INTEGRABILITY · Analysis of a novel class of integrable partial differential equations
FP6 — Marie Curie Actions (Human Resources and Mobility)
- Duration
- 2007-07-02 → 2009-07-01
- EU contribution
- €152,242
- Participants
- 1
- Scheme
- EIF
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Results in brief
Final Activity Report Summary - INTEGRABILITY (Analysis of a novel class of integrable Partial Differential Equations)
Many different type phenomena in physics are described by nonlinear partial differential equations. Some of these equations have a very special structure tied into them which is called integrability. Several powerful methods are available for analysing and constructing solutions for integrable equations. In this project we studied numerous problems, using such equations, which described: 1. the propagation of waves, either in water or optic fibres, and 2. rotating disks in Einstein relativity theory. In particular, we considered problems with a boundary, for which the solution data was known on the boundary of some domain. The problem consisted of constructing the solution in the interior. By applying a novel method based on spectral theory, we showed how several boundary value problems for physically relevant equations could be solved. The solutions could be used to understand how electromagnetic waves evolved in optical fibres and how an astrophysical disk of dust rotated in space.
Data: CORDIS, © European Union
Project objective
The so-called integrable evolution equations possess several remarkable properties. In particular, their initial value problem can be solved using a nonlinear version of the Fourier transform method, called inverse scattering (spectral) method. For evolution equations in one and two spatial dimensions this method involves the Riemann-Hilbert and the d-bar formalisms, respectively.An important advantage of these formulations is that they can be used for the explicit evaluation of the long time behaviour of the solution. Among the most important integrable evolution equations in one space dimension is the Camassa-Holm (CH) equation, which is a certain generalization of the celebrated Korteweg-de Vries equation.The main objective of this project is to evaluate the long time behaviour of the solution of the CH equation using the Riemann-Hilbert formalism, to implement the inverse scattering method to equations analogous with the CH equation which are generalizations of the nonlinear Schroedinger and of the sine-Gordon equations, and to extend these results to integrable generalizations of the above equations in two and three spatial variables.
Original text from CORDIS.
Participants
- University of Cambridge · CambridgeCoordinatorUnited Kingdom
Links
Data: CORDIS, © European Union