Special discontinuities in solving the self-similar problem of waves in a rod

The self-similar problem of long nonlinear longitudinally torsional waves of small amplitude arising in a semi-infinite elastic rod with a change of stresses at its end is considered. The peculiarity of this problem is that among the possible discontinuities in its solution, in addition to shock waves, there may be a special discontinuity with an additional boundary condition expressing the condition for the existence of a stationary viscous structure of this discontinuity.