Shock wave evolution and discontinuity propagation for relativistic superfluid hydrodynamics with spontaneous symmetry breaking

In this Letter, we have studied the shock wave and discontinuity propagation for relativistic superfluid with spontaneous U(1)U(1) symmetry breaking in the framework of hydrodynamics. General features of shock waves are provided, the propagation of discontinuity and the sound modes of shock waves are also presented. The first sound and the second sound are identified as the propagation of discontinuity, and the results are in agreement with earlier theoretical studies. Moreover, a differential equation, called the growth equation, is obtained to describe the decay and growth of the discontinuity propagating along its normal trajectory. The solution is in an integral form and special cases of diverging waves are also discussed.