Biorthogonal Covariance Amplification Near the Breathing Threshold of Dissipative Kerr Solitons

<title>Abstract</title> <p>A linearly stable dissipative Kerr soliton can be substantially more sensitive to stochastic forcing than its eigenvalues alone imply. We quantify this effect by combining phase-fixed Lyapunov covariance with the biorthogonal geometry of the critical Hopf mode. For the spatial Lugiato–Lefever equation, the stationary soliton loses stability through a complex pair at Fc = 3.15194 and angular frequency ωc = 5.80567. The covariance of the phase-fixed internal dynamics diverges on the stable side as | Re λc|−1, as expected from critical slowing down, but its excess over a normal isospectral reference converges to a finite Petermann-like factor: Rrms → √ Kc = 3.11485. Direct nonlinear integration above threshold yields a breather frequency 5.84482, within 0.50% of the linear eigenfrequency at the same pump value. We also dissect a scalar auxiliary thermal coordinate into optical-only, dimension-matched decoupled, auxiliary-to-optical, optical-to-auxiliary, and closed-loop variants. In both a homogeneous reduction and the spatial soliton, the covariance change is carried almost entirely by the auxiliary-to-optical path, whereas closing the return path changes the result negligibly. Thus the observed amplification is intrinsic to the Kerr optical operator rather than generated by thermal feedback. Finally, the adjoint translational Goldstone mode gives the source-dependent timing-diffusion coefficient Dq = 1 2wT 0Qw0, clarifying which noise channels can drive soliton position. The results provide a resolution-robust modal diagnostic of nonnormal fluctuation amplification near breathing and a structural test for separating auxiliary-state coupling from genuine feedback-induced memory effects.</p>