The Lefschetz Center for Dynamical Systems at Brown University sponsors a workshop on "Lattice dynamical systems"
Our intention is to bring together colleagues who work on different aspects of dissipative and Hamiltonian lattice differential equations using a variety of approaches such as dynamical-systems methods, dispersive estimates, and computational tools.
Schedule
Talks will take place in the Barus & Holley building (Corner of George and Hope Street) on Friday and Sunday and in the Division of Applied Mathematics (182 George Street) on Saturday. Barus and Holley is adjacent to the Division of Applied Mathematics: maps are posted below. To view abstracts in the schedule below, move your mouse over a title.
| Friday 6 Nov in Room BH 190 |
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| Time | Speaker | Title |
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| 9-10 | Kening Lu |
Chaos in Differential Equations Driven by a Brownian MotionWe investigate the chaotic behavior of ordinary differential equations with a homoclinic orbit to a saddle fixed
point under an unbounded random forcing driven by a Brownian motion. We prove that, for almost all sample pathes of the
Brownian motion in the classical Wiener space, the forced equation admits a topological horseshoe of infinitely many branches. This result is then applied to the randomly forced Duffing equation and the pendulum equation. |
| 10-11 | Coffee break | |
| 11-12 | John Mallet-Paret |
TBA |
| 12-2 | Lunch break | |
| 2-3 | Erik van Vleck |
Patterns in some lattice models with negative discrete diffusionWe consider a prototypical discrete model of phase transitions. The model
consists of a chain of particles, each interacting with its nearest and
next-to-nearest neighbors. The long-range interaction between next-to-nearest
neighbors is assumed to be harmonic, while the nearest-neighbor interactions are
nonlinear and bistable. After suitable rescaling we obtain a discrete
reaction-diffusion equation with a negative diffusion coefficient. By introducing
new variables, we study both analytically and numerically the behavior of traveling wave like solutions. |
| 3-4 | Hermen Jan Hupkes |
Travelling Pulses for the Discrete FitzHugh-Nagumo SystemThe existence of fast travelling pulses of the discrete FitzHugh-Nagumo equation is obtained in the weak-recovery regime. This result extends to the spatially discrete setting the well-known theorem that states that the FitzHugh-Nagumo PDE exhibits a branch of fast waves that bifurcates from a singular pulse solution. The key technical result that allows for the extension to the discrete case is the Exchange Lemma that we establish for functional differential equations of mixed type. |
| 4-5 | Coffee break | |
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| Saturday 7 Nov in Room DAM 110 |
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| Time | Speaker | Title |
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| 9-10 | Panos Kevrekidis |
Discrete Breathers in Nonlinear Schrodinger and Klein-Gordon Lattices: Some Universal FindingsMany physical systems, ranging from optical waveguide arrays in nonlinear optics to Bose-Einstein condensates in optical lattices in atomic physics, are governed, in appropriate regimes, by discrete nonlinear Schrodinger type equations. On
the other hand, a wide variety of systems ranging from simple coupled torsion pendula in mechanics, to nonlinear electrical transmission lines and even dusty plasmas are governed by nonlinear Klein-Gordon equations. In this talk, we present some unifying features of the fundamental nonlinear excitations of these nonlinear dynamical lattices, such as discrete solitons and breathers, as well as vortices, necklaces and other complex structures in higher dimensional settings. We present a general formulation of the existence and stability of the relevant waveforms near the, so-called, anti-continuum limit and connect their dynamical evolution, whenever possible, with experimental findings and observations in both square and non-square, isotropic, as well as highly anisotropic lattices for both focusing and defocusing nonlinearities, and for short- as well as
long-range interactions. |
| 10-11 | Coffee break | |
| 11-12 | Peter Bates |
Traveling pulse solutions to the lattice Klein-Gordon equation |
| 12-2 | Lunch break | |
| 2-3 | Dmitry Pelinovsky |
Breathers from infinity in the anti-continuum limit of the discrete Klein-Gordon equationExistence of large-amplitude time-periodic breathers localized at a single site is proved in the anti-continuum limit of the Klein--Gordon lattice with a bounded potential. These breathers cannot be continued from the MacKay--Aubry theory because the amplitude of oscillations diverge
to infinity in the anti-continuum limit. We use normal form transformations and the Implicit Function Theorem to develop continuations of the large-amplitude breathers. This is a joint work with G. James (Grenoble). |
| 3-4 | Yingfei Yi |
Quasi-periodic breathers in Hamiltonian networks Hamiltonian networks form an important
class of infinite dimensional Hamiltonian systems arising in solid state physics, cell biology, and many other areas of science and technology. They also arise naturally in the discretization of Hamiltonian PDEs but the physical interest in
Hamiltonian networks mainly lies in dynamics which are far away from those of Hamiltonian PDEs. Among interesting dynamics of a Hamiltonian network, of physical importance is a robust coherent
structure known as breathers or quasi-periodic breathers which are self-localized, time periodic or quasi-periodic solutions. In this lecture, several models of Hamiltonian networks of long-range, weakly coupled anharmonic oscillators will be considered. It will be shown that corresponding to any fixed number of sites in such a Hamiltonian network, there is a positive Lebesgue measure set of linear stable, quasi-periodic breathers with the number of oscillating frequencies equal to the
number of excited sites. |
| 4-5 | Coffee break | |
| 6:30-9 | Conference dinner | Kabob & Curry on 261 Thayer Street |
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| Sunday 8 Nov in Room BH 190 |
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| Time | Speaker | Title |
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| 9-10 | Michael Herrmann |
Microscopic Oscillations and Macroscopic Waves in Hamiltonian
LatticesHamiltonian lattice equations arise naturally from atomistic models
for solids and crystals but can also be viewed as
semi-discretisations of Hamiltonian PDEs. The two-scale analysis of
such lattices is a longstanding and intriguing mathematical problem
and aims in deriving reduced macroscopic models that govern the
effective dynamics on large spatial and temporal scales.
This talk concerns the macroscopic dynamics of Hamiltonian lattices
on the hyperbolic scale. We report on some recent results on
periodic, homoclinic, and heteroclinic travelling waves for FPU
chains. Afterwards we review the notion of modulated travelling
waves and discuss the self-thermalisation of Hamiltonian lattices.
The key observation is that macroscopic shock phenomena create
strong microscopic oscillations which have a natural interpretation
as temperature and can be described by so called dispersive shocks.
Finally, we combine all results and characterise macroscopic Riemann
solvers for FPU chains. |
| 10-11 | Coffee break | |
| 11-12 | Aaron Hoffman |
Long time behavior of interacting solitary waves in the Fermi-Pasta-Ulam latticeWe study the interaction of small amplitude, long-wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish the existence and stability of "asymptotic two-soliton solutions" and "nearly two-soliton solutions." Asymptotic two-soliton solutions are solutions which approach the linear superposition of two solitary waves as t→∞. Nearly two-soliton solutions are close to one asymptotic two-soliton solution as t → −∞ and the combination of some dispersive modes and another, lower energy, asymptotic two-soliton solution as t → ∞. These solutions are stable in ℓ2 and asymptotically stable with respect to perturbations which decay exponentially at spatial ± ∞. |
| 12-2 | Lunch break | |
Accommodation and travel information
We have made reservations for all speakers at the Marriott Courtyard in downtown Providence. The Brown campus is approximately 20 minutes walk from the hotel; alternatively, you can take a taxi or one of the buses that run from Kennedy Plaza to the Brown campus. Here are links to two maps with the hotel and the conference site indicated: Map of downtown Providence and Brown campus and Conference site on Brown campus.
If you arrive at T.-F. Green airport (PVD), then it is easiest to take a taxi to the hotel. From Logan airport, take the Peter Pan bus to Providence: this bus stops at each of the terminals on the airport, and you can get off at Kennedy Plaza in Providence, which is a two-minute walk from the hotel.
