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 [25] 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.

我们在满足相应跳跃条件的局部时域存在和不连续性的唯一性意义上，研究了浅水磁流体动力学（SMHD）中冲击波和涡流片的结构稳定性。SMHD方程形成一个对称的双曲系统，形式上类似于特定非物理变形的2D可压缩弹性动力学系统。使用这种类比和[25]中2D可压缩弹性力学中冲击波的最新结果，我们证明，当且仅当流体高度沿整个冲击前沿增加时，SMHD中的冲击波才是结构稳定的。对于电流涡流板，流体高度是连续的，而速度和磁场的切向分量可能会有跳跃。