  We study the structural stability of shock waves and current-vortex sheets in
shallow water magnetohydrodynamics (SMHD) in the sense of the local-in-time
existence and uniqueness of discontinuous solutions satisfying corresponding
jump conditions. The equations of SMHD form a symmetric hyperbolic system which
is formally analogous to the system of 2D compressible elastodynamics for
particular nonphysical deformations. Using this analogy and the recent results
in [Morando A., Trakhinin Y., Trebeschi P. Math. Ann. (2019),
https://doi.org/10.1007/s00208-019-01920-6] for shock waves in 2D compressible
elastodynamics, we prove that shock waves in SMHD are structurally stable if
and only if the fluid height increases across the shock front. For
current-vortex sheets the fluid height is continuous whereas the tangential
components of the velocity and the magnetic field may have a jump. Applying a
so-called secondary symmetrization of the symmetric system of SMHD equations,
we find a condition sufficient for the structural stability of current-vortex
sheets.
