Multidimensional integrable systems: deformations of dispersionless limits

Integrable multidimensional PDEs appear in many areas of modern Mathematics and Nonlinear Science as universal models. Recently there has been significant progress in understanding and developing the integrability theory of 3-dimensional first order quasilinear systems, led by the Loughborough team. Such systems appear in a wide range of applications, including shallow wave theory, general relativity, differential geometry. Moreover, the further novel approach to the integrability of multidimensional soliton equations was proposed by the team in Loughborough, based on the method of dispersive deformations of hydrodynamic reductions – the method of deformed hydrodynamic reductions. The ultimate goal of the project is to obtain the complete description of integrable dispersive multidimensional systems. This project relates to the interconnection of Integrable Systems and Differential Geometry.

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Full Project Detail:
Integrable multidimensional PDEs appear in many areas of modern Mathematics and Nonlinear Science as universal models. There is the rich theory of integrable systems in 2-dimensions, while the theory of integrability in dimensions higher than 2 remains much less developed. Moreover, one distinguishes two types of integrable systems in higher dimensional case: dispersionless and dispersive integrable systems. Our team in Loughborough proposed a novel technique of studying integrability in higher dimensional case, which extends the definition of integrability in dispersionless case to fully dispersive systems – the method of deformed hydrodynamic reductions. This project will apply the method of deformed hydrodynamic reductions to multi-component systems of Davey-Stewartson type.

The method of deformed hydrodynamic reductions is also applicable to differential-difference and fully discrete discrete systems. The discrete systems of Davey-Stewartson type will also be studied.

Find out more:
http://www.lboro.ac.uk/departments/maths/staff/academic/vladimir-novikov/
http://www.lboro.ac.uk/science/study/postgraduate-research/studentships/

Entry requirements:
Applicants should have, or expect to achieve, at least a 2:1 Honours degree (or equivalent) in Mathematical Sciences or a related subject. A relevant Master’s degree and/or experience in one or more of the following will be an advantage: pure mathematics, differential geometry, integrable systems.

How to apply:
Applications should be made online at http://www.lboro.ac.uk/study/apply/research/. Under programme name, select Mathematics

Please quote reference number: VN/MA/2018