How does Hawking emission change the analog horizon that produces it?

For a quasi-1D BEC sonic black hole, the back-reaction of the Hawking flux on the flow, to first order in the quantum depletion 1/(n_1D $\xi$) (n_1D the linear density, xi the healing length), has been computed as a stationary shift of the density, velocity and Mach-number profiles. Determine whether the surface gravity $\kappa$ also drifts in time (sign and rate of $\mathrm{d}\kappa/\mathrm{d}t$, and whether the evolution follows an evaporation law analogous to $\mathrm{d}M/\mathrm{d}t$ proportional to $-T_H^2$), and measure the predicted stationary or time-dependent profile change. An answer is a measured profile change that agrees with the computed one after three-body loss, heating and stimulated emission are subtracted.

A measurement of the density, velocity or Mach-number profile change in a stationary BEC analog that matches the computed back-reaction after atom loss and heating are subtracted.