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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 agenda
The propagation of waves in nonlinear media is in many physical applications described by Maxwell's Equations. In the special case of standing polarized waves in infinitely extended Kerr-type media these equations reduce to Helmholtz equations of the form

$  -\Delta u+ V(x)u - \lambda  u = \Gamma(x)|u|^{p-2}u \qquad\text{in }\mathbb{R}^n, \qquad u(x)\to 0\quad (|x|\to\infty)$

that describe the corresponding spatial profile. In our project we are interested in the existence and qualitative properties of localized solutions of this equation under physically relevant assumptions on the material parameters V,\Gamma\in L^\infty(\mathbb{R}^n) and p>2 as well as on the parameter \lambda\in\mathbb{R} that depends on the frequency of the wave. While the case \lambda\notin \sigma(-\Delta+V(x)) is nowadays widely understood, only little is known for \lambda\in\sigma_{ess}(-\Delta+V(x)). Classical variational and degree theoretical methods available in the former case do not apply here due to oscillatory behaviour and slow decay rates of solutions. Our main interest is to combine methods from the calculus of variations, spectral theory and harmonic analysis to overcome the prevailing difficulties and to establish new methods allowing to study localized solutions.

Results (see below for the references (1)-(7))
During the first funding period we first thoroughly investigated the case of radially symmetric and monotonic potentials V,\Gamma where ODE techniques could be applied even for much more general nonlinearities. In (1) it became apparent that such equations possess uncountably many small solutions that oscillate and decay like |x|^{(1-n)/2} as |x|\to\infty. Next, we applied dual variational methods (following ideas of Evequoz and Weth) in order to find vector solutions of two weakly coupled nonlinear Helmholtz equations (2) and solutions of Helmholtz equations in the presence of higher order dispersion (3). In (4) we extended the dual variational method, which is based on a Limiting Absorption Principle in L^p(\R^n), to the case of periodic materical laws that serve as a model for photonic crystals. Our main tools were Floquet-Bloch theory and L^p-estimates for oscillatory integrals. One fundamental insight is that crystals with sufficiently regular and positively curved Fermi surfaces at level \lambda\in\sigma_{ess}(-\Delta+V(x)) admit standing waves with frequency $\lambda$ having the same spatial decay rates as in vacuum.

Fermi surfaces for -\Delta + 10\sin(2\pi x)^2\cos(2\pi y) at level \tau

In the current preprint (5) we extend our analysis of nonlinear Helmholtz systems initiated in (2) and prove new existence results for vector solutions via bifurcation theory. In (6) a fixed point approach for nonlinear Helmholtz equations (introduced by S. Gutierrez) was refined and eventually extended to curl-curl equations that are also studied in Project A6. In (7) we investigated the corresponding equation in the hyperbolic space. As a byproduct, we obtained the nonvalidity of Strichartz estimates for the Schrödinger equation with initial data u_0\in L^q(\mathbb{H}^n),q>2.

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 2020 Lecture
Summer Semester 2018 Lecture
Winter Semester 2017/18 Lecture
Summer Semester 2017 Lecture

Guests in our group

Publications of the Junior Research Group since July 1st 2016

  1. link 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. link R.Mandel, D.Scheider: Dual variational methods for a nonlinear Helmholtz system. NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, 25:13.
  3. link R.Mandel: The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations. Erscheint in CIMP.
  4. link D.Bonheure, J.-B.Casteras, R.Mandel: On a fourth order nonlinear Helmholtz equation, Journal of the London Mathematical Society 99 (2019), no. 3, 831-852.
  5. link R.Mandel, D.Scheider: Bifurcations of nontrivial solutions of a cubic Helmholtz system, Advances in Nonlinear Analysis, 9(1), 1026–1045.
  6. link R.Mandel: Uncountably many solutions for nonlinear Helmholtz and curl-curl equations with general nonlinearities, Adv. Nonlinear Stud. 19 (2019), no. 3, 569-593.
  7. link (Preprint) J.-B.Casteras, R.Mandel: On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space.
  8. link (Preprint) R.Mandel: Dispersive estimates, blow-up and failure of Strichartz estimates for the Schrödinger equation with slowly decaying initial data
  9. link (Preprint) R.Mandel, D.Scheider: A Limiting absorption principle for linear and nonlinear Helmholtz equations with a step potential

Publications in Projekt B3 of our CRC1173 since 01.07.2016

  1. link 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.
  2. link R.Mandel: Global secondary bifurcation, symmetry breaking and period-doubling.
  3. link J. Gärtner, P. Trocha, R. Mandel, C. Koos, T. Jahnke, W. Reichel: Bandwidth and conversion efficiency analysis of dissipative Kerr soliton frequency combs based on bifurcation theory, Phys. Rev. A 100, 033819
  4. link (Preprint) J.Gärtner, R.Mandel, W.Reichel: The Lugiato-Lefever equation with nonlinear damping caused by two photon absorption.

Further research activities