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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


Current preprints of the Junior Research Group

  1. link R. Mandel, D. Oliveira E Silva: The Tomas-Stein Inequality under the effect of symmetries.
  2. link R. Mandel: On Gagliardo-Nirenberg Inequalities with vanishing symbols.
  3. link R. Mandel: A simple variational approach to Nonlinear Maxwell equations.
  4. link C. Carvalho, Z. Moitier: Scattering resonances in unbounded transmission problems with sign-changing coefficient.

Publications of the Junior Research Group since July 1st 2016

  1. link R. Mandel, Z. Moitier, B. Verfürth: Nonlinear Helmholtz equations with sign-changing diffusion coefficient. C. R. Math. Acad. Sci. Paris 360 (2022), 513–538.
  2. link R. Griesmaier, M. Knöller, R. Mandel: Inverse medium scattering for a nonlinear Helmholtz equation. J. Math. Anal. Appl. 515 (2022), no. 1, Paper No. 126356, 27 pp.
  3. link A. Fernandez, L. Jeanjean, R.Mandel, M. Maris: Non-homogeneous Gagliardo-Nirenberg inequalities in RN and application to a biharmonic non-linear Schrödinger equation. J. Differential Equations 330 (2022), 1–65.
  4. link 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. Ann. Henri Poincaré 23 (2022), no. 5, 1831–1882.
  5. link C. Carvalho, A. D. Kim, L. Lewis, Z. Moitier: Quadrature by parity asymptotic expansions (QPAX) for scattering by high aspect ratio particles. Multiscale Model. Simul. 19 (2021), no. 4, 1857–1884.
  6. link R. Mandel: Ground states for Maxwell's equations in nonlocal nonlinear media. Partial Differ. Equ. Appl. 3 (2022), no. 2, Paper No. 22, 16 pp.
  7. link R. Mandel: A uniqueness result for the Sine-Gordon breather. Partial Differ. Equ. Appl. 2 (2021), no. 2, Paper No. 26, 8 pp.
  8. link L. Cossetti, R. Mandel: A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations. J. Funct. Anal. 281 (2021), no. 11, Paper No. 109233, 41 pp.
  9. link R. Mandel, D. Scheider: Variational methods for breather solutions of nonlinear wave equations. Nonlinearity 34 (2021), no. 6, 3618–3640.
  10. link R. Mandel, D. Scheider, T. Yesil: Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity. Calc. Var. Partial Differential Equations 60 (2021), no. 4, Paper No. 133, 13 pp.
  11. link D. Scheider: Breather solutions of the cubic Klein-Gordon equation. Nonlinearity 33 (2020), no. 12, 7140–7166.
  12. link J.-B. Casteras, R. Mandel: On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space. Int. Math. Res. Not. IMRN 2021, no. 7, 4838–4863.
  13. link R. Mandel, D. Scheider: An annulus multiplier and applications to the Limiting absorption principle for Helmholtz equations with a step potential. Math. Ann. 379 (2021), no. 1-2, 865–907.
  14. 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.
  15. link R. Mandel, D. Scheider: Bifurcations of nontrivial solutions of a cubic Helmholtz system. Advances in Nonlinear Analysis, 9(1), 1026–1045.
  16. 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.
  17. 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.
  18. 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.
  19. link R. Mandel, D. Scheider: Dual variational methods for a nonlinear Helmholtz system. NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, 25:13.
  20. 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.



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. Topol. Methods Nonlinear Anal. 53 (2019), no. 2, 779–800.
  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 J.Gärtner, R.Mandel, W.Reichel: The Lugiato-Lefever equation with nonlinear damping caused by two photon absorption. J. Dynam. Differential Equations 34 (2022), no. 3, 2201–2227.

Further research activities