# Associated Project 2 • Nonlinear Helmholtz equations and systems

## Summary

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 $$$\label{eq:NLH} -\Delta u+ V(x)u - \lambda u = \Gamma(x)|u|^{p-2}u \qquad\text{in }\mathbb{R}^n \tag{NLH}$$$ 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 of \eqref{eq:NLH}.

## Results

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 [MMP17] 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 [MS18] and solutions of Helmholtz equations in the presence of higher order dispersion [BCM19]. In [Man19a] we extended the dual variational method, which is based on a Limiting Absorption Principle in $L^p(\mathbb{R}^n)$, 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.

In [Man19b] we extend our analysis of nonlinear Helmholtz systems initiated in [MS18] and prove new existence results for vector solutions via bifurcation theory. In [MS19] 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 [CM19] we investigated the analogon of \eqref{eq:NLH} in the hyperbolic space. As a byproduct, we obtained the nonvalidity of Strichartz estimates for the Schrödinger equation in $\mathbb{H}^n$ with initial data $u_0\in L^q(\mathbb{H}^n),q>2$.

## Publications

1. and . Variational methods for breather solutions of nonlinear wave equations. Nonlinearity, 34(6):3618–3640, June . [preprint] [bibtex]

2. , , and . Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity. Calc. Var. Partial Differ. Equ., 6(4):133, June . [preprint] [bibtex]

3. . A uniqueness result for the sine-Gordon breather. SN Partial Differ. Equ. Appl., 2:26, 8pp., March . [preprint] [bibtex]

4. and . An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential. Math. Ann., 1–43, October . Online first. [preprint] [bibtex]

5. . 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 . [preprint] [bibtex]

6. and . On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space. Int. Math. Res. Not. IMRN, rnz389, January . [preprint] [bibtex]

7. and . Bifurcations of nontrivial solutions of a cubic Helmholtz system. Adv. Nonlinear Anal., 9(1):1026–1045, January . [preprint] [bibtex]

8. . Uncountably many solutions for nonlinear Helmholtz and curl-curl equations. Adv. Nonlinear Stud., 19(3):569–593, August . [preprint] [bibtex]

9. , , and . On a fourth-order nonlinear Helmholtz equation. J. Lond. Math. Soc. (2), 99(3):831–852, June . [preprint] [bibtex]

10. . The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations. Comm. Math. Phys., 368(2):799–842, June . [preprint] [bibtex]

11. and . Dual variational methods for a nonlinear Helmholtz system. NoDEA Nonlinear Differential Equations Appl., 25(2):Art. 13, 26, March . [preprint] [bibtex]

12. , , and . Oscillating solutions for nonlinear Helmholtz equations. Z. Angew. Math. Phys., 68(6):Art. 121, 19, October . [preprint] [bibtex]

## Preprints

1. and . Fourier transform of surface-carried measures of two-dimensional generic surfaces and applications. CRC 1173 Preprint 2021/37, Karlsruhe Institute of Technology, August . [bibtex]

2. , , and . Nonlinear Helmholtz equations with sign-changing diffusion coefficient. CRC 1173 Preprint 2021/35, Karlsruhe Institute of Technology, August . [bibtex]

3. and . The Tomas–Stein inequality under the effect of symmetries. CRC 1173 Preprint 2021/27, Karlsruhe Institute of Technology, June . [bibtex]

4. , , , and . Quadrature by parity asymptotic expansions (QPAX) for scattering by high aspect ratio particles. CRC 1173 Preprint 2021/22, Karlsruhe Institute of Technology, May . [files] [bibtex]

5. and . Time-harmonic solutions for Maxwell's equations in anisotropic media and Bochner–Riesz estimates with negative index for non-elliptic surfaces. CRC 1173 Preprint 2021/13, Karlsruhe Institute of Technology, April . [files] [bibtex]

6. . Resolvent estimates for the time-harmonic Maxwell's equations in the partially anisotropic case. CRC 1173 Preprint 2021/11, Karlsruhe Institute of Technology, March . [bibtex]

7. , , , and . Some non-homogeneous Gagliardo–Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation. CRC 1173 Preprint 2020/27, Karlsruhe Institute of Technology, October . [bibtex]

8. and . A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equation. CRC 1173 Preprint 2020/26, Karlsruhe Institute of Technology, September . [bibtex]

9. . Breather solutions of the cubic Klein–Gordon equation. CRC 1173 Preprint 2020/1, Karlsruhe Institute of Technology, January . To be published in Nonlinearity. [bibtex]

## Theses

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

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

Former staff
Name Title Function
Dr. Doctoral and Postdoctoral researcher