Integrable nonlinear PDEs on graphs and applications to solitons on networks

Supervisor(s)

Contact Dr Vincent Caudrelier to discuss this project further informally.

Project description

The discovery of integrable partial differential equations(PDEs) solvable by the inverse scattering method, starting with the pioneering paper by Gardner, Greene, Kruskal and Miura, has given rise to a myriad of mathematical discoveries (Poisson-Lie geometry, infinite dimensional Lie algebras of symmetries, etc.) and physical applications (soliton dynamics in fluids, optics, cosmology, etc.). This rich area of Mathematical Physics and Applied Mathematics is generically called ``Integrable Systems''. The inverse scattering method is a generalization of the Fourier transform triggered by the study of integrable nonlinear PDEs. More recently, the Fokas method emerged as a generalization of the inverse scattering method triggered by the study of these integrable PDEs in the presence of general boundary conditions. In retrospect, it shed new light on the inverse scattering method and the classical Fourier transform itself! It also happens to have had great impact in applications such as medical imaging. An account of the method can be found in the book by A. Fokas, A Unified Approach to Boundary Value Problems, CBMS-SIAM (2008). The proposed research is based on a further generalization of the Fokas method to construct solutions for integrable PDEs on graph structures. The framework was introduced in V. Caudrelier, "On the On the Inverse Scattering Method for Integrable PDEs on a Star Graph", Comm. Math. Phys. 338 (2015), 893. The aim is to develop those results and establish a theory of nonlinear waves and coherent structures (like solitons) propagating on networks.

An important motivation for this PhD proposal is its potential application to networks supporting nonlinear wave phenomena, for instance optical fibers supporting solitons. Examples of questions raised by this motivation include:

- What happens to an incoming soliton on a graph after it is reflected and transmitted through the nodes of the network? Can we quantify the reflection and transmission coefficients exactly? Do the resulting waves still present some amount of coherence?

- Can we use special boundary conditions to ``pump'' solitons into a graph or control their behaviour as their propagate (logical gates)? The primary focus of the project will be to establish analytical results on simple configurations (star-graphs) and work our way up to more complicated and realistic networks.

A strong taste for complex analysis and some inclination into functional analysis are required. Numerical simulations are also possible if the candidate has a strong passion. Integrable Systems are systems that, albeit highly nontrivial and nonlinear, are amenable to exact and rigorous techniques for their solvability. They can take many shapes or forms: nonlinear evolution equations, partial and ordinary differential equations and difference equations, Hamiltonian many-body systems, quantum systems and spin models in statistical mechanics.

A large number of mathematical techniques have been developed to unravel the rich structures behind these systems. Integrable systems in Leeds. The Integrable Systems group in Leeds, one of the leading groups worldwide working on integrable systems, is part of a bigger research group, Algebra, Geometry and Integrable Systems (AGIS) and there is considerable cross-disciplinary research across these areas. This group represents a wide range of research activities into integrable nonlinear systems, their symmetries, solution techniques and the underlying mathematical structures, as well as more mathematical aspects of physical systems, for example quantum systems. The models comprise ordinary and partial differential and difference equations, dynamical mappings, discrete Painleve equations, Hamiltonian and many-particle systems and systems of hydrodynamic type. The theory and its specific models have wide-ranging applications, for example, in nonlinear optics, theory of water waves, integrable quantum field theory, statistical mechanics and combinatorics random matrix theory and nanotechnology.

Entry requirements

Applicants should have, or expect to obtain, a minimum of a UK upper second class honours degree in Mathematics or a related discipline (such as Theoretical Physics), or equivalent.

If English is not your first language, you must provide evidence that you meet the University’s minimum English Language requirements.

How to apply

Formal applications for research degree study should be made online through the university's website. Please state clearly in the research information section that the PhD you wish to be considered for is 'Integrable nonlinear PDEs on graphs and applications to solitons on networks’ as well as Dr Vincent Caudrelier as your proposed supervisor.

We welcome scholarship applications from all suitably-qualified candidates, but UK black and minority ethnic (BME) researchers are currently under-represented in our Postgraduate Research community, and we would therefore particularly encourage applications from UK BME candidates. All scholarships will be awarded on the basis of merit.

If you require any further information please contact the Graduate School Office, e: maps.pgr.admissions@leeds.ac.uk