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Integrability and solvability of nonlinear odes and pdes in the complex domain

Objective

The purpose of this project is:
(a) To investigate the relation between the singularity structure of a number of physically important nonlinear ODEs and PDEs in the complex domain and the analytical methods needed to solve these equations and
(b) to solve boundary value problems for some novel nonlinear PDEs describing wave propagation. More specifically, we intend to study integrable and solvable ODEs possessing only algebraic singularities, i.e. ODEs whose solutions are all finitely sheeted (FSS property). We attempt either to validate or disprove the conjecture that all FSS systems can be mapped by nonlinear transformations to ones that possess the full Painleve property. We also plan to elucidate the breakdown of the FSS property beyond a certain contour size in the complex plane. Thus, we aim to justify rigorously our numerical results which connect non-integrability with the occurrence of infinitely sheeted solutions (the ISS property) and singularity clustering in the complex domain. We also plan to study the ISS property in near-integrable PDEs whose physically interesting solutions (e.g. solitary waves) are expected to have different dynamics from that of integrable PDEs. Finally, we will study boundary value problems, for these near-integrable PDEs, using the new method for solving boundary value problems for integrable PDEs recently introduced by Fokas.

Funding Scheme

RGI - Research grants (individual fellowships)

Coordinator

University of Cambridge
Address
Silver Street
CB3 9EW Cambridge
United Kingdom