Principal investigators
Prof. Dr. Michael Plum  (7/2015  )  
Prof. Dr. Wolfgang Reichel  (7/2015  ) 
Project summary
The aim of this project is to investigate the existence and the nature of timeperiodic solutions of the Maxwell 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, Xrays 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.
Waves, which are fully localized in all directions of space, have finite energy and would be amazing objects. Even more astonishing would be a standing fully localized wave: it may be interpreted as "stopped light". We do not know yet if such waves exist. But if they do, it is certainly a consequence of the nonlinear material laws. Continue reading. Collapse content.
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{alignat*}{2} \nabla\cdot\mathbf D &= 0, \qquad\qquad \nabla \times \mathbf E &=\partial_t \mathbf B, \qquad\qquad\ \nabla\cdot\mathbf B &= 0, \qquad\qquad \nabla \times \mathbf H&& = \partial_t \mathbf D, \end{alignat*} 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 \[ \mathbf B =\mu \mathbf H \] with a constant permeability \(\mu\). Nonlinear optical materials have the property that the displacement field \[ \mathbf D= \mathbf E +\mathbf P(\mathbf E) \] depends nonlinearly on the electric field \(\mathbf E\) through the polarization field \(\mathbf P(\mathbf E)\). This leads to a quasilinear curlcurl equation for the electric field \(\mathbf E\): \[ \nabla\times\nabla \times \mathbf E + \mu\partial_t^2 \left(\mathbf E + \mathbf P(\mathbf E)\right)=0. \] 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 Kerrtype term, and possibly quintic terms. All (or just some) of these terms may be local or nonlocal in space or time. Nonlocalintime material laws model a retardation in the reaction of atoms in their interaction with photons. Continue reading. Collapse content.
Semilinear Maxwell equations
A nonlinear constitutive relation for the polarization field transforms the Maxwell equations to a quasilinear curlcurl equation. The analysis is much more involved than in the linear case and requires novel approaches. As a first step towards understanding the nature of timeperiodic solutions in the quasilinear case, we consider a semilinear approximation given by \[ \nabla\times \nabla\times \mathbf E + V(\mathbf x)\partial_t^2\mathbf E +\Gamma(\mathbf x)\mathbf E^2\mathbf E=0. \] Here, \(V\) and \(\Gamma\) represent material functions and are given. The above equation is a reasonable approximation when considering electric fields which are slowly varying in time. Continue reading. Collapse content.
We used different approaches to study the existence of standing and traveling waves for certain classes of potentials \(V\) and \(\Gamma\):

ODE methods: Restricting to radially symmetric electric fields, the question of existence of timeperiodic solutions of the seminlinear curlcurl equation reduces to studying and solving a second order ordinary differential equation ([PR16]).

Variational methods: The existence of timeperiodic, localized solutions for various classes of periodic potentials including Delta, step, and continuous material functions \(V\) was established ([BTPR16], [HR17], [HR19], and [Hir17]).

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 steppotentials \(V\). Continue reading. Collapse content.
Quasilinear Maxwell equations
One of our main goals is to gain a better understanding of the quasilinear curlcurl equation \[ \nabla\times \nabla\times \mathbf E +\partial_t^2 \left(V(\mathbf x)\mathbf E +\Gamma(\mathbf x)\mathbf E^2\mathbf E\right)=0. \] This equation can be derived from the Maxwell equations under the assumption of a cubic polarization field. In contrast to the semilinear case, the nonlinearity now appears in the highest order timederivatives. This causes severe mathematical difficulties not only with regard to wellposedness 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 our project we try to unveil some insight into the nature of solutions by investigating standing and traveling waves. This is done by variational as well as bifurcation methods. Continue reading. Collapse content.
Profiting from our studies of the semilinear case, we investigate, which assumptions on the potentials \(V\) and \(\Gamma\) guarantee the existence of special solutions in the quasilinear case. This is ongoing work.
So far we have been able to prove the existence of traveling waves unde the assumption that \(V\) is a Deltapotential as a result of the bifurcation theory approach ([Idz17]). Continue reading. Collapse content.
Future goals and ongoing work

Existence of traveling and standing waves for the quasilinear Maxwell equations: As mentioned before, part of our ongoing work relies on the existence of traveling and standing waves for the quasilinear Maxwell equations by means of both, variational methods and bifurcation theory. Having already achieved some results, we now consider different types of potentials, and also try to see numerical simulations of standing localized waves. We expect interesting interactions with project A4.

Stability of traveling and standing waves: Roughly speaking, stability of a specific solution means, that starting from an initial datum close by, the evolution of this initial datum over time stays forever close. A more refined version of stability (orbital stability) means that the evolution of the initial datum over times stays forever close to the orbit (i.e. the specific solution together with all its translations and/or multiples by complex units).
The first challenge in studying stability of our traveling/standing waves, relies in need for a wellposedness theory for the quasilinear equation. This is an issue addressed by project A5. 
Retarded material laws:Including retardation effects means assuming that the material needs some time to react to an incoming wave, just like humans need time to process incoming information (listening) before they can show a reaction (speaking). Mathematically, retardation effects lead to nonlocal in time equations, which certainly adds a new level of interesting challenges to our studies of timeperiodic solutions and their stability.