PhD project, Starting October 2010

Multi-symplectic structures and symplectic pattern formation


Project Supervisor: Prof T.J. Bridges
University of Surrey
An understanding of nonlinear partial differential equations (PDEs) is one of the great challenges for the twenty-first century. In the analysis of nonlinear ordinary differential equations (ODEs), the use of geometry methods has led to great progress. An example is the use of Hamiltonian structure and symplectic geometry. While Hamiltonian systems are a special case of nonlinear ODEs, they do occur widely in applications such as robotics, fluid mechanics, meteorology, and oceanography. Therefore it is natural to consider a geometric approach to nonlinear PDEs. The purpose of this project is to pursue this agenda for nonlinear elliptic and hyperbolic PDEs with a multi-symplectic structure.

The proposed project has three parts.

  • Part I. Develop the geometry of the total exterior algebra bundle for smooth two-dimensional manifolds and combine with multi-symplecticity. The starting point for this part of the project is Bridges (2006), where the case of flat manifolds was considered. The extension to curved manifolds leads to a class of Dirac operators on manifolds.
  • Part II. Multi-symplectic PDEs are generated by a form of Hamilton's principle and therefore the use of the calculus of variations to find critical points is appealing. However, the functionals that appear in this variational principle are indefinite. The idea of this part of the project is to use the Mountain Pass Theorem (cf. Jabri 2003) to find saddle points. The theory is to be applied to the PDEs found in Part I, and to multi-symplectic PDEs on flat n-tori, which are of interest in pattern formation.
  • Part III. Apply numerical methods to either part I or to part II. Discretisation of part I involves the theory of discrete differential forms to derive a numerical scheme for multi-symplectic PDEs on 2-manifolds. The numerical implementation of part II involves numerical implementation of the mountain pass theorem (MPT). The numerical implementation of MPT for semilinear elliptic and hyperbolic PDEs was introduced by Choi & McKenna (1993). The idea would be to extend the numerical MPT to multi-symplectic Dirac operators on tori.

Students with a background in mathematics, numerical analysis or physics will be natural candidates for this studentship.

Funding: British applicants are eligible for EPSRC DTG funding which provides a scholarship for full fees and a bursary to cover living expenses. Candidates can also apply for top-up scholarships with a value of up to 3K pounds per annum. EU candidates are eligible for EPSRC funded fees-only scholarships, and can apply for bursaries and top-up scholarships. Non-EU overseas students are not eligible for EPSRC funding but can apply for fully-funded scholarships offered by the Department on a competitive basis. For further information about the project contact the project supervisor, and for general information about the postgraduate programme in mathematics at Surrey, and application information, contact Dr Gianne Derks.

 
References
  • T.J. Bridges. Multi-symplectic structures and wave propagation Math. Proc. Camb. Phil. Soc. 121 147-190 (1997).
  • T.J. Bridges. Canonical multi-symplectic structure on the total exterior algebra bundle, Proc. Royal Soc. Lond. A 462 1531-1551 (2006).
  • T.J. Bridges & S. Reich. Numerical methods for Hamiltonian PDEs, J. Phys. A: Math. Gen. 39 5287-5320 (2006).
  • Y.S. Choi & P.J. McKenna. A mountain pass method for the numerical solution of semilinear elliptic problems, Nonlinear Analysis 20 417-437 (1993).
  • Y. Jabri. The Mountain Pass Theorem, Encyclopedia of Mathematics and its Applications 95 Cambridge University Press (2003).
 
Project Supervisor Contact Information:

 
Address: Professor Thomas J. Bridges
Department of Mathematics
University of Surrey
Guildford
Surrey GU2 7XH
ENGLAND
Telephone: 01483 68 9634
Fax: 01483 68 6071
Mail to: T.Bridges@surrey.ac.uk
TJB Web page: http://www.maths.surrey.ac.uk/
 
Director of Postgraduate Admissions:

 
Address: Dr Gianne Derks
Department of Mathematics
University of Surrey
Guildford
Surrey GU2 7XH
ENGLAND
Telephone: 01483 68 9643
Fax: 01483 68 6071
Mail to: G.Derks@surrey.ac.uk
GD Web page: http://www.maths.surrey.ac.uk/