mathematics, centered around a single equation or method, viewed from an integrative, multidisciplinary perspective encompassing nonlinear PDE's, mathematical physics, computational simulations, as well as applications in physics and engineering.

Focus topics addressed in these thematic years tentatively include nonlinear Schrodinger equations, wave propagation in random media, Vlasov and Boltzmann equations, Euler equations, and multiscale and renormalization group methods. The mathematical physics component will address the derivation of these equations from quantum dynamics.

Some of the main educational goals are:
  • To provide graduate students in analysis, applied mathematics, and mathematical physics with an broad insight into areas close to their research fields.

  • To help advanced undergraduate students compare the styles of research in various disciplines, possibly to identify directions for their future studies.
Presentations and lectures will be held by experts in pure and applied mathematics, physics, and engineering, invited from UT Austin and other institutions (in part joint invitations with Natasa Pavlovic).

This program is supported by the NSF CAREER grant DMS-1151414.
Organizer: Thomas Chen.

Recommended Reading


T. Cazenave, Semilinear Schrodinger equations, Courant lecture notes 10, Amer. Math. Soc. (2003).

T. Chen, N. Pavlovic, On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies, Discr. Contin. Dyn. Syst. A, 27 (2), 715 - 739, 2010.

L. Erdos, B. Schlein, H.-T. Yau, Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems, Invent. Math. 167, 515 - 614, 2007.

S. Klainerman, M. Machedon, On the uniqueness of solutions to the Gross-Pitaevskii hierarchy, Commun. Math. Phys. 279 (1), 169 - 185, 2008.

E.H. Lieb, R. Seiringer, J.P. Solovej, J. Yngvason, The mathematics of the Bose gas and its condensation, Birkhauser, 2005.

B. Schlein, Derivation of Effective Evolution Equations from Microscopic Quantum Dynamics, Lecture notes, CMI 2008 Summer School on Evolution Equations.

G. Staffilani, The theory of nonlinear Schrodinger equations I + II, Lecture notes, CMI 2008 Summer School on Evolution Equations.

C. Sulem, P.-L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Springer, 1999.

T. Tao, Nonlinear dispersive equations. Local and global analysis, CBMS 106, eds: AMS, 2006.