Whitham approach to Generalized Hydrodynamics

The formation of dispersive shock waves in one-dimensional Bose gas represents a limitation of Generalized Hydrodynamics (GHD) due to the coarse -grained nature of the theory. Nevertheless, GHD accurately captures long -wavelength behavior, thus indicating an implicit knowledge of the underlying microscopic physics. Such representations are already known through the Whitham modulation theory, where dispersionless equations describe the evolution of the slowly varying shock wave parameters. Here we study the correspondence between Whitham's approach to the Gross-Pitaevskii equation and GHD in the semiclassical limit and beyond. Our findings enable the recovery of the shock wave solution directly from GHD simulations, which we demonstrate for both zero and finite temperature. Additionally, we study how free expansion protocols affect the shock wave density and their implications for experimental detection. The combined picture of Whitham and GHD lends itself to additional physical interpretation regarding the formation of shock waves.