On some properties of elliptic and parabolic functional differential operators arising in nonlinear optics

Quasilinear parabolic functional differential equations containing multiple transformations of spatial variables are considered with the Neumann boundary-value conditions. Sufficient conditions of the Andronov-Hopf bifurcation of periodic solutions are obtained along with expansions of the solutions in powers of a small parameter. Spectral properties of the linearized elliptic operator of this problem are investigated. Necessary and sufficient conditions of normality are obtained for such operators. Examples illustrating their properties are given. © 2008 Springer Science+Business Media, Inc.