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Junior Research Group "Nonlinear Helmholtz 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

Junior Research Group "Nonlinear Helmholtz Equations"

This is the webpage of the Junior Research Group "Nonlinear Helmholtz equations", existent since July 1st 2016 as an associated project (AP2) with the Collaborative Research Center 1173.

Research aims:

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 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.

Recent results:

  • For certain radially symmetric potentials such as V(x)=-1 and a large class of nonlinearities we could apply ODE methods in order to classify all radially symmetric solutions and describe their oscillations along with their asymptotic behaviour as |x|\to\infty . See publication 1. below or here.
  • In certain nonlinear optical Materials one encounters the effect of birefringence, which is modeled via vectorial (i.e. fully nontrivial) solutions of weakly coupled semilinear Schrödinger or (in the case of resonant frequencies) Helmholtz systems. We found first existence results for such a solutions. See publication 2. below or here.
  • Periodic structures like photonic crystals come with periodic material laws so that the Nonlinear Helmholtz Equation has to be studied for periodic potentials V. Using Floquet-Bloch-theory and harmonic analysis we managed to prove a limiting absorption principle allowing to prove existence of solutions in nonlinear periodic Media verwenden. The fundamental observation is that crystals with sufficiently regular Fermi surfaces with nonvanishing Gaussian curvature admit standing waves with the same decay rates as in the vacuum. See publication 3. below or here.
  • We studied the effect of fourth order dispersion and proved the existence of nontrivial solutions to a kind of Helmholtz equation of fourth order. See publication 4. below or here.

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
Summer Semester 2018 Lecture
Winter Semester 2017/18 Lecture
Summer Semester 2017 Lecture

Publications of the Junior Research Group since July 1st 2016

  1. R.Mandel, E.Montefusco, B.Pellacci: Oscillating solutions for nonlinear Helmholtz equations. Z. Angew. Math. Phys. 68 (2017), no. 6, Art. 121, 19 pp.
  2. R.Mandel, D.Scheider: Dual variational methods for a nonlinear Helmholtz system. NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, 25:13.
  3. R.Mandel: The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations (preprint)
  4. D.Bonheure, J.-B.Casteras, R.Mandel: On a fourth order nonlinear Helmholtz equation, accepted for publication in JLMS.
  5. R.Mandel, D.Scheider: Bifurcations of nontrivial solutions of a cubic Helmholtz system (preprint)

Publications in Project B3 of CRC1173 since July 1st 2016

Further research activities