An envelope system with third order dispersion: ‘Unconventional’ modulation instability and Floquet analysis

Abstract

Modulation instability of plane waves of the Hirota equation, an ‘integrable’ system with third order dispersion, arises from the interplay of dispersive and nonlinear effects. The conventional analysis cannot account for finite amplitude disturbances and suffers from a scenario of indefinite growth. In an attempt to remove these constraints, we consider, as an illustrative example, the exact doubly periodic solutions expressed via the Jacobi elliptic functions. Wavy profiles at the minima of the intensity are interpreted as finite amplitude disturbances on a plane wave background. The profiles are amplified and will saturate at the maxima of the intensity. Such periodic states and breathers can be generated from finite amplitude disturbances with wavenumbers falling outside the linear instability band. This growth phase thus qualifies as ‘unconventional’ or ‘extraordinary’ modulation instability. Floquet analysis is performed to investigate the stability of the periodic patterns.