LMS/EPSRC Short Course on Nonlinear Wave Phenomena
July 4-9, 2005
Organiser: Dr. Beatrice Pelloni
!!!!!!!!!!!!!!!!!!!!!! NEW: PROVISIONAL PROGRAMME here !!!!!!!!!!!!!!!!!!!!!! NOTE: The week following the course, a meeting on Nonlinear dispersive waves, funded by the EC, will be held in Anogeia (Crete) The principal invited speakers are Proff. Bona, Fokas, Sulem and Saut. Funding is available for students and young researchers, please see the announcement for further details.
Nonlinear wave phenomena are of great importance in the physical world, and have been for a long time a challenging topic of research for both pure and applied mathematicians. This course will focus on analytical and physical aspects of nonlinear wave phenomena. This important area of research has traditionally been strong in the UK in what concerns application to fluid dynamics. However there are other interesting aspects of the theory of nonlinear waves, especially as described by one and two space-dimensional integrable PDEs, and inverse problems relating to this area. All of these topics have seen significant advances in recent years, and research is very active.
The present course focuses more specifically on nonlinear waves and recent related techniques, presenting nonlinear wave propagation models and specific properties. The course will also include the classical inverse scattering transforms and some recent advances in this field. It aims to describe various different aspects of the relevant theory to an audience of postgraduate students and young postdoctoral researchers in applied mathematics.
The programme will consist of two five-hour courses on Nonlinear models of water wave propagation (Prof. Jerry Bona, University of Illinois at Chicago), Wave propagation and the NLS equation (Prof. Catherine Sulem, University of Toronto), and one six-hour course on Integrable nonlinear PDEs and the inverse scattering transform (Prof. Thanasis Fokas, University of Cambridge, and Dr. Beatrice Pelloni, University of Reading). The lecturers are world expert in the fields, and well known as very good speakers. The first two come from overseas, and will provide a good opportunity to learn about topics not usually discussed in lecture courses at UK universities.
The classical water wave problem and derivation of model equations
One-way models; elementary theory and comparisons with experiments
Two-way models and their relation to the one-way descriptions
Fully three-dimensional models
Wave sediment models and their use in engineering practice
Derivation of canonical equations of mathematical physics from the water wave problem, with focus on weakly nonlinear dispersive waves : introduction to multiple scale analysis, the nonlinear Schr\"odinger equation as an envelope equation.
Mean field generation: Multiple scale formalism; a few examples; derivation of the Davey-Stewartson system;
The nonlinear Schr\"odinger equation : Basic dynamical effects: Modulational instability; Solitons in one space dimension; Soliton Instability for tranverse perturbation.
Structural properties of the NLS equation: Lagrangian and Hamiltonian structure, Noether theorem, invariances and conservation laws.
The initial value problem: Existence theory, Long-time behavior; finite-time blowup.
Analysis of the blow-up: self-similarity; ; modulation analysis; rate of blow-up.
W. Strauss: Nonlinear Wave Equations, CBMS, Volume 73, American Mathematical Society, 1989.
C. Sulem and P.-L. Sulem: The Nonlinear Schroedinger Equation: Self-focusing and Wave Collapse, Appl. Math. Sciences, Volume 139, 1999, Springer
Thierry Cazenave : Semilinear Schroedinger equations, AMS, Lecture Notes of the Courant Institute, vol 10, 2003.
Jean Bourgain: Global solutions of Nonlinear Schroedinger equation, AMS, Colloquium Series, Vol 46, 1999.
Mathematical Preliminaries: the Riemann-Hilbert and D-bar problems
The derivation of one and two-dimensional Fourier Transform via RH and d-bar
Lax pair formulation and inverse scattering transform for the solution of the Cauchy problem: the nonlinearisation of Fourier transform
Direct linearising transform and the dressing method
Extension of the inverse scattering transform to boundary value problems: the linear case
The nonlinear case: solution of NLS and KdV on the half line
Accommodation has been arranged in single rooms with ensuite at Whiteknights Hall, on the university campus. In addition to accommodation, breakfast and dinner will be provided at Whiteknights Hall for course residents. Lunch and coffee breaks will also be provided for course participants. The lectures will start on Monday July 4th at 1:30pm, but registration will open at 11:30am, and a buffet lunch will be provided. The course ends on Saturday, July 9th at 12:30, with lunch for those who want it until 2:00pm.
The number of participants will be limited and those interested are encouraged to make an early application. Interested participant are strongly encouraged to express their interest informally to the organiser as soon as possible.
An online application form is available from the London Mathematical Society.
Dr. Beatrice Pelloni
Department of Mathematics
University of Reading
Reading RG6 6AX
Tel. +44 (0)118 3788990