|Summer Semester 2020||Seminar Partial Differential Equations and Waveguides (Plum)||Seminar|
|Winter Semester 2019/20||Proseminar|
|Summer Semester 2019||Seminar (Analysis and Differential Equations)||Seminar|
|Winter Semester 2018/19||Seminar||Seminar|
|Summer Semester 2018||Proseminar|
|Winter Semester 2017/18||Variational and Topological Methods in Bifurcation Theory||Lecture|
|Summer Semester 2017||Bifurcation Theory||Lecture|
In my PhD project, I analyse systems of two coupled nonlinear Helmholtz equations.
The goal is to prove the existence of solutions using tools from, e.g., variational calculus or bifurcation theory.
Why the Helmholtz equation? It is a fundamental equation in physics describing the propagation and scattering of waves. For more information, please visit the webpage of our research group.
In a current project, I focus on the construction of breather (that is, time-periodic and spatially localized) solutions of nonlinear wave-type equations.
Here the application of methods from the theory of stationary Helmholtz equations promises new existence results. Indeed, using a Fourier series ansatz, this leads to an infinite system of (stationary, coupled, nonlinear) Helmholtz equations for the coefficients . This strategy, first applied in the final chapter of my dissertation thesis, seems to promise (new) families of weakly localized breathers.
Publications and Preprints
Here you can find my dissertation thesis.
R. Mandel, D. Scheider: Dual Variational Methods for a Nonlinear Helmholtz System. NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, 25:13.
R. Mandel, D. Scheider: Bifurcations of nontrivial solutions of a cubic Helmholtz system. ANONA Advances in Nonlinear Analysis 9 (2019), no. 1, 1026 - 1045.
D. Scheider: Breather Solutions of the Cubic Klein-Gordon Equation, Preprint, to appear in Nonlinearity.
R. Mandel, D. Scheider: An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential, Preprint.
R. Mandel, D. Scheider: Variational methods for breather solutions of Nonlinear Wave Equations, Preprint.
Grundzustände einer nichtlinearen Schrödingergleichung mit Deltapotentialen (in German)
DMV-OeMG Students' Conference, Salzburg, Sept 2017
Vector Solutions of a Nonlinear Helmholtz System
Nonlinear PDEs in Braga, Braga, June 2019
Time-Periodic Solutions of a Cubic Wave Equation
WAVES, Vienna, Aug 2019
Bifurcations of a Cubic Helmholtz System
DMV Congress, Karlsruhe, Sept 2019
Über ein nichtlineares Helmholtz-System (in German)
PhD Defense, Karlsruhe, Oct 2019
How to construct Breather Solutions using Nonlinear Helmholtz Systems
Stuttgart, Jan 2020