Workshop on Nonlinear Structures Arising in Dispersive Partial Differential Equations
March 2-6, 2009
Universität Bonn, Bonn, Germany
A photo of a series of Bose-Einstein condensate solitons courtesy of the American Institute of Physics website and a team at Rice University.
Topic
We will host a five day workshop on nonlinear dispersive equations. Specifically, we will look at existence, description, stability and scattering results for phenomenological objects resulting from nonlinear effects in dispersive PDE. Such objects include solitons, blow-up profiles, breathers, as well as other interesting objects existing for long times in the dynamics of an equation. Topics ranging from analytical descriptions to numerical simulations and asymptotics are all welcome aspects to the workshop.
Confirmed Participants
- Ramona Anton (University of Paris-Sud, XI)
- Marius Beceanu (University of Chicago)
- Nicolas Burq (University of Paris-Sud)
- Adrian Constantin (University of Vienna)
- Joachim Escher (University of Hannover)
- Boris Ettinger (University of California, Berkeley)
- Gadi Fibich (Tel Aviv University)
- Axel Grünrock (University of Bonn)
- Justin Holmer (Brown University)
- Oana Ivanovici (University of Paris-Sud)
- Markus Kunze (University of Duisburg-Essen)
- Yvan Martel (University of Versailles-Saint-Quentin-en-Yvelines)
- Luc Molinet (University of Paris-Nord)
- Fabrice Planchon (University of Paris 13)
- Jean-Claude Saut (University of Paris-Sud)
- Daniel Tataru (University of California, Berkeley)
- Nikolay Tzvetkov (University of Lille)
- Erik Wahlen (Lund University)
- Gang Zhou (ETH - Zurich)
- Maciej Zworski (University of California, Berkeley)
Conference Location and Schedule
The talks will take place at Wegelerstrasse 10 at the Applied Mathematics Institute in Bonn. The official arrival date for the conference is March 1st, 2009 and the departure date is March 7th, 2009. Registration will begin at 9 AM on Monday, March 2nd, 2009.
- Monday:
9:00 AM - 9:30 AM: Registration
9:30 AM - 10:30 AM: Nicolas Burq
10:30 AM - 11:00 AM: Coffee
11:00 AM - 12:00 PM: Axel Grünrock
12:00 PM - 2:30 PM: Lunch/Afternoon Break
2:30 PM - 3:30 PM: Luc Molinet
3:30 PM - 4:30 PM: Adrian Constantin
- Tuesday:
9:30 AM - 10:30 AM: Nikolay Tzvetkov
10:30 AM - 11:00 AM: Coffee
11:00 AM - 12:00 PM: Yvan Martel
12:00 PM - 2:30 PM: Lunch/Afternoon Break
2:30 PM - 3:30 PM: Daniel Tataru
3:30 PM - 4:30 PM: Gadi Fibich
- Wednesday:
9:00 AM - 10:00 AM: Jean-Claude Saut
10:00 AM - 11:00 AM: Erik Wahlen
11:00 AM - 11:30 AM: Coffee
11:30 AM - 12:30 PM: Joachim Escher
12:30 PM - 2:30 PM: Lunch/Afternoon Break
2:30 PM - 5:30 PM: Excursion in Bonn
- Thursday:
9:30 AM - 10:30 AM: Oana Ivanovici
10:30 AM - 11:00 AM: Coffee
11:00 AM - 12:00 PM: Markus Kunze
12:00 PM - 2:30 PM: Lunch/Afternoon Break
2:30 PM - 3:30 PM: Fabrice Planchon
3:30 PM - 4:30 PM: Maciej Zworski
4:30 PM - 6:00 PM: Afternoon Break
6:00 PM - 7:00 PM: Justin Holmer in the Intercontinental Video Seminar
- Friday:
9:30 AM - 10:30 AM: Gang Zhou
10:30 AM - 11:00 AM: Coffee
11:00 AM - 12:00 PM: Marius Beceanu
12:00 PM - 2:30 PM: Lunch/Afternoon Break
2:30 PM - 3:30 PM: Ramona Anton
Talk Abstracts
- Ramona Anton
Title: Global existence for Gross-Pitaevskii equation on three dimensional exterior domains
Abstract: We prove global existence in the energy space for the Gross-Pitaevskii equation on exterior domains of dimension three. We use a Strichartz estimate adapted to the domain. This estimate follows from a semi-classical dispersive estimate combined with a smoothing effect.
- Marius Beceanu
Title: A center-stable manifold in $H^{1/2}$ for the $H^{1/2}$ critical NLS
Abstract: Available here in PDF.
- Nicolas Burq
Title: Invariant measure for the focusing nonlinear harmonic oscillator
- Adrian Constantin
Title: Global conservative and global dissipative solutions to the Camassa-Holm equation
Abstract: The Camassa-Holm equation is a nonlinearly dispersive model equation for the propagation of waves in shallow water. The equation can be written in the form of a scalar conservation law plus an integral source term that preserves the H^1 norm of the solution. Smooth initial data can lose regularity in finite time in the form of breaking waves (the solution remains bounded but its slope becomes unbounded in finite time). We present a method for constructing a continuous semigroup of global conservative solutions. By introducing a new set of independent and dependent variables, the equation is transformed into a semilinear system, whose global solutions are obtained as fixed points of a contractive transformation. These new variables resolve all singularities due to possible wave breaking. Returning to the original variables, we obtain a semigroup of global solutions, depending continuously on initial data. These solutions are conservative, in the sense that the total energy equals a constant, for almost every time. A modification of the approach by considering a semilinear hyperbolic system with a non-local source term which is discontinuous but has bounded directional variation permits the construction of global dissipative solutions, ensuring that energy loss occurs only through wave breaking. This is joint work with A. Bressan.
- Joachim Escher
Title: Wave breaking and shock waves for the periodic Degasperis-Procesi Equation
Abstract: The Degasperis-Procesi equation is a recently derived shallow water wave equation, which is - similar as the Camassa-Holm equation - embedded in a family of spatially periodic third order dispersive conservation laws. The coexistence of globally in time defined classical solutions, wave breaking solutions, and spatially periodic peakons and shock waves is evidenced in the talk, and the precise blow-up scenario, including blow-up rates and blow-up sets, is discussed in detail. Finally several conditions on the initial profile are presented, which ensure the occurence of a breaking wave. This is joint work with Zhaoyang Yin.
- Gadi Fibich
Title: Collapsing vortex solutions and the NGO method
Abstract: In the first part of this talk I will present some recent results on singular vortex solutions of the Nonlinear Schrödinger equation (NLS). These solutions have the unique property that they vanish identically at the singularity.
In the second part of this talk I will present a novel Nonlinear Geometrical Optics (NGO) method that predicts the self-focusing dynamics of high-power solutions of the NLS.
- Axel Grünrock
Title: Bilinear space-time estimates for linearized KP-type equations with semiperiodic and periodic data
Abstract: Available here in PDF.
- Justin Holmer
Title: Effective dynamics for KdV type equations
- Markus Kunze
Title: Global asymptotic stability for a rotating charged sphere in the Abraham model
Abstract: The Abraham model can be used to describe the dynamics of a classical charged particle under the influence of its self-generated Maxwell fields. If the particle is fixed at the origin, then the dynamical quantities are the angular velocity and the fields. For a charged sphere, it is shown that every solution approaches the set of stationary solutions in the long-time limit. Since for this particular charge model the usual non-resonance condition ("Wiener condition") is violated, it is much harder to exploit the dispersive mechanism induced by the local energy decay.
- Yvan Martel
Title: Collision of solitons for the generalized KdV equations
- Luc Molinet
Title: Global attractor and asymptotic smoothing effects for the weakly damped cubic Schrödinger equation in L^2(T).
Abstract: We prove that the weakly damped cubic Schrödinger flow in L^2(T) provides a dynamical system that possesses a global attractor. The proof relies on a sharp study of the behavior of the associated flow-map with respect to the weak L^2(T)-convergence. Combining the compactness in L^2 of the attractor with the approach developed by O. Goubet we show that the attractor is actually a compact set of H^2(T). This asymptotic smoothing effect is optimal in view of the regularity of the steady states.
- Fabrice Planchon
Title: On the nonlinear Schrödinger equation on domains
- Jean-Claude Saut
Title: The transonic limit of the Gross-Pitaevskii equation
- Daniel Tataru
Title: Concentration and dispersion in large data wave maps
- Nikolay Tzvetkov
Title: Instability of line solitary water waves
- Erik Wahlen
Title: Stability of solitary water waves with weak surface tension
Abstract: Available here in PDF.
- Gang Zhou
Title: Equipartition of energy in nonlinear Schrödinger/Gross-Pitaevskii equations
Abstract: In this talk I will present the recent result, joint with M. I. Weinstein, on the mass transfer between neutral modes and ground states of weakly nonlinear NLS. Our result confirms rigorously what was observed in physical experiments, in which it was observed that the ground states grow by half of the mass of the neutral modes as the solution reaches equilibrium.
- Maciej Zworski
Title: Breathing patterns in nonlinear relaxation
Abstract: In numerical experiments involving nonlinear solitary waves propagating through nonhomogeneous media one observes "breathing" in the sense of the amplitude of the wave going up and down on a much faster time scale than the motion of the wave. In this paper we investigate this phenomenon in the simplest case of stationary waves in which the evolution corresponds to relaxation to a nonlinear ground state. The particular model is the popular $\delta_0$ impurity in the cubic nonlinear Schrödinger equation on the line. We give asymptotics of the amplitude on a finite but relevant time interval and show their remarkable agreement with numerical experiments. We stress the nonlinear origin of the "breathing patterns" caused by selection of the ground state depending on the initial data, and by the non-normality of the linearized operator (joint work with Justin Holmer).
Accommodations
Travel Information
There will be full reimbursement of travel expenses for all workshop invitees. Please fill out the HCM Travel Expense Form and attach your travel receipts. You can either submit them during the workshop or mail them in after your departure from Bonn.
If you come by train, the Bonn Hauptbahnhof is centrally located and close to the hotels. If you come by plane, we suggest flying into either the Frankfurt or Köln-Bonn Airports. For travel information regarding getting to Bonn by train (including from Frankfurt Airport), see the Deutsche Bahn site to make train reservations and check schedules. If you arrive to the Cologne-Bonn airport, you may take the SB 60 Bus to the Bonn Hauptbahnhof. For information about Bonn and surrounding areas, see the Universität Bonn site.
A web-site with further information, more links and detailed maps is the HCM Events Link for the conference.
Excursion
There will be a trip on Wednesday afternoon, March 4th, 2009 including a walk along the Rhine River and a visit to a local historical/political site in Bonn. There will be a small cost to participate.
Contact Information