Principal investigators
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Prof. Dr. Michael Plum |
(7/2015 - 6/2023) |
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Prof. Dr. Wolfgang Reichel |
(7/2015 - ) |
Project summary
The aim of this project is to investigate the existence and the nature of localized solutions for Maxwell and wave-type equations.
Motivation
About 150 years ago J. C. Maxwell discovered that electric and magnetic radiation are in fact two sides of the same coin -- called electromagnetism. Depending on their wavelength, electromagnetic waves comprise for instance radio waves, microwaves, infrared, visible light, ultraviolet, X-rays and gamma rays. Our investigation focuses on the existence and (ultimately) the stability of localized standing and traveling waves of the Maxwell equations. Our investigation is based on nonlinear material laws. With some simplification this could be described as follows: the refractive index of an object made from such a material depends in a nonlinear fashion on the amplitude of a wave hitting the object. Localization means that most of the electromagnetic wave is confined to a small region in space. Waves can be localized in all or only some directions of space. A stable traveling wave, which is localized in the directions complementary to the propagation direction could be used as an optical signal. Of particular interest are waves that are peridioc in time. Waves, which are fully localized in all directions of space, have finite energy and would be amazing objects. Even more astonishing would be standing fully localized waves: they may be interpreted as "stopped light", but we do not know yet if they exist. If so, it is certainly a consequence of the nonlinear material laws.
Denoting by $\mathbf E$ the electric and by $\mathbf B$ the magnetic field, which depend on time and space, Maxwell postulated the following law of electromagnetism (here reduced to the special case of no externally applied currents and no external charges): \begin{align*} \nabla \cdot \mathbf D = 0, \qquad \nabla \times \mathbf E = - \partial_t \mathbf B, \qquad \nabla \cdot \mathbf B = 0, \qquad \nabla \times \mathbf H = \partial_t \mathbf D, \end{align*} where $\mathbf D$ and $\mathbf H$ represent the electric displacement and the magnetic intensity field, respectively. For many materials, a linear relation between the magnetic field $\mathbf B$ and its intensity field $\mathbf H$ can be assumed, that is \begin{align*} \mathbf B = \mu \mathbf H \end{align*} with a constant permeability $\mu$. Nonlinear optical materials have the property that the displacement field \begin{align*} \mathbf D = \mathbf E + \mathbf P(\mathbf E) \end{align*} depends nonlinearly on the electric field $\mathbf E$ through the polarization field $\mathbf P(\mathbf E)$. This leads to a quasilinear curl-curl equation for the electric field $\mathbf E$: \begin{align*} \nabla \times \nabla \times \mathbf E + \mu \partial_t^2 (\mathbf E + \mathbf P(\mathbf E)) = 0. \end{align*} In contrast to the theory of linear Maxwell equations, nonlinear relations make the existence theory for solutions mathematically much more challenging. Typically one considers $\mathbf P(\mathbf E)$ as a sum of linear and cubic Kerr-type terms, and possibly quintic terms. All (or just some) of these terms may be local (instantaneous) or nonlocal in time. Nonlocal-in-time material laws model a retardation in the reaction of atoms in their interaction with photons.
Semilinear wave-type equations
As a first step towards understanding the nature of time-periodic solutions in the quasilinear case, we consider a semilinear approximation given by \begin{align}\label{eq:semwe} \nabla\times \nabla\times \mathbf E + V(\mathbf x)\partial_t^2\mathbf E +q(\mathbf x) \mathbf E +\Gamma(\mathbf x)|\mathbf E|^{p-1}\mathbf E=0 \end{align} for some $p >1$ and given functions $q, V$ and $\Gamma$. Continue reading. Collapse content.
We used different approaches to study the existence of different types of solutions for certain classes of potentials $V$ and $\Gamma$ :
- Variational methods: During the first funding period the existence of time-periodic, localized solutions for various classes of periodic potentials including Delta-, step-, and continuous material functions $V$ was established ([HR17, BRPR16, Hir17]). In the second funding period the methods were further developed and extended to cover not only time-harmonic (monochromatic) but in particular real-valued (polychromatic) solutions. A variety of new examples for $V$ were constructed ([HR19, MRS22, Koh21]). This includes new wave guide scenarios leading to sign changing $V$ ([Koh21]). Moreover, improvements concerning regularity of solutions were made due to a refined Sobolev-type embedding ([MRS22]).
- ODE methods: By restricting to radially symmetric fields, the question of existence of time-periodic solutions of the semilinear curl-curl equation reduces to studying and solving a second order ordinary differential equation ([PR16]). In the second funding period this work was extended under a more generalized symmetry assumption on the coefficients and the existence of gradient-type breather solutions and also space-time localized solutions (rogue waves) was proven ([PR22]).
- Bifurcation theory: In the doctoral thesis [Idz17] the existence of traveling waves, which are periodic in the direction of propagation and localized in one orthogonal spatial direction, was proven for Delta- as well as for step-potentials $V$ .
- Approach via Helmholtz equations: A new approach during the second funding period addressing breather solutions in higher dimensions originates from project AP2. A time-periodic solution ansatz leads to an infinite system of nonlinear coupled Helmholtz equations which can be treated with bifurcation methods (see project AP2) and variational methods (see [Man21]).
- Related approaches The study of time-harmonic solutions $\mathbf E(\mathbf x,t)= \mathbf U(\mathbf x)\mathrm{e}^{\mathrm{i}k \omega t}$ leads to a nonlinear elliptic curl-curl problem for the profile $\mathbf{U}$. Besides many other papers in the literature, this problem was studied in [Man22b, Man22b] where the existence of ground states was shown and also nonlocal nonlinearities were treated.
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Quasilinear Maxwell equations
One of our main goals is to understand the quasilinear curl-curl equation \begin{align*} \nabla \times \nabla \times \mathbf E + \partial_t^2 \left( V(\mathbf x) \mathbf E + \Gamma(\mathbf x) \left|\mathbf E\right|^2 \mathbf E \right) = 0. \end{align*} This equation can be derived from the Maxwell equations under the assumption of a cubic and instantaneous polarization field, which model Kerr-type optical materials. In contrast to the semilinear case, the nonlinearity now appears in the highest order time-derivatives. This causes severe mathematical difficulties not only with regard to well-posedness and regularity of solutions, which is subject to project A5, but also from a numerical point of view, which is investigated in project A4. In this project we try to unveil some insight into the nature of solutions by investigating standing and traveling waves. Continue reading. Collapse content.
Using different approaches, we so far have been able to prove existence of traveling waves for various classes of material coefficients $V$ and $\Gamma$:
| nonlinear potential $\Gamma$ |
linear potential $V$ |
nonlinearity |
references |
| $\Gamma(\mathbf x)=\pm\delta_0(x_1)$ |
$V(\mathbf x)=V_0(x_1)+\alpha\delta_0(x_1)$ |
$\partial_t^2(\left|\mathbf E \right|^2 \mathbf E)$ |
[BIR22, KR22] |
| $\Gamma\in L^\infty$ |
$V(\mathbf x)=V_0(x_1)+\alpha\delta_0(x_1)$ |
$\partial_t^2(\left| \mathbf E \right|^2 \mathbf E)$ |
[BIR22] |
| $\inf\Gamma>0$ |
$\inf V>0$ |
$\partial_t^2(\langle \left| \mathbf E \right|^2 \rangle \mathbf E)$ |
[MR23] |
- For examples of bounded $V$ and $\Gamma$ a delta potential on an infinitesimally thin slab waveguide, using variational methods traveling polarized waves were shown to exist ([KR22]) and well-posedness of the associated initial value problem was discussed ([ORS22]).
- For various classes of potentials $V$, $\Gamma$ including delta potentials and bounded potentials (see table above), existence of traveling polarized waves was shown using a bifurcation approach ([BIR22]).
- A retarded material law of the form \begin{align*} \mathbf P(\mathbf E) = V(\mathbf x) \mathbf E + \langle |\mathbf E|^2\rangle \mathbf E = V(\mathbf x) \mathbf E + \frac{1}{T}\int_{t - T}^{t} \left|\mathbf E\right|^2 \,\mathrm d \tau \,\mathbf E \end{align*} results in an elliptic problem for the wave profile of the traveling wave, which is amenable to variational methods ([MR23]).
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