- E-mail: J.Lenells@damtp
- Research Group:
Nonlinear Dynamical Systems - Office Tel: 01223 337904
- Office: F1.20
- Address:
DAMTP,
Centre for Mathematical Sciences,
Wilberforce Road,
Cambridge,
CB3 0WA,
United Kingdom - More Info (internal):
lookup.cam.ac.uk
Dr Jonatan Lenells
Career
- 2007 - present Marie-Curie Intra-European Research Fellow, DAMTP, University of Cambridge
- 2006 - 2007 Assistant Professor, Department of Mathematics, University of California Santa Barbara
- 2002 - 2006 PhD Student, Department of Mathematics, Lund University
Research
Jonatan Lenells is interested in problems arising in mathematical physics. In particular, he studies differential equations modeling the propagation of nonlinear waves in optical fibers and fluid mechanics. More recently, he has also been working on problems related to quantum field theory and low-dimensional topology.
Current research projects include:
Solutions of initial-boundary value problems via inverse scattering
We analyze solutions of nonlinear integrable evolution equations in plane domains with boundary by means of inverse scattering techniques. Recent work have focused on nonlinear Schrödinger type equations on the half-line and their soliton solutions.
Transform methods and spectral theory for PDEs in 3D
The goal is to generalize the methods that are currently so successful for integrable equations in 2D to higher dimensions. At the moment we consider the application of inverse spectral theory to equations in 3D.
Geometric methods
Some nonlinear partial differential equations can be viewed geometrically as describing geodesic flow on spaces of diffeomorphisms. We utilize this geometric viewpoint to deduce interesting properties of the systems. Equations studied include the Camassa-Holm and Hunter-Saxton equations.
Supersymmetric integrable systems
Most integrable equations admit at least one supersymmetric extension. Many of the standard techniques used for integrable systems can be generalized to this setting. By exploiting the supersymmetric structure, interesting things can also be said about the underlying bosonic systems.
Topological quantum field theory and low-dimensional topology
The interplay between mathematics and physics has in recent years led to many new results in low-dimensional topology. By using the ideas of gauge theory and quantum field theory which are so prevalent in particle physics, new topological invariants can be constructed. Together with people at the theoretical physics department at Caltech, I try to understand one of these invariants for hyperbolic three-dimensional manifolds and its relation to 3D quantum gravity and conformal field theory.
Selected Publications
- J. Lenells, The derivative nonlinear Schrödinger equation on the half-line, Physica D: Nonlinear Phenomena (2008) DOI 10.1016/j.physd.2008.07.005.
- J. Lenells, A bi-Hamiltonian supersymmetric geodesic equation, Lett. Math. Phys. (2008) DOI 10.1007/s11005-008-0257-4.
- B. Khesin, J. Lenells, and G. Misiolek, Generalized Hunter-Saxton equation and the geometry of the group of circle diffeomorphisms, Mathem. Annalen (2008) DOI 10.1007/s00208-008-0250-3.
- J. Lenells, The Hunter-Saxton equation describes the geodesic flow on a sphere, J. Geom. Phys. 57 (2007) 2049--2064.
- J. Lenells, Traveling wave solutions of the Camassa-Holm equation, J. Differential Equations 217 (2005), 393--430.