13. Mai 2026
Maximilian Hanisch (MLU)
Polychromatic Localized Waves in Nonlinear Maxwell Equations with Time Delayed Polarization
We study the Maxwell equations with a cubically nonlinear and nonlocal in time dependence of the polarization on the electric field. In an essentially one-dimensional wave-guide geometry, we deduce a method to obtain so called polychromatic solutions with complex frequencies for the generally non-selfadjoint problem.
These solutions are in the form of a Fourier series of travelling waves along the wave-guide, combined with a series over arbitrarily large decay rates in time. The leading frequency is given as a complex eigenvalue of the corresponding linear operator pencil and the leading order terms of the solution are given by a corresponding eigenfunction. Our method allows us to construct such polychromatic solutions by repeatedly solving linear ODEs under some assumptions on the spectrum and resolvent estimates of the linear operator.