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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, Wed, Thu 10-13 and by email

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

Photo of Simon Kohler M. Sc. Simon Kohler

Office hour for students: Come in whenever the door is open
Room: 3.038 Kollegiengebäude Mathematik (20.30)
Tel.: +49 721 608 46174
Email: simon.kohler@kit.edu

Englerstraße 2
76131 Karlsruhe

Current List of Courses
Semester Titel Typ
Summer Semester 2020 Lecture
Winter Semester 2019/20 Lecture
Summer Semester 2019 Seminar
Winter Semester 2018/19 Seminar
Summer Semester 2018 Proseminar
Winter Semester 2017/18 Lecture

Additional teaching: Special help for starters in math at KIT

Since summer 2018 I am part of the team for special help for starters in math at KIT. Further information is only available in german:
Lernraum-Betreuung SS19 for Lineare Algebra II and Analysis II
Lernraum-Betreuung WS18/19 for Lineare Algebra I and Analysis I
Lernraum-Betreuung SS18 for Lineare Algebra II and Analysis II

Research interest

I am part of Projekt A6 Time-periodic solutions for nonlinear Maxwell equations of the Collaborative Research Center (CRC) 1173 »Wave phenomena: analysis and numerics«.

Together with my adviser Prof. Dr. Wolfgang Reichel I investigate nonlinear wave equations for timeperiodic solutions. A prototype is

$\qquad g(x)u_{tt}-u_{xx}=\Gamma(x)(u_t^3)_t \qquad (x,t)\in\mathbb{R}\times\mathbb{T}_T.$

The equation is physically motivated by nonlinear polarisation effects. In particular I look at a polarisation of the form P(E)=\chi_1E+\chi_3\abs{E}^2E.

My main tool is the calculus of variations. Its basic idea is the fact that some solutions of differential equations are minimizers of suitable functionals. Such solutions are often calles ground states. These are of special physical interest. In addition I use the concept of truncated Fourier series to generate aproximative solutions. These (hopefully) converge to an exact solution in some sense. The advantage of this concept is not only the fact that the problem is much easier for series of finite length, but it also generates a method for numerical approximation.

First results are obtained for \Gamma(x)=\gamma\,\delta_0(x),\,\gamma\in\mathbb{R}\backslash\lbrace0\rbrace and special step-potentials g\in L^\infty(\mathbb{R}). A publication is work in progress. Another promising model is g\in L^2(\mathbb{R}),\Gamma\in L^\infty(\mathbb{R}) with g>0, \inf\Gamma>0.


Work in progress.