Associated Project 2 • Nonlinear Helmholtz equations and systems (7/2016 - 6/2023)

Summary

The propagation of waves in nonlinear media is 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 \[\begin{equation}\label{eq:NLH} -\Delta u+ V(x)u - \lambda u = \Gamma(x)|u|^{p-2}u \qquad\text{in }\mathbb{R}^n \tag{NLH} \end{equation}\] 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 frequency 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 \(\lambda\in\sigma(-\Delta+V(x))\). Classical variational and degree theoretical methods 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 of \eqref{eq:NLH} and related equations describing the propagation of waves.

Results

We started to analyze the nonlinear Helmholtz equation in a radially symmetric setting [MMP17] and showed that, in contrast to standard elliptic problems, these equations admit a whole family of localized solutions. During the following years we generalized our analysis in various directions, for instance Helmholtz systems [MS20, MS18], sign-changing coefficients [MSY21, MMV22] or periodic potentials $V,\Gamma$ [Man19]. In the latter context, which is relevant for wave propagation in photonic crystals, it became apparent that frequencies $\lambda\in\sigma(-\Delta+V(x))$ with positively curved Fermi surfaces allow for an existence theory that is comparable to the vacuum case up to replacing Fourier analysis by (new) Floquet-Bloch analysis.

Relevant techniques for Nonlinear Helmholtz equations are equally useful in other contexts. In the case of Maxwell's equations we investigated the time-harmonic isotropic [CM21] and anisotropic case [MS22] from the viewpoint of a Limiting Absorption Principle. Existence proofs for nonlinear Maxwell's equations by variational methods were established in [Man22a, Man22c]. We are also interested in breather solutions of nonlinear wave equations in $\mathbb{R}^N$. By definition, such solutions are real-valued, periodic in time and localized in space. Existence and uniqueness results were proved in [Sch20, MS21, Man21].

The related functional analytical foundations have ultimately gained importance in our research: we proved new Gagliardo–Nirenberg inequalities [Man22b, FJMM22] and Fourier restriction estimates [MO21].

Publications

1. R. Mandel and D. Oliveira e Silva. The Tomas–Stein inequality under the effect of symmetries. J. Anal. Math., 150(2):547–582, September 2023. URL https://doi.org/10.1007/s11854-023-0282-3. [preprint] [bibtex]

2. C. Carvalho and Z. Moitier. Scattering resonances in unbounded transmission problems with sign-changing coefficient. IMA J. Appl. Math., 88(2):215–257, April 2023. URL https://doi.org/10.1093/imamat/hxad005. [preprint] [bibtex]

3. R. Griesmaier, M. Knöller, and R. Mandel. Inverse medium scattering for a nonlinear Helmholtz equation. J. Math. Anal. Appl., 515(1):126356, November 2022. URL https://doi.org/10.1016/j.jmaa.2022.126356. [preprint] [files] [bibtex]

4. J.-C. Cuenin and R. Schippa. Fourier transform of surface-carried measures of two-dimensional generic surfaces and applications. Commun. Pure Appl. Anal., 21(9):2873–2889, September 2022. URL https://doi.org/10.3934/cpaa.2022079. [preprint] [bibtex]

5. A. Fernández, L. Jeanjean, R. Mandel, and M. Mariş. Non-homogeneous Gagliardo–Nirenberg inequalities $\mathbb{R}^N$ and application to a biharmonic non-linear Schrödinger equation. J. Differential Equations, 330:1–65, September 2022. URL https://doi.org/10.1016/j.jde.2022.04.037. [preprint] [bibtex]

6. R. Mandel, Z. Moitier, and B. Verfürth. Nonlinear Helmholtz equations with sign-changing diffusion coefficient. C. R. Math., 360:513–538, May 2022. URL https://doi.org/10.5802/crmath.322. [preprint] [bibtex]

7. R. Mandel and 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(5):1831–1882, May 2022. URL https://doi.org/10.1007/s00023-021-01144-y. [preprint] [files] [bibtex]

8. R. Mandel. Ground states for Maxwell's equations in nonlocal nonlinear media. Partial Differ. Equ. Appl., 3(2):22, 16pp., April 2022. URL https://doi.org/10.1007/s42985-022-00159-2. [preprint] [files] [bibtex]

9. C. Carvalho, A. D. Kim, L. Lewis, and Z. Moitier. Quadrature by parity asymptotic expansions (QPAX) for scattering by high aspect ratio particles. SIAM J. Numer. Anal., 19(4):1857–1884, December 2021. URL https://doi.org/10.1137/21M1416801. [preprint] [files] [bibtex]

10. L. Cossetti and R. Mandel. A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equation. J. Funct. Anal., 281(11):109233, December 2021. URL https://doi.org/10.1016/j.jfa.2021.109233. [preprint] [bibtex]

11. R. Mandel and D. Scheider. Variational methods for breather solutions of nonlinear wave equations. Nonlinearity, 34(6):3618–3640, June 2021. URL https://doi.org/10.1088/1361-6544/abed38. [preprint] [bibtex]

12. R. Mandel, D. Scheider, and T. Yeşil. Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity. Calc. Var. Partial Differ. Equ., 60:133, 13pp., June 2021. URL https://doi.org/10.1007/s00526-021-02006-5. [preprint] [bibtex]

13. R. Mandel. A uniqueness result for the sine-Gordon breather. SN Partial Differ. Equ. Appl., 2:26, 8pp., March 2021. URL https://doi.org/10.1007/s42985-021-00084-w. [preprint] [bibtex]

14. R. Mandel and D. Scheider. An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential. Math. Ann., 379(1):865–907, February 2021. URL https://doi.org/10.1007/s00208-020-02093-3. [preprint] [bibtex]

15. D. Scheider. Breather solutions of the cubic Klein–Gordon equation. Nonlinearity, 33(12):7140–7166, December 2020. URL https://doi.org/10.1088/1361-6544/abb78b. [preprint] [bibtex]

16. R. Mandel. Dispersive estimates, blow-up and failure of Strichartz estimates for the Schrödinger equation with slowly decaying initial data. Pure Appl. Anal., 2(2):519–532, May 2020. URL https://doi.org/10.2140/paa.2020.2.519. [preprint] [bibtex]

17. J.-B. Casteras and R. Mandel. On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space. Int. Math. Res. Not. IMRN, rnz389, January 2020. URL https://doi.org/10.1093/imrn/rnz389. [preprint] [bibtex]

18. R. Mandel and D. Scheider. Bifurcations of nontrivial solutions of a cubic Helmholtz system. Adv. Nonlinear Anal., 9(1):1026–1045, January 2020. URL https://doi.org/10.1515/anona-2020-0040. [preprint] [bibtex]

19. R. Mandel. Uncountably many solutions for nonlinear Helmholtz and curl-curl equations. Adv. Nonlinear Stud., 19(3):569–593, August 2019. URL https://doi.org/10.1515/ans-2019-2050. [preprint] [bibtex]

20. D. Bonheure, J.-B. Casteras, and R. Mandel. On a fourth-order nonlinear Helmholtz equation. J. Lond. Math. Soc. (2), 99(3):831–852, June 2019. URL https://doi.org/10.1112/jlms.12196. [preprint] [bibtex]

21. R. Mandel. The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations. Comm. Math. Phys., 368(2):799–842, June 2019. URL https://doi.org/10.1007/s00220-019-03363-1. [preprint] [bibtex]

22. R. Mandel and D. Scheider. Dual variational methods for a nonlinear Helmholtz system. Nonlinear Differ. Equ. Appl., 25(2):article no. 13, March 2018. URL https://doi.org/10.1007/s00030-018-0504-z. [preprint] [bibtex]

23. R. Mandel, E. Montefusco, and B. Pellacci. Oscillating solutions for nonlinear Helmholtz equations. Z. Angew. Math. Phys., 68(6):article no. 121, October 2017. URL https://doi.org/10.1007/s00033-017-0859-8. [preprint] [bibtex]

Preprints

24. R. Mandel. A simple variational approach to nonlinear Maxwell equations. CRC 1173 Preprint 2022/82, Karlsruhe Institute of Technology, December 2022. [bibtex]

25. R. Mandel. On Gagliardo–Nirenberg inequalities with vanishing symbols. CRC 1173 Preprint 2022/10, Karlsruhe Institute of Technology, February 2022. [bibtex]

Theses

26. R. Mandel. Nonlinear Helmholtz equations and systems. Habilitation thesis, Karlsruhe Institute of Technology (KIT), December 2019. [bibtex]

27. D. Scheider. On a nonlinear Helmholtz system. PhD thesis, Karlsruhe Institute of Technology (KIT), October 2019. [bibtex]