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Junior Research Group "Nonlinear Helmholtz Equations"

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 3.029

Address
Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe
Germany

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

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

Junior Research Group "Nonlinear Helmholtz Equations"

Welcome on the site of the junior research group "Nonlinear Helmholtz equations". If you are interested in our research topics ... that is what we are interested in:

The aim of our group is to better understand nonlinear Helmholtz equations and systems which are frequently used to describe the propagation behaviour of electromagnetic waves in nonlinear optical media. We intend to analytically investigate both locally and (spatially) nonlocally interacting materials. One model equation we are interested in is given by

$  - \Delta u + V(x)u - \lambda u = \mathcal{G}(x,u) \qquad\text{in }\mathbb{R}^n, \qquad u(x)\to 0\quad (|x|\to\infty)$

where  V(x),\mathcal{G}(x,\cdot) incorporate the material properties of the waveguide. In the nonlocal case  \mathcal{G}(x,\cdot) may, as an example, involve integrals over powers of  u. Contrary to numerous contributions about strongly localized solutions of Nonlinear Schrödinger equations we will assume that  \lambda lies in the spectrum of  -\Delta+V(x) which typically is purely continuous. In this case a totally different solution theory is expected and different difficulties have to be overcome; above all wave-like oscillations and low decay rates of the solutions have to be dealt with. Our results aim at a better understanding of these phenomena.



Staff in the Junior Research Group Nonlinear Helmholtz Equations
Name Tel. E-Mail
+49 721 608 46178 rainer.mandel@kit.edu
0049 721 608 42046 dominic.scheider@kit.edu

Current Offering of Courses
Semester Titel Typ
Winter Semester 2017/18 Lecture
Seminar
Summer Semester 2017 Lecture


Publications of the this workgroup since 01.01.2017

  • (Projekt B3) R.Mandel, W.Reichel: A priori bounds and global bifurcations results for frequency combs modeled by the Lugiato-Lefever equation. SIAM J. Appl. Math. 77 (2017), no. 1, 315–345.
  • (Projekt AP2) R.Mandel, E.Montefusco, B.Pellacci: Oscillating solutions for nonlinear Helmholtz equations. (erscheint in ZAMP)
  • (Projekt AP2) D. Scheider, R.Mandel: Dual variational methods for a nonlinear Helmholtz system (preprint)
  • (Projekt AP2) R.Mandel: The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations (preprint)

Please consult the following page for a description of these projects.
The preprints can be found here.