Bogoliubov-de Gennes spectrum, Cherenkov radiation and recoil of solitons with third-order dispersion underlying optical analogue horizons

Abstract

Third-order dispersion turns the soliton of the nonlinear Schrödinger equation, which underlies optical analogues of event horizons in fibers, into a weakly radiating solitary wave. We compute three of its properties without assuming a thermal emission law. First, the linearized (Bogoliubov-de Gennes) problem must be posed about the stationary state of the dispersive equation: linearizing about the integrable profile produces a complex quartet that is converged in grid and box, grows as the square root of the dispersion strength and is spurious. About the stationary state, on a periodic grid, the zero-mode sector consists of two Jordan blocks of size two and the spectrum is real at the operating point, including negative-energy box modes of the resonant branch. Second, we measure the steady Cherenkov emission of adiabatically prepared solitons from the flux of the radiation behind them, over eight decades. The radiation is phase-matched in the frame of the drifting soliton, and the first-order rate falls short of the measured one by factors of 29 to 45. Expressed through a scaled dispersion coefficient of the local soliton, the measured rate describes launched solitons of three amplitudes; with norm and momentum balance, and no further fitted parameter, it reproduces their recoil and the slow decay of their loss rate; at weak dispersion the loss of a launched soliton is the shedding of its launch mismatch. Third, a census of the asymptotic channels shows that long-wavelength upstream emission requires the chemical potential of the soliton, lowered by the comoving-frame Doppler shift of a drifting soliton, to lie inside the analysis band, which bounds the soliton amplitude. In a kinematic model of the flow the surface gravity has a closed form; the thermal relation and the entanglement built on it are imposed inputs of that model. The construction applies to Raman-free Kerr media.