Conservation laws of shallow water theory and the Galilean relativity principle

By the example of a model of shallow water theory, it is shown that the compatibility analysis of the Hugoniot conditions for various basic systems of conservation laws in the coordinate system moving together with a strong discontinuity can lead to erroneous results. It is connected with the hierarchy of conservation laws in shallow water theory with respect to the Galilean transformation, according to which the conservation law for total energy is unconditionally noninvariant with respect to this transformation, which leads to the dependence of the corresponding Hugoniot condition on the velocity of the inertial reference frame. It is shown that the specified shortcoming of the classical shallow water theory is absent in the model of vortex shallow water suggested by V. M. Teshukov.