Clarence E. Wayne - US grants

9803164
Wayne

This project will involve the study of the long-time behavior of solutions
of partial differential equations and other infinite dimensional dynamical
systems. Of particular interest are systems defined on unbounded domains
and infinite dimensional Hamiltonian systems. In such systems it is often
not obvious what the principal modes and mechanisms governing the
stability or instability of solutions are. A main thrust of the research
project will be to identify invariant, finite-dimensional objects in the phase
space of these systems which govern the long-time behavior of solutions.
Among the specific examples to be studied are dissipative partial
differential equations like the Cahn-Hilliard equation or systems of
reaction-diffusion equations, which will be analyzed using invariant
manifold theorems, and infinite dimensional Hamiltonian systems to
which will be attempted to extend the Kolmogorov-Arnold-Moser
Theorem. Finally, the project will include attempts to give rigorous
estimates of the validity of some of the modulation equations used to
approximate traveling waves in fluid mechanics.

The sorts of equations that will be studied arise in many applications such
as the formation of patterns in biological and physical systems,
crystallization, and the motion of waves in the ocean. They are all
characterized by the fact that even though it is impossible to solve the
equations which govern their motion explicitly, applications require at
least a qualitative understanding of the behavior of their solutions. This
research is aimed both at providing such a qualitative understanding, and
also at explaining the limits of such approximate solutions. A prime
example of this situation is the study of tsunamis. It is experimentally
observed that disturbances of the ocean surface generate large, solitary
waves which travel enormous distances over the sea surface without
change of form. It is impossible to solve the equations which describe
such waves exactly, and while a number of model equations which
describe such traveling waves exist, the mathematical justification of such
models is lacking. One of the goals of the research in this proposal is to
give a rigorous derivation of the approximate equations that govern wave
motion of fluid surfaces. This will both help to clarify which of the various
physical models gives the best description of the actual waves, and should
also shed light on how better models can be developed.

Professor Wayne will study the long-time behavior of partial
differential equations like the Navier-Stokes, Euler, and Maxwell's
equations which arise in the study of fluid dynamics and optics. He
will use methods and techniques from dynamical systems theory to make
qualitative and quantitative predictions about the behavior of the
solutions of these equations and will focus on five main areas: (i)
The derivation, justification and experimental validation of model
equations for waves on fluid surfaces; (ii) The long-time behavior of
solutions of the Navier-Stokes equations; particularly vortex
solutions; (iii) The modeling of very short pulses in optical media;
(iv) The existence and stability of pulse solutions in coupled optical
systems; and (v) The approximation of the motion of thin elastic
media. In particular, he will attempt to extend the insights gained
from understanding the invariant geometrical objects in the phase
space of ordinary differential equations to the infinite dimensional
setting of partial differential equations, focusing particularly on
the case of problems on unbounded spatial domains where the presence
of continuous spectrum may cause qualitative differences between the
behavior of the finite and infinite dimensional systems.

The types of systems that Professor Wayne will study arise in many
applications, including nonlinear optical communication, fluid
mechanics and the behavior of elastic materials. For instance,
attempts to obtain faster and faster transmission of information
through optical fibers often utilize extremely short optical pulses.
While there is a well understood and much studied method for
approximating the transmission of long pulses through such fibers
optical technology is now reaching the point where this approximation
breaks down and new models are needed. Point (iii) in the list of
projects above aims to develop such approximations. This problem,
like the others that will be studied in this project, is characterized
by the fact that while it is impossible to solve the equations
governing the system exactly, applications require at least a
qualitative understanding of the behavior of solutions. Professor
Wayne's research has three goals: First, the derivation of model
equations which can be used to approximate the behavior of the true
physical system -- whether it arises in optics, fluids, or elsewhere.
Secondly, the computation of accurate estimates to control how much
the behavior of the true system can deviate from that of the model
system, and finally the development of geometrical insights which can
provide a qualitative means of understanding and predicting the
behavior of this type of complicated physical system even if the
actual solution can not be computed.

Abstract DMS 0543432

C E Wayne, Boston University

Workshop on Mathematical Hydrodynamics at the Steklov Institute


This is a proposal for support for US-based researchers to attend the Workshop on Mathe-
matical Hydrodynamics at the Steklov Institute that is being planned for the week of June
12 - 17, 2006.

Intellectual merit of the proposed activity: The subject matter of the workshop is
mathematical aspects of the theory of hydrodynamics, which includes the study of Euler's
equations and the Navier - Stokes equations of fluid dynamics, as well as the equations for
the dynamics of waves on free surfaces. In the past several years, there has been progress on
a number of the most basic and difficult questions in this field. These advances further the
mathematical understanding of the fundamental, nonlinear, partial differential equations
that describe these systems and also shed light on the basic physical processes they model.

Broader impact of the proposed activity: The purpose of this meeting is to present
the state of the art to the diverse international community of researchers who have an
interest in these topics, and to discuss the perspectives for the next advances in the field.
This should aid in the dissemination of the current state of the knowledge in this area and
also highlight avenues ripe for further progress. Furthermore, the funds from this grant
will allow both junior and underfunded US researchers to participate in the event who
might not otherwise have the opportunity to do so.

This project will advance the creation and support of a community of scholars, from undergraduate to faculty, working at the interfaces among dynamical systems and biological applications. The three main areas of focus are: 1. Analysis of systems with multiple length and time scales, including applications to pattern formation; 2. Mathematical neuroscience, including analytical methods for working with small networks and reduction of dimension techniques; 3. Gene regulatory networks, including the development of RNA switches, transcriptional bursting and programmable cells. These areas have major applications to issues concerning health and medicine. The project will build on the previous research and training experience of the Center for BioDynamics, co-directed by the Principal Investigator and one of the other senior faculty members. Trainees will be pre- and post-doctoral students who will take part in a wide variety of formal and informal activities, including special seminars, working groups, mini-symposia, laboratory work, journal clubs and social events, which will enable them to acquire the multiple scientific cultures needed to work in a trans-disciplinary manner. The pre-doctoral students will be from the departments of Mathematics or Biomedical Engineering; the postdoctoral associates will be drawn from a wide range of backgrounds, with a focus on applied math. In addition to their research activities, trainees will obtain experience teaching at different levels. Math department faculty and trainees will be involved in the construction of new interdisciplinary curricula for undergraduates in other departments, including Biology; the faculty will mentor the trainees in teaching the new curricula.

The focus of the thematic program semester of winter 2008 at the CRM is on dynamical systems, interpreted in a broad sense so as to include applications to fundamental problems in differential geometry as well as in mathematical physics. Topics that are considered include:(1) the interplay between dynamical systems and PDE, in particular in the context of Hamiltonian systems,
(2) geometric evolution equations such as Ricci flows and extrinsic curvature flows, (3) spectral theory and its relationship to Hamiltonian dynamics, and
(4) Floer theory and Hamiltonian flows. In the past several years there have been dramatic achievements in these four areas, representing progress on a number of the most basic and difficult questions in this field. These advances have had a broad impact on recent progress in geometry and topology, and they also shed light on basic physical processes, such as nonlinear wave phenomena, that are modeled by ordinary and partial differential equations.
The purpose of this program semester is to bring together members of the diverse international community of researchers who have an interest in these topics, to give a series of advanced-level courses on relevant subject matter so as to make the topic accessible to new researchers in the field, and to bring into discuss the perspectives and general indications for the next advances and directions of progress in the area.


The central focus of the theme semester of winter 2008 at CRM is dynamical systems. The theory of dynamical systems is concerned with the description of the evolution of systems depending on time. Such systems are fundamental, and appear very commonly in the modeling of physical, chemical and biological phenomena, as well as in geometry and many other areas of mathematics. In the past several years there have been dramatic achievements in the area of dynamical systems, including the proof of Poincaré conjecture by G. Perelman (an event known to the public through the drama of the Fields medal awards in 2006). These advances have had a broad impact on recent progress in geometry and topology. The modern theory of dynamical systems has also been fundamental in the study of many basic physical processes, and their modeling by ordinary and partial differential equations. The purpose of this program semester is to bring together representatives of the diverse international community of researchers who have an interest in these topics. Its activities will comprise (1) a series of advanced-level courses on the subject matter, so as to make the field available to students and new researchers, (2) to host discussions of the perspectives and future directions for the next advances and areas of progress in the field.

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).

Professor Wayne will study the behavior of infinite dimensional dynamical systems such as the Fermi-Pasta-Ulam model, the Navier-Stokes equations and the Euler equations. He will use methods from dynamical systems theory to make qualitative and quantitative predictions about the solutions of these systems and will focus on four main areas: (i) The stability and interactions of solitary wave solutions in infinite dimensional dispersive Hamiltonian systems, and the geometry of the phase space of such systems; (ii) The derivation and justification of approximate equations for the evolution of wave packets on a fluid surface and the relation of such results to normal form theorems for Hamiltonian systems in which the linear part has continuous spectrum; (iii) Metastable behavior in the nearly inviscid Navier-Stokes equations and other weakly dissipative systems; and (iv) The use of invariant manifold theorems to analyze singularly perturbed partial differential equations. The geometrical properties of objects like invariant manifolds have been a great aid in illuminating the behavior of finite dimensional dynamical systems and this project will develop similar methods and insights into the behavior of infinite dimensional systems, particularly those defined on unbounded spatial regions where the linear problem has continuous spectrum.

The differential equations that Professor Wayne will study arise in a variety of different physical circumstances and are characterized by the fact that while the equations themselves are well known they are too complicated to solve except in special and/or unrealistic cases. Nonetheless, applications require at least a qualitative understanding of the behavior of their solutions and this project will develop such an understanding for the equations enumerated above. As an example related to point (i) in the preceding paragraph, the equations that describe waves on the ocean, which are of importance both for understanding climate and weather and for predicting events such as tsunamis, have a family of solutions known as ``solitary waves'' which represent a single wave traveling across theocean. In practice, however, many waves are inevitably present and it becomes necessary to understand how these waves interact with each other. This project will study the types of interactions that can occur in such systems and their consequences. The remaining three sections of the research project will aim, in a similar fashion, to extend the understanding of special or limiting cases of equations of physical importance to more realistic conditions.

Abstract for DMS-1311553

Professor Wayne will study how methods such as invariant manifold theory and the Kolmogorov-Arnold-Moser (KAM) theory, which were originally developed to understand finite dimensional dynamical systems, can be adapted to yield insight into the qualitative and quantitative behavior of solutions of partial differential equations. He will concentrate primarily on equations arising in physical applications such as fundamental fluid equations, e.g the Navier-Stokes equation, the equations for vortex sheets, and equations from nonlinear optics. His research will focus on four main problems: (A) Metastable behavior in two-dimensional fluids; (B) Periodic solutions of the vortex sheet equations; (C) Breathers in periodic media; and (D) Normal forms and invariant manifolds in dispersive Hamiltonian systems. Dynamical systems methods have yielded many insights into the qualitative behavior of finite dimensional systems for which no closed form solutions exist. The existence theory of many of the infinite dimensional systems to be studied is now well established and this research will attempt to derive more detailed information about the behavior of the solutions on various physically relevant time scales, as well as illuminating the origin of these time scales.

The differential equations that Professor Wayne will study arise in a variety of different physical contexts and are characterized by the fact that while the equations themselves are well known, they are too complicated to solve explicitly except in very special or physically unrealistic cases. Nevertheless, applications require at least a qualitative understanding of the behavior of their solutions and this research project will aim to develop such an understanding for the systems described above. As an example related to point (C) in the preceding paragraph, consider light pulses of the types that are used in fiber optic cables currently used for telecommunications. In a homogeneous medium, such as glass, such pulses rapidly spread out or disperse. However, in periodic media, like certain crystals, such pulses may become trapped. Trapped pulses are of great current interest because of the hope that they might serve as the basis for a purely optical computational system. Professor Wayne's research will examine the conditions that the medium must satisfy in order to support these ``trapped'' pulses as well as investigating how common such media are likely to be. The other three projects will also aim to develop new fundamental insights into these important physical systems.