Simple wave in magnetohydrodynamics ($A_x{\mathbf{U}}_q=v(q){\mathbf{U}}_q$)

An ideal-MHD simple wave traces a one-parameter state-space curve whose tangent is a right eigenvector of the characteristic matrix. Its parameter obeys $q_t+v(q)q_x=0$ and its characteristic speed obeys the Inviscid Burgers equation. Nonlinear Alfvén waves are a linearly degenerate example.