# 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 [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 $$L^p(\mathbb{R}^n)$$, 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 the current preprint [6] we extend our analysis of nonlinear Helmholtz systems initiated in [2] and prove new existence results for vector solutions via bifurcation theory. In [5] 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 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. Oscillating solutions for nonlinear Helmholtz equations,
R. Mandel, E. Montefusco, B. Pellacci,
Z. Angew. Math. Phys., 68 (2017), no. 6, Art. 121, 19 pp.
2. Dual variational methods for a nonlinear Helmholtz system
R. Mandel, D. Scheider,
NoDEA Nonlinear Differential Equations Appl., 25 (2018), no. 2, 25:13.
3. On a fourth order nonlinear Helmholtz equation,
D. Bonheure, J.-B. Casteras, R. Mandel,
J. Lond. Math. Soc., 99 (2019), no. 3, 831–852.
4. The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations,
R. Mandel,
Commun. Math. Phys., 368 (2019), no. 2, 799–842.
5. Uncountably many solutions for nonlinear Helmholtz and curl-curl equations with general nonlinearities,
R. Mandel,
Adv. Nonlinear Stud., 19 (2019), no. 3, 569–593.

## Preprints

Project-specific staff
Name Phone E-Mail
+49 721 608-46178 rainer mandelWfq6∂kit edu
+49 721 608-42046 dominic scheiderJzt5∂kit edu