| Non-focusing instabilities in coupled, integrable, nonlinear Schrödinger PDEs (2007) | |||||||||||||||
Abstract | |||||||||||||||
| AMS Subject Classification: 35Q55, 58F07. The nonlinear coupling of two scalar nonlinear Schrodinger (NLS) fields results in non-focusing instabilities that exist independently of the well-known modulational instability of the focusing NLS equation. The focusing versus defocusing behavior of scalar NLS fields is a well-known model for the corresponding behavior of pulse transmission in optical fibers in the anomalous (focusing) versus normal (defocusing) dispersion regime [19, 20]. For fibers with birefringence (induced by an asymmetry in the cross-section,) the scalar NLS fields for two orthogonal polarization modes couple nonlinearly [26]. Experiments by Rothenberg [32, 33] have demonstrated a new type of modulational instability in a birefringent normal dispersion fiber, and he proposes this cross-phase coupling instability as a mechanism for the generation of ultra-fast, terahertz optical oscillations. In this paper the non-focusing plane wave instability in an integrable coupled nonlinear Schrodinger (CNLS) partial differential equation system is contrasted with the focusing instability from two perspectives: traditional linearized stability analysis and integrable methods based on periodic inverse spectral theory. The latter approach is a crucial first step toward a nonlinear, nonlocal understanding of this new optical instability analogous to that developed 1 | |||||||||||||||
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