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)
• 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 /li>

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
• 12:00 PM - 2:30 PM: Lunch/Afternoon Break
• 2:30 PM - 5:30 PM: Excursion in Bonn
• 6:00 PM - 9:00 PM: Conference Party at Herbert Koch's House

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 - 6:00 PM: Afternoon Break
• 6:00 PM - 7:00 PM: Justin Holmer in 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 ^{1/2}$ for the ^{1/2}$ critical NLS

Abstract: Available here in PDF.

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.

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. Co mbining the compactness if 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.

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 Scrödinger/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 ob served 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).

Accommodation

Hotel Deadline 28 January: We have reserved blocks of rooms at the Hotel Krug and the Hotel Mozart for all invitees who have responded that they will be attending. Please send us your travel information by January 28th so that we may finalize all reservations accordingly.

Please use the travel reimbursement form to claim reimbursement of your travel costs and per diem.

Hotel Mozart
Mozartstr.1
53115 Bonn
Telefon: 0228/ 659071
Fax: 0228/ 659075
Emails: info@hotel-mozart-bonn.com, hotel.mozart@web.de
Website: http://hotel-mozart-bonn.com/

Hotel Krug
Sternenburgstrasse 15
53115 Bonn
Telefon: (0228) 22 58 68
Fax: (0228) 24 15 44
Email: hotel-krug@online.de
Website: www.hotelkrug.de

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.

Arriving from Colgone

The closest airport is Cologne/Bonn (CGN). The express bus SB60 shuttles between the airport and Bonn city center (ZOB). The ride takes approx. 30 minutes, and a one-way ticket costs approx. Euro 7,-. For details see (German version) here. English version: Here.

The bus terminal (ZOB) in Bonn is opposite the central railway station (Hauptbahnhof) in the centre of town. If you are flying into Frankfurt or Düsseldorf, we recommend taking a train from the airport to Bonn Central station (Hauptbahnhof). There are trains every hour (IC or EC trains) to Bonn. You can check the schedules on the website of the German railway company DB www.bahn.de (remove the tick from the box: prefer fast connections!). This website can be switched to English by clicking on "Internat. Guests." If you take the ICE from Frankfurt to Siegburg/Bonn, you then take tram 66 from Siegburg to Bonn. The tram stop is just below the train platform (take the elevator) in the same building. It is about a 15 minute tram ride to Bonn. Map here (Nordrhein-Westfalen>Bonn).

Arriving from Frankfurt

If you are arriving at Franfurt airport please take a closer look at the documents Frankfurt Airport rail connections and AIRail Service to Cologne.

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.

Further Information

• Informative links
• Restaurants

Contact Information

For further information contact either Herbert Koch or Jeremy Marzuola.