Research Interests Overview
My current research area concerns mathematical physics and dynamical systems. I am interested in direct and inverse spectral theory for differential and difference operators in connection with completely integrable nonlinear wave equations. In particular, I am studying Schrödinger and Jacobi operators and their connections with the Korteweg-de Vries and Toda hierarchies of nonlinear evolution equations.
My research is funded by the Austrian Science Fund (FWF). Further information can be found on the corresponding project pages:
- FWF project P17762
- START project Y330
- Doctoral Program Dissipation and Dispersion in Nonlinear Differential Equations (archived)
More precisely, my interests can be split up as follows:
Applied Functional Analysis
- Spectral theory for Schrödinger, Sturm-Liouville, Dirac, and Jacobi operators
- Spectral deformations (Darboux type transformations)
- Inverse spectral theory, trace formulas
- Direct and inverse scattering theory
- Weyl asymptotics
- Measure theory
- Fourier analysis
Ordinary Differential Equations
- Oscillation theory
Nonlinear Dynamics
- Bäcklund transformations (N-soliton solutions on arbitrary background)
- Inverse scattering transform
- Stability/Long time asymptotics (nonlinear steepest descent)
Applied Algebraic Geometry
- Applications of the theory of hyperelliptic curves (in particular Jacobi's inversion problem on Riemann surfaces) to completely integrable wave equations
- Hierarchies of completely integrable wave equations (Korteweg-de Vries, Toda, Ablowitz-Ladik)
- Algebro-geometric solutions (quasi periodic solutions)
- Riemann-Hilbert problems on Riemann surfaces
Linear and Nonlinear Partial Differential Equations
- Abstract wave equations
- Kinetic equations (Vlasov equation coupled to field equations)
- Nonlinear wave equations (Korteweg-de Vries, Camassa-Holm)
Biomathematics
- Modeling of the respiratory and cardiovascular system