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].
