The Dynamical Systems seminar is held on Monday afternoons at
4:00 PM in MCS 148. Tea
beforehand is at 3:45 PM in MCS 144.
September 11; José Antonio Carrillo
(Imperial College London)
Title: Swarming, Interaction Energies and PDEs
Abstract: I will present a survey of the main results about first and second
order models of swarming where repulsion and attraction are modeled
through pairwise potentials. We will mainly focus on the stability of
the fascinating patterns that you get by random particle simulations,
flocks and mills, and their qualitative behavior. Qualitative properties
of local minimizers of the interaction energies are crucial in order to
understand these complex behaviors. Compactly supported global
minimizers determine the flock patterns whose existence is related to
the classical H-stability in statistical mechanics and the classical
obstacle problem for differential operators.
September 18; John Bush
Title: Pilot-wave hydrodynamics.
Abstract: A decade ago, Yves Couder and Emmanuel Fort discovered that droplets walking on a vibrating fluid bath exhibit
several features previously thought to be exclusive to the microscopic, quantum realm. These walking droplets propel
themselves by virtue of a resonant interaction with their own wavefield, and so represent the first macroscopic realization
of a pilot-wave system of the form proposed for microscopic quantum dynamics by Louis de Broglie in the 1920s.
New experimental and theoretical results in turn reveal and rationalize the emergence of quantization and quantum-like
statistics from this hydrodynamic pilot-wave system in a number of settings. The potential and limitations of the walking
droplet system as a hydrodynamic quantum analog are discussed.
September 25; Max Hess (University of Stuttgart)
Title: Validity of the Nonlinear Schrödinger approximation
for quasilinear dispersive equations
We consider a nonlinear dispersive equation with a quasilinear quadratic term.
We derive the Nonlinear Schrödinger (NLS) equation as a formal
approximation equation describing the evolution of the envelopes of
oscillating wave packet-like solutions to the quasilinear dispersive
equation, and justify the NLS approximation with the help of error
estimates between exact solutions to the quasilinear dispersive
equation and their formal approximations obtained via the NLS
equation. The proof relies on estimates of an appropriate energy whose
construction is inspired by the method of normal-form
transforms. Moreover, we have to overcome difficulties caused by the
occurrence of resonances. The method of proof has been generalized to justify the validity of the NLS approximation for a larger class of quasilinear dispersive systems with resonances.
Some ideas of this generalized method can be used
to justify the NLS approximation
for the arc length formulation of the two-dimensional water wave problem
in case of finite depth of water and with and without surface tension.
This is a joint work with Wolf-Patrick Düll.