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Junior Research Group Nonlinear Helmholtz Equations

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. Please have a look at these pages for more detailed information.


Staff in the Junior Research Group Nonlinear Helmholtz Equations
Name Tel. E-Mail
PD Dr. Rainer Mandel +49 721 608 46178 rainer.mandel@kit.edu

Current Offering of Courses
Semester Titel Typ
Winter Semester 2022/23 Bifurcation Theory Lecture
Summer Semester 2021 Sobolev spaces Lecture
Winter Semester 2020/21 Seminar (Nichtlineare Funktionalanalysis) Seminar
Summer Semester 2020 Nichtlineare Funktionalanalysis Lecture
Winter Semester 2018/19 Seminar Seminar
Summer Semester 2018 Boundary and Eigenvalue Problems Lecture
Winter Semester 2017/18 Variational and Topological Methods in Bifurcation Theory Lecture
Seminar Bifurcation Theory Seminar
Summer Semester 2017 Bifurcation Theory Lecture
Winter Semester 2013/14 Analysis III Lecture
Proseminar (Ungleichungen) Proseminar
Summer Semester 2013 Analysis II Lecture
Proseminar ( Analysis) Proseminar
Winter Semester 2012/13 Analysis I Lecture
Summer Semester 2012 Numerische Methoden (Elektrotechnik, Meteorologie, Geodäsie, Geoinformatik) Lecture
Höhere Mathematik II (Analysis) für die Fachrichtung Informatik Lecture
Winter Semester 2011/12 Höhere Mathematik I (Analysis) für die Fachrichtung Informatik Lecture
Summer Semester 2011 Seminar (Differentialgleichungen der Mathematischen Physik) Seminar
Winter Semester 2010/11 Nichtlineare Randwertprobleme/Nonlinear boundary value problems Lecture
Seminar (Randwertprobleme/boundary value problems) Seminar
Summer Semester 2010 Boundary and Eigenvalue Problems Lecture
Seminar Partial Differential Equations Seminar
Winter Semester 2009/10 Partial Differential Equations Lecture
Proseminar(Ungleichungen) Proseminar
Numerische Mathematik für die Fachrichtung Elektroingenieurwesen Lecture
Summer Semester 2009 Proseminar (Analysis: Zahlen) Proseminar

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, Commun. Math. Phys. 368 (2019), no. 2 799-842.
  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 J.-B.Casteras, R.Mandel: On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space.
  8. link R.Mandel: Dispersive estimates, blow-up and failure of Strichartz estimates for the Schrödinger equation with slowly decaying initial data, Pure And Applied Analysis, vol. 2 (2020), No. 2, 519-532.
  9. link R.Mandel, D.Scheider: An annulus multiplier and applications to the Limiting absorption principle for Helmholtz equations with a step potential, Mathematische Annalen.
  10. link D.Scheider: Breather Solutions of the Cubic Klein-Gordon Equation, Nonlinearity.
  11. link (Preprint) R.Mandel, D.Scheider: Variational methods for breather solutions of Nonlinear Wave Equations.
  12. link (Preprint) L.Cossetti, R.Mandel: A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations.
  13. link (Preprint) A. Fernandez, L. Jeanjean, R.Mandel, M. Maris: Some non-homogeneous Gagliardo-Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation.
  14. link (Preprint) R.Mandel, D. Scheider, T. Yesil: Dual variational methods for an indefinite nonlinear Helmholtz equation.
  15. link (Preprint) R. Mandel, R. Schippa: Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner-Riesz estimates with negative index for non-elliptic surfaces.
  16. link (Preprint) R. Mandel, D. Oliveira E Silva: The Tomas-Stein Inequality under the effect of symmetries.
  17. link (Preprint) C. Carvalho, A. D. Kim, L. Lewis, Z. Moitier: Quadrature by parity asymptotic expansions (QPAX) for scattering by high aspect ratio particles.



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