Existence of traveling breather solutions to cubic nonlinear Maxwell equations in waveguide geometries

TL;DR

This paper proves the existence of localized, time-periodic breather solutions in nonlinear Maxwell waveguides, demonstrating their properties through variational methods and numerical simulations, with implications for monochromatic and polychromatic wave behavior.

Contribution

It introduces a variational approach to establish infinitely many breather solutions in nonlinear Maxwell equations within waveguides, including bifurcation analysis and numerical validation.

Findings

Existence of infinitely many breather solutions in nonlinear waveguides.

Conditions under which monochromatic breathers can exist.

Numerical simulations illustrating the solutions and open questions.

Abstract

We consider the full set of Maxwell equations in a slab or cylindrical waveguide with a cubically nonlinear material law for the polarization of the electric field. The nonlinear polarization may be instantaneous or retarded, and we assume it to be confined inside the core of the waveguide. We prove existence of infinitely many spatially localized, real-valued and time-periodic solutions (breathers) propagating inside the waveguide by applying a variational minimization method to the resulting scalar quasilinear elliptic-hyperbolic equation for the profile of the breathers. The temporal period of the breathers has to be carefully chosen depending on the linear properties of the waveguide. As an example, our results apply if a two-layered linear axisymmetric waveguide is enhanced by a third core region with low refractive index where also the nonlinearity is located. In this case we can also connect our existence result with a bifurcation result. We illustrate our results with numerical simulations. Our solutions are polychromatic functions in general, but for some special models of retarded nonlinear material laws, also monochromatic solutions can exist. In this case the numerical simulations raise an interesting open question: are the breather solutions with minimal energy monochromatic or polychromatic?