Collective-Coordinate Fluctuations of Driven-Dissipative Solitons

Fluctuations of nonequilibrium localized waves are shaped not only by direct stochastic forcing but also by deterministic transfer among coupled collective degrees of freedom. We develop a pathway-resolved stochastic collective-coordinate theory that makes this transfer explicit for stationary driven-dissipative solitons of the generalized Lugiato--Lefever equation with Raman response. The reduction yields a refined stationary phase-locking relation, providing a fixed point for the subsequent stochastic theory. Projecting field-level fluctuations onto four soliton coordinates: amplitude, frequency shift, temporal position, and global phase, yields a reduced Langevin model and, after linearization about a stable stationary state, an analytic power-spectral-density matrix. This framework separates direct stochastic injection from deterministic inter-coordinate conversion and thereby resolves how each observable spectrum is assembled from distinct internal fluctuation pathways. It shows that timing jitter is governed primarily by Gordon--Haus-type frequency-to-timing conversion, while phase noise is often dominated by amplitude-to-phase transfer rather than by direct phase diffusion. Raman response opens additional cascaded pathways, and the low-detuning hump in the intensity and phase spectra is traced to the driven response of an underdamped amplitude--phase subsystem preceding the breathing instability. Comparisons with stochastic simulations of both the reduced model and the full generalized Lugiato--Lefever equation show good agreement throughout most of the stable stationary single-soliton regime, with systematic deviations mainly near the Hopf boundary. The theory provides a general route for connecting internal fluctuation-transfer mechanisms of dissipative solitons to measurable noise observables.