Travelling breather solutions in waveguides for cubic nonlinear Maxwell equations with retarded material laws
Sebastian Ohrem, Wolfgang Reichel
TL;DR
The paper studies travelling breather solutions to nonlinear Maxwell equations with time-retarded polarization in slab and cylindrical waveguides, focusing on TE modes that are localized orthogonally to the propagation direction. It builds a time-periodic variational framework by decomposing the polarization into linear and cubic retarded parts with kernels G and N, and uses a small travelling speed to render the E-equation essentially elliptic, enabling a mountain-pass critical-point construction. Under broad assumptions on the kernels (even in time, with decay of Fourier coefficients) and sign/structure conditions, the authors prove the existence of nontrivial breathers and, when the set of regular frequencies is infinite, infinitely many such solutions, along with regularity results for the fields. Concrete kernel examples illustrate the method, and the results extend rigorous understanding of polychromatic breathers in Maxwell systems with time-retarded media, providing a versatile approach for waveguide photonics with nonlocal temporal responses.
Abstract
For Maxwell's equations with nonlinear polarization we prove the existence of time-periodic breather solutions travelling along slab or cylindrical waveguides. The solutions are TE-modes which are localized in space directions orthogonal to the direction of propagation. We assume a magnetically inactive and electrically nonlinear material law with a linear $χ^{(1)}$- and a cubic $χ^{(3)}$-contribution to the polarization. The $χ^{(1)}$-contribution may be retarded in time or instantaneous whereas the $χ^{(3)}$-contribution is always assumed to be retarded in time. We consider two different cubic nonlinearities which provide a variational structure under suitable assumptions on the retardation kernels. By choosing a sufficiently small propagation speed along the waveguide the second order formulation of the Maxwell system becomes essentially elliptic for the $\mathbf{E}$-field so that solutions can be constructed by the mountain pass theorem. The compactness issues arising in the variational method are overcome by either the cylindrical geometry itself or by extra assumptions on the linear and nonlinear parts of the polarization in case of the slab geometry. Our approach to breather solutions in the presence of time-retardation is systematic in the sense that we look for general conditions on the Fourier-coefficients in time of the retardation kernels. Our main existence result is complemented by concrete examples of coefficient functions and retardation kernels.