Integrable coupling of optical waves in higher-order nonlinear Schrödinger equations

Integrable higher-order generalizations of the nonlinear Schrödinger equation that describes the propagation of multi-mode optical pulses in a fiber are presented. We construct the coupled higher-order nonlinear Schrödinger equation (CHONSE) in association with each Hermitian symmetric spaces and demonstrate its integrability by deriving the Lax pair. We show that two distinct types of higher-order generalizations are possible, which we call as the 'type-I' and the 'type-II' CHONSE. The type-I and the type-II CHONSE generalize the Hirota and the Sasa-Satsuma equations respectively and it is shown that the type-II CHONSE can be obtained via a consistent reduction of the type-I CHONSE based on the AIII symmetric spaces.