# Details of Grant

EPSRC Reference: EP/H022201/1
Title: Integrable Nonlinear Evolution PDEs on the Interval
Principal Investigator: Fokas, Professor A
Other Investigators:
Researcher Co-Investigators:
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Department: Applied Maths and Theoretical Physics
Organisation: University of Cambridge
Scheme: Standard Research
Starts: 18 January 2010 Ends: 17 April 2010 Value (£): 15,982
EPSRC Research Topic Classifications:
 Non-linear Systems Mathematics
EPSRC Industrial Sector Classifications:
 No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
During the three-month research visit of Professor Deconinck to Cambridge, we will investigate two problems which arise in the analysis of the celebrated nonlinear Schr\ odinger equation (NLS) formulated on a finite interval: (a) The reduction of the general theory for solving intial-boundary value problems on the finite interval in the particular case that the boundary values are periodic in space. (b) The asymptotic behaviour of the solution in the case that the given Dirichlet boundary conditions are periodic in time. Regarding the first problem, we note that the related beautiful algebraic-geometric theory characterises the particular finite-gap case in terms of theta functions. Our goal is to characterise the general case in terms of a matrix Riemann-Hilbert problem and to rederive the results of the finite-gap representation as a particular case. Regarding the second problem, we note that for the linearized version of the NLS it has been shown recently that the asymptotic behaviour of the solution depends on the commensurability of the time period $T$ of the boundary data with $L^2/ \pi$, where $L$ is the length of the finite interval. Our goal is to obtain an analogous result for the NLS.
Key Findings
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