Modulation Instability and Loss-Tolerant Optical Frequency Combs from Exact Breathers in the ℂ P 2 Sine-Gordon System

Optical frequency combs underpin precision metrology and broadband photonics, yet analytic models that tie their formation directly to modulation instability remain scarce. We address this gap in the ℂP 2 sine-Gordon system, a coupled optical model in which a real field and a complex field evolve on a nonzero background. We show that this background is linearly unstable across a finite band of modulation wavenumbers, and we construct in closed form, through a Darboux transformation, the breather that carries the instability into its nonlinear stage. The spatial spectrum of this breather is a discrete frequency comb whose growth rate coincides exactly with the linear gain evaluated at the comb spacing, providing a transparent analytic bridge between the linear and nonlinear regimes. We further find that the fastest growing mode and the widest comb arise at distinct wavenumbers, a separation that directly constrains comb design. Iterating the transformation produces a second order breather with a richer spectral structure. Guided by the requirements of practical optical platforms, we translate the dimensionless results into physical comb line spacings and gain lengths, we show that a weak dissipative perturbation lowers the instability gain by its own rate and sets a threshold below which the comb still forms, and we show that the comb spacing that emerges from broadband noise is fixed by the gain peak rather than by the noise itself. These exact results establish the ℂP 2 sine-Gordon system as a tractable platform for supercontinuum generation and optical frequency comb engineering.