Abstract
Named after the Irish physicist, astronomer, and mathematician W. R. Hamilton (1805-1875), Hamiltonian systems are an important class of dynamical systems with certain conservation laws and rigidity features. A well-known classical example is the n-body problem (`movement of the planets around the sun'). In fact, Hamiltonian systems appear in many shapes throughout mathematics, physics, chemistry, biology, and engineering.
Classical Hamiltonian problems are formulated as systems of ordinary differential equations on finite dimensional spaces. Nevertheless, there are also equations that can be reformulated as Hamiltonian systems, but this time on infinite dimensional spaces. Such systems are called Hamiltonian partial differential equations, in short Hamiltonian PDEs. Examples are the Korteweg-de Vries equation, the Sine-Gordon equation, the nonlinear Schrödinger equation, nonlinear sigma models etc.
This project has two main aspects:
- On the one hand, we start with a `triholomorphic' Dirac-type equation on a so-called hyperkähler manifold that can be transformed into a Hamiltonian PDE on the infinite dimensional loop space of the manifold, and then studies conservation laws, integrability (`extra symmetries'), and features from modern symplectic geometry (`non-squeezing' properties, symplectic capacities etc.) of this Hamiltonian PDE.
- On the other hand, we investigate the occurrence and bifurcation behavior of singularities with hyperbolic components in 4-dimensional integrable Hamiltonian systems and classify the associated fibers.
Researcher(s)
Research team(s)
Project type(s)