Homoclinic connections of unstable plane waves of the modified nonlinear Schrödinger equation

The modified nonlinear Schrödinger (MNLS) equation is a model equation for the propagation of ultra-fast pulses in long-haul optical fiber transmission systems. Here we identify the linearized instabilities of plane wave solutions to both the defocusing and the focusing MNLS equation. Extending the results of Doktorov and Rothos [Phys. Lett. A 314 (2003) 59] for spatially independent plane waves of the focusing equation, the entire homoclinic manifold of both focusing and defocusing unstable plane waves of arbitrary wave number is explicitly constructed using an auto-Bäcklund transformation obtained via the dressing method [Theory of Solitons and the Inverse Scattering Method, Consultant Bureau, New York, 1984]. The MNLS equation is integrable via the Wadati–Konno–Ichikawa quadratic spectral problem [J. Phys. Soc. Jpn. 46 (1979) 1965] and there is a correspondence between pairs of elements of the solution basis of the linearized equation and quartets of multiple points of the Floquet spectral curve, similar to the NLS and sine-Gordon equations [Phys. D 43 (1990) 349, J. Nonlinear Sci. 10 (2000) 291, Phys. D 141 (2000) 104].