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Regularization of fluctuations near the sonic horizon due to the quantum potential and its influence on the Hawking radiation

We consider dynamics of fluctuations in transonically accelerating Bose-Einstein condensates and luminous liquids (coherent light propagating in a Kerr nonlinear medium) using the hydrodynamic approach. It is known that neglecting the quantum potential (QP) leads to a singular behavior of quantum and classical fluctuations in the vicinity of the Mach (sonic) horizon, which in turn gives rise to the Hawking radiation. The neglect of QP is well founded at not too small distances $|x| \gg l_h$ from the horizon, where $l_h$ is the healing length. Taking the QP into account we show that a second characteristic length $l_r > l_h$ exists, such that the linear fluctuation modes become regularized for $|x| \ll l_r$. At $|x| \gg l_r$ the modes keep their singular behavior, which however is influenced by the QP. As a result we find a deviation of the high frequency tail of the spectrum of Hawking radiation from Planck's black body radiation distribution. Similar results hold for the wave propagation in Kerr nonlinear media where the length $l_h$ and $l_r$ exist due to the nonlinearity.