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Dispersionless integrable systems in 2+1 dimensions


About This PhD Project

Project Description

Dispersionless integrable systems occur in a wide range of applications in various areas of pure and applied mathematics. In 2+1 dimensions, there exists an efficient approach to the classification of integrable systems of this kind based on the method of hydrodynamic reductions pioneered by the Loughborough group [1-3]. This method has lead to extensive classification results within particularly interesting classes revealing deep relations with generalised conformal geometry, theory of hypergeometric functions and modular forms.

The goal of this project is twofold:
1. Classification of dispersionless integrable systems in 2+1 dimensions with an emphasis on multi-component systems of hydrodynamic type.
2. Reconstruction of dispersive deformations of dispersionless integrable systems based on the method of dispersive deformations of hydrodynamic reductions proposed in [4-5]. This programme is expected to lead to a new class of integrable soliton PDEs with remarkable properties and potential applications.

Entry requirements

You should have, or expect to achieve, a good 2:1 honours degree (or equivalent international qualification) in mathematics.

The project will require basic knowledge of differential equations, differential geometry, and familiarity with symbolic computations (Mathematica/Maple).

How to apply

All applications should be made online: https://www.lboro.ac.uk/study/postgraduate/apply/research-applications/

Under programme name, select 'Mathematical Sciences'. Please quote reference: MA/EVF-Un1/2020.

The deadline for applications is 30 September 2020.

Supervisors' online staff profile pages:
https://www.lboro.ac.uk/departments/maths/staff/academic/jenya-ferapontov/
https://www.lboro.ac.uk/departments/maths/staff/academic/vladimir-novikov/

References

[1] E.V. Ferapontov and K.R. Khusnutdinova, On integrability of (2+1)-dimensional quasilinear systems,
Comm. Math. Phys. 248 (2004) 187-206.

[2] E.V. Ferapontov and B. Kruglikov, Dispersionless integrable systems in 3D and Einstein-Weyl geometry,
J. Diff. Geom. 97 (2014) 215-254.

[3] B. Doubrov, E.V. Ferapontov, B. Kruglikov, V.S. Novikov,
On integrability in Grassmann geometries: integrable systems associated with fourfolds in Gr(3, 5), Proc. London Math. Soc. (3) 116, no. 5 (2018) 1269-1300.

[4] E.V. Ferapontov, A. Moro and V.S. Novikov, Integrable equations in 2+1 dimensions: deformations of dispersionless limits, J. Phys. A: Math. Theor. 42 (2009) (18pp).

[5] E.V. Ferapontov, V. Novikov and I. Roustemoglou, On the classification of discrete Hirota-type equations in 3D, IMRN no. 13 (2015) 4933-4974.

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