  By constructing a hydrodynamic canonical formalism, we show that the
occurrence of an arbitrary density-dependent gauge potential in the meanfield
Hamiltonian of a Bose-condensed fluid invariably leads to nonlinear
flow-dependent terms in the wave equation for the phase, where such terms arise
due to the explicit dependence of the mechanical flow on the fluid density. In
addition, we derive a canonical momentum transport equation for this class of
nonlinear fluid and obtain an expression for the stress tensor. Further, we
study the hydrodynamic equations in a particular nonlinear fluid, where the
effective gauge potential results from the introduction of weak contact
interactions in an ultracold dilute Bose gas of optically-addressed two-level
atoms. In the Cauchy equation of mechanical momentum transport of the
superfluid, two non-trivial terms emerge due to the density-dependent vector
potential. A body-force of dilation appears as a product of the gauge potential
and the dilation rate of the fluid, while the stress tensor features a
canonical flow pressure term given by the inner-product of the gauge potential
and the canonical current density. By numerical simulation, we illustrate an
interesting effect of the nonlinear gauge potential on the groundstate
wavefunction of a superfluid in the presence of a foreign impurity. We find
that the groundstate adopts a non-trivial local phase, which is antisymmetric
under reversal of the gauge potential. The phase profile leads to a
canonical-flow or phase-flow dipole about the impurity, resulting in a skirting
mechanical flow. As a result, the pressure becomes asymmetric about the object
and the condensate undergoes a deformation.
