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Workgroup Nonlinear Partial Differential Equations

Kollegiengebäude Mathematik (20.30)
Room 3.029

Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe

Office hours:
Mon-Fri 10-12, Tue+Wed 14-16

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

Photo of Dominic Scheider Dr. Dominic Scheider

Office hour for students: on appointment
Room: -1.021 Kollegiengebäude Mathematik (20.30)
Tel.: 0049 721 608 42046
Fax.: 0049 608 46530
Email: dominic.scheider@kit.edu

Englerstraße 2
76131 Karlsruhe

Current List of Courses
Semester Titel Typ
Winter Semester 2019/20 Proseminar
Summer Semester 2019 Seminar
Winter Semester 2018/19 Seminar
Summer Semester 2018 Proseminar
Winter Semester 2017/18 Lecture
Summer Semester 2017 Lecture


In my PhD project, I analyse systems of two coupled nonlinear Helmholtz equations such as

$  - \Delta u - \mu u = u \: (u^2 + b \: v^2) \quad\text{in }\mathbb{R}^3, \qquad - \Delta v - \nu v = v \: (v^2 + b \: u^2) \quad\text{in }\mathbb{R}^3, 
 \qquad  u, v \in L^4(\mathbb{R}^3)   $

where  \mu, \nu > 0 and  b denotes a (constant or periodic) coupling.
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, e.g.

$  \partial_t^2 U - \Delta U = \Gamma(x) |U|^{p-2} U \quad\text{on } \mathbb{R} \times \mathbb{R}^N.   $

Here the application of methods from the theory of stationary Helmholtz equations promises new existence results. Indeed, using an ansatz of the form

$  U(t, x) = \sum_{k} \cos(k t) \: u_k(x),   $

this leads to a system of (stationary, coupled, nonlinear) Helmholtz equations for the coefficients  u_k . This strategy, first applied in the final chapter of my dissertation thesis, seems to promise (new) families of weakly localized breathers.

Publications and Preprints

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.

Here you can find my dissertation thesis.


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