Hamiltonian Methods and Asymptotic Dynamics
Institute for Computational and Experimental Research in Mathematics (ICERM)
December 6, 2021  December 10, 2021
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Monday, December 6, 2021

8:55  9:00 am ESTWelcome11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University

9:00  9:45 am ESTAsymptotic stability of the SineGordon kink under odd perturbations via supersymmetry11th Floor Lecture Hall
 Wilhelm Schlag, Yale University
Abstract
We will describe the recent asymptotic analysis with Jonas Luehrmann of the SineGordon evolution of odd data near the kink. We do not rely on the complete integrability of the problem in a direct way, in particular we do not use the inverse scattering transform.

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTTimedependent BogoliubovdeGennes and GinzburgLandau equations11th Floor Lecture Hall
 Virtual Speaker
 Rupert Frank, LMU Munich
Abstract
We study the timedependent BogoliubovdeGennes equations for generic translationinvariant fermionic manybody systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a timedependent GinzburgLandau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time.

11:30 am  12:15 pm ESTOn the wellposedness of the derivative nonlinear Schr\"odinger equation11th Floor Lecture Hall
 Maria Ntekoume, Rice University
Abstract
We consider the derivative nonlinear Schr\"odinger equation in one space dimension, posed both on the line and on the circle. This model is known to be completely integrable and $L^2$critical with respect to scaling. However, not much is known regarding the wellposendess of the equation below $H^{\frac 12}$. In this talk, we prove that this problem is globally wellposed for initial data in the Sobolev spaces $H^s$ for $\frac 1 6\leq s<\frac 12$. The key ingredient in our argument is proving that ensembles of orbits with $L^2$equicontinuous initial data remain equicontinuous under evolution. This is joint work with Rowan Killip and Monica Visan.

12:30  2:30 pm ESTLunch/Free Time

2:30  3:15 pm ESTQuantitative derivation and scattering of the 3D cubic NLS in the energy space11th Floor Lecture Hall
 Justin Holmer, Brown University
Abstract
We consider the derivation of the {defocusing cubic nonlinear Schr\"{o}dinger equation (NLS) on $\mathbb{R}^{3}$ from quantum $N$body dynamics. We reformat the hierarchy approach with KlainermanMachedon theory and prove a biscattering theorem for the NLS to obtain convergence rate estimates under $H^{1}$ regularity. The $H^{1}$ convergence rate estimate we obtain is almost optimal for $H^{1}$ datum, and immediately improves if we have any extra regularity on the limiting initial oneparticle state. This is joint work with Xuwen Chen (University of Rochester).

3:30  4:15 pm ESTTBD11th Floor Lecture Hall
 Hong Wang, Institute for Advanced Study (IAS)

4:30  6:00 pm ESTWelcome ReceptionReception  11th Floor Collaborative Space
Tuesday, December 7, 2021

9:00  9:45 am ESTHighOrder Rogue Waves and Solitons, and Solutions Interpolating Between Them11th Floor Lecture Hall
 Virtual Speaker
 Peter Miller, University of Michigan
Abstract
A family of exact solutions to the focusing nonlinear Schrödinger equation is presented that contains fundamental rogue waves and multiplepole solitons of all orders. The family is indexed with a continuous parameter representing the "order" that allows one to continuously tune between rogue waves and solitons of different integer orders. In this scheme, solitons and rogue waves of increasing integer orders alternate as the continuous order parameter increases. For example, the Peregrine solution can be viewed as a soliton of order threehalves. We show that solutions in this family exhibit certain universal features in the limit of high (continuous) order. This is joint work with Deniz Bilman (Cincinnati).

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTTBD11th Floor Lecture Hall
 Igor Rodnianski, Princeton University

11:30 am  12:15 pm ESTGround state in the energy supercritical GrossPitaevskii equation with a harmonic potential11th Floor Lecture Hall
 Virtual Speaker
 Dmitry Pelinovsky, McMaster University
Abstract
In order to prove the existence of a ground state (a positive, radially symmetric solution in the energy space), we develop the shooting method and deal with a oneparameter family of classical solutions to an initialvalue problem for the stationary equation. We prove that the solution curve (the graph of the eigenvalue parameter versus the supremum norm) is oscillatory below a threshold and monotone above a threshold. Compared to the existing literature, rigorous asymptotics are derived by constructing families of solutions to the stationary equation with functionalanalytic rather than geometric methods. The same analytical technique allows us to characterize the Morse index of the ground state.

12:30  2:00 pm ESTLunch/Free Time

2:00  2:45 pm ESTRigidity for solutions to the quintic NLS equation at the ground state level11th Floor Lecture Hall
 Benjamin Dodson, John Hopkins University
Abstract
In this talk, we will prove rigidity for solutions to the quintic nonlinear Schrodinger equation in one dimension, at the level of the ground state. Specifically, we show that the only solutions that fail to scatter are the solitons and the pseudoconformal transformation of the solitons.

3:00  4:30 pm ESTLightning Talks followed by Coffee Break and discussionsLightning Talks  11th Floor Lecture Hall
Wednesday, December 8, 2021

9:00  9:45 am ESTTBD11th Floor Lecture Hall
 Virtual Speaker
 Nataša Pavlovic, University of Texas at Austin

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTTBD11th Floor Lecture Hall
 Tadahiro Oh, The University of Edinburgh

11:30 am  12:15 pm ESTMathematical Construction for Gravitational Collapse11th Floor Lecture Hall
 Yan Guo, Brown University
Abstract
We will discuss recent constructions of blowup solutions for describing gravitational collapse for EulerPoisson system.

12:30  12:40 pm ESTGroup Photo11th Floor Lecture Hall

12:40  2:30 pm ESTLunch/Free Time

2:30  3:15 pm ESTInternal Modes and Radiation Damping for 3d KleinGordon equations11th Floor Lecture Hall
 Virtual Speaker
 Fabio Pusateri, University of Toronto
Abstract
We consider quadratic KleinGordon equations with an external potential $V$ in $3+1$ spacetime dimensions. We assume that $V$ is generic and decaying, and that the operator $\Delta + V + m^2$ has an eigenvalue $\lambda^2 < m^2$. This is a socalled ‘internal mode’ and gives rise to timeperiodic localized solutions of the linear flow. We address the question of whether such solutions persist under the full nonlinear flow. Our main result shows that small nonlinear solutions slowly decay as the energy is transferred from the internal mode to the continuous spectrum, provided a natural Fermi golden rule holds. Moreover, we obtain very precise asymptotic information including sharp rates of decay and the growth of weighted norms. These results extend the seminal work of SofferWeinstein for cubic nonlinearities to the case of any generic perturbation. This is joint work with Tristan Léger (Princeton University).

3:30  4:00 pm ESTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm ESTSolutions to the KdV and Related Equations With Almost Periodic Initial Data11th Floor Lecture Hall
 David Damanik, RICE University
Abstract
We discuss recent work concerning the existence, uniqueness, and structure of solutions to the KdV equation, as well as related ones, with almost periodic initial data. The talk is based on several joint works with a variety of coauthors, including Ilia Binder, Michael Goldstein, Yong Li, Milivoje Lukic, Alexander Volberg, Fei Xu, and Peter Yuditskii.
Thursday, December 9, 2021

9:00  9:45 am ESTThe Quartic Integrability and Long Time Existence of Steep Water Waves in 2D11th Floor Lecture Hall
 Virtual Speaker
 Sijue Wu, University of Michigan
Abstract
Abstract. It is known since the work of Dyachenko & Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3wave interactions and all of the 4wave interaction coefficients vanish on the non trivial resonant manifold. In this talk, I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals Ej (t), directly in the physical space, which are explicit in the Riemann mapping variable and involve material derivatives of order j of the solutions for the 2d water wave equation, so that ddtEj (t) is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than ε, then the lifespan of the solution for the 2d water wave equation is at least of order O(ε−3), and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size ε, then the lifespan of the solution is at least of order O(ε−5/2). Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTGlobal wellposedness of the Zakharov System below the ground state11th Floor Lecture Hall
 Virtual Speaker
 Sebastian Herr, Bielefeld University
Abstract
We consider the Cauchy problem for the Zakharov system with a focus on the energycritical dimension d = 4 and prove that global wellposedness holds in the full (nonradial) energy space for any initial data with energy and wave mass below the ground state threshold. The result is based on a uniform Strichartz estimate for the Schr ̈odinger equation with potentials solving the wave equation. This is joint work with Timothy Candy and Kenji Nakanishi.

11:30 am  12:15 pm ESTTBD11th Floor Lecture Hall
 Svetlana Roudenko, Florida International University

12:30  2:30 pm ESTLunch/Free Time

2:30  3:15 pm ESTTBD11th Floor Lecture Hall
 Daniel Tataru, University of California, Berkeley

3:30  4:00 pm ESTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm ESTSimple motion of stretchlimited elastic strings11th Floor Lecture Hall
 Virtual Speaker
 Casey Rodriguez, University of North Carolina
Abstract
Perfectly flexible strings are among the simplest onedimensional continuum bodies and have a rich mechanical and mathematical theory dating back to the derivation of their equations of motion by Euler and Lagrange. In classical treatments, the string is either completely extensible (force produces stretching) or completely inextensible (every segment has a fixed length, regardless of the motion). However, common experience is that a string can be stretched (is extensible), and after a certain amount of force is applied the stretch of the string is maximized (becoming inextensible). In this talk, we discuss a simple model for these stretchlimited elastic strings, in what way they model ``elastic" behavior, the wellposedness and asymptotic stability of certain simple motions, and (many) open questions.
Friday, December 10, 2021

9:00  9:45 am ESTTBD11th Floor Lecture Hall
 Pierre Germain, NYU  Courant Institute

10:00  10:30 am ESTCoffee Break11th Floor Collaborative Space

10:30  11:15 am ESTNontrivial selfsimilar blowup in energy supercritical wave equations11th Floor Lecture Hall
 Birgit Schoerkhuber, University of Innsbruck, Austria
Abstract
Selfsimilar solutions play an important role in the dynamics of nonlinear wave equations as they provide explicit examples for finitetime blowup. This talk will be concerned with the focusing cubic and the quadratic wave equation, respectively, in dimensions where the models are energy supercritical. For both equations, we present new selfsimilar solutions with nontrivial profiles, which are completely explicit in all supercritical dimensions. Furthermore, we analyse their stability locally in backward light cones without symmetry assumptions. This involves a delicate spectral problem that we are able to solve rigorously only in particular space dimensions. In these cases, we prove that the solutions are codimension one stable modulo translations in a backward light cone of the blowup point. This is joint work with Irfan Glogić (Vienna) and Elek Csobo (Innsbruck).

11:30 am  12:15 pm ESTThe stability of charged black holes11th Floor Lecture Hall
 Virtual Speaker
 Elena Giorgi, Columbia University
Abstract
Black hole solutions in General Relativity are parametrized by their mass, spin and charge. In this talk, I will motivate why the charge of black holes adds interesting dynamics to solutions of the Einstein equation thanks to the interaction between gravitational and electromagnetic radiation. Such radiations are solutions of a system of coupled wave equations with a symmetric structure which allows to define a combined energymomentum tensor for the system. Finally, I will show how this physicalspace approach is resolutive in the most general case of KerrNewman black hole, where the interaction between the radiations prevents the separability in modes.

12:30  2:00 pm ESTLunch/Free Time

2:00  2:45 pm ESTInvariance of the Gibbs measures for the periodic generalized KdV equations11th Floor Lecture Hall
 Virtual Speaker
 Andreia Chapouto, UCLA
Abstract
In this talk, we consider the periodic generalized Kortewegde Vries equations (gKdV). In particular, we study gKdV with the Gibbs measure initial data. The main difficulty lies in constructing localintime dynamics in the support of the measure. Since gKdV is analytically illposed in the L2based Sobolev support, we instead prove deterministic local wellposedness in some FourierLebesgue spaces containing the support of the Gibbs measure. New key ingredients are bilinear and trilinear Strichartz estimates adapted to the FourierLebesgue setting. Once we construct localintime dynamics, we apply Bourgain's invariant measure argument to prove almost sure global wellposedness of the defocusing gKdV and invariance of the Gibbs measure. Our result completes the program initiated by Bourgain (1994) on the invariance of the Gibbs measures for periodic gKdV equations. This talk is based on joint work with Nobu Kishimoto (RIMS, University of Kyoto).

3:00  3:45 pm ESTTBD11th Floor Lecture Hall
 Sameer Iyer, UC Davis

4:00  4:30 pm ESTCoffee Break11th Floor Collaborative Space
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