Nonlinear Waves in PDEs

This London Dynamical Systems Group Graduate School will provide an introduction to nonlinear waves and pattern formation in nonlinear dissipative and conservative systems. It is meant to be accessible to entry-level PhD students and thus has no non-standard prerequisites. The primary goal of the three lecture series is to give a broad overview of the main phenomena, ideas and concepts in this area and to the techniques that are used to study them.

The Graduate School will comprise a total of 30 hours worth of lectures and tutorials. Registration forms can be found at LDSG Graduate School website. Please check our accommodation page for hotel information; Campus maps of the University of Surrey are available from our Visitor Information Page.

The Graduate School on Nonlinear Waves in PDEs will run from Monday, 9 January 2006, to Friday, 13 January 2006. All lectures will be in Room 10AA04 in Building AA at the University of Surrey. Lectures are scheduled as follows:

TimeCourse and Lecturer
9:00-11:00Pattern formation (Sandstede)
11:00-11:30Coffee break
11:30-12:30Nonlinear waves (Derks)
12:30-2:00Lunch break
2:00-4:00Differential forms (Bridges)
4:00-4:30Coffee break
4:30-5:30Nonlinear waves (Derks)

Differential forms in dynamical systems and pattern formation
Thomas J. Bridges (University of Surrey)

The lectures will develop the theory of exterior algebra and differential forms and their application in dynamical systems and pattern formation. No prior knowledge of exterior algebra or differential forms will be assumed.

Given a vector space V of dimension n, there are a number of other vector spaces that can be built on it: the dual space, the spaces of k-vectors, and k-forms, for k=0,...,n. Given a linear ODE on V it is often of interest to numerically integrate the induced systems on exterior algebra spaces. Such systems arise in the linearization of nonlinear ODEs about homoclinic and other trajectories and in the linearization of nonlinear PDEs about solitary waves, where V is a model for the tangent space of the phase space. These lectures will discuss the theory behind such equations and the implementation of numerical algorithms for their integration.

The use of differential forms for PDEs will also be discussed. In this case one has (horizontal) differential forms on the base manifold (the independent variables, i.e. space and time) and (vertical) differential forms on the fibre (the dependent variables). The theory for such differential forms and their application to nonlinear PDEs will be introduced.

The lectures will be applications oriented, with examples taken from the stability of solitary waves and fronts, the linearization about homoclinic orbits of ODEs, solution of boundary value problems, hydrodynamic stability, and the numerical solution of nonlinear PDEs.

Nonlinear waves in Hamiltonian systems
Gianne Derks (University of Surrey)

The lectures will highlight some aspects of nonlinear waves in Hamiltonian systems like solitary waves/fronts in one dimensional systems and vortices in two dimensional systems. Symmetries or Casimir functionals often play an important role in the analysis of such nonlinear waves, especially in the linear and nonlinear stability analysis. It is planned that the lectures cover the following areas:

Pattern formation
Björn Sandstede (University of Surrey)

The lectures will give an overview of various pattern-forming systems, their mathematical models, and some of the techniques used to investigate the emerging patterns. The following topics will be covered: