Second order reductions of \(N\)-wave interactions related to low-rank simple Lie algebras. (English) Zbl 0999.35037

Gerdjikov, Vladimir; Grahovski, Georgi; Kostov, Nikolay

Mladenov, I. M. (ed.) et al., Proceedings of the international conference on geometry, integrability and quantization, Varna, Bulgaria, September 1-10, 1999. Sofia: Coral Press Scientific Publishing. 55-77 (2000).

Summary: We study the propagation of monochromatic fields in a layered medium. The mathematical model is derived from Maxwell’s equations. It consists of a nonlinear eigenvalue problem on the real axis with coefficients depending on the various layers.
A systematic analysis is carried out to uncover the various mechanisms leading to the bifurcation of asymmetric solutions even in a completely symmetric setting. We derive two particular simple conditions for the occurence of asymmetric bifurcation from the symmetric branch. One of these conditions occurs at a matching of the refractive indices across the interface while the other corresponds to a switching of the peak from the core to the cladding.
The rich bifurcation structure is illustrated by numerical calculations. Further stability considerations are included.
For the entire collection see [Zbl 0940.00039].

MSC:

35K15	Initial value problems for second-order parabolic equations
37K30	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures
17B80	Applications of Lie algebras and superalgebras to integrable systems
78A60	Lasers, masers, optical bistability, nonlinear optics

Keywords:

propagation of monochromatic fields; Maxwell’s equations; nonlinear eigenvalue problem; numerical calculations; stability; bifurcation