Integrable Nonlinear Evolution PDEs on the Interval
Lead Research Organisation:
University of Cambridge
Department Name: Applied Maths and Theoretical Physics
Abstract
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.
Planned Impact
The results of this project can be used by researchers working in the area of nonlinear evolution PDEs, or those areas where such equations are applied. This includes all areas of science, such as nonlinear optical communication, rogue wave modeling, Bose-Einstein condensation, etc.
Organisations
People |
ORCID iD |
Athanassios Fokas (Principal Investigator) |
Publications
Ashton A
(2011)
A non-local formulation of rotational water waves
in Journal of Fluid Mechanics
Charalambopoulos A
(2010)
Laplace's equation in the exterior of a convex polygon. The equilateral triangle
in Quarterly of Applied Mathematics
Fokas A
(2014)
Perturbative and Exact Results on the Neumann Value for the Nonlinear Schrödinger on the Half-line
in Journal of Physics: Conference Series
Lenells J
(2010)
On a novel integrable generalization of the sine-Gordon equation
in Journal of Mathematical Physics
Spence E
(2010)
A new transform method I: domain-dependent fundamental solutions and integral representations
in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Spence E
(2010)
A new transform method II: the global relation and boundary-value problems in polar coordinates
in Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
Description | For example, several researchers including David Smith and Natalie Sheils have used the results of the so called Unified Transform (or Fokas Transform) to solve new mathematical problems. This method was further developed in the collaborative efforts of PI and Bernard Deconinck. |
Sector | Education,Other |
Impact Types | Cultural |