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Gravity-analogy in one-dimensional ideal Fermi fluids and Burgers' equation

An hydrodynamic description of a one-dimensional flow of an ideal Fermi fluid is constructed from a semiclassical approximation. For an initially fully degenerate fluid, Euler and continuity hydrodynamic equations are dual to two uncoupled inviscid Burgers' equations. Yet the price for the initial simplicity of the description is paid by the complexity of non-linear instabilities towards possible turbulent evolutions. Nevertheless, it is shown that linear long-wavelength density perturbations on a stationary flow are generically stable. Consequently, linear sound obeys a wave equation with analogy to gravity. The results have applications for ultra-cold atomic gases.