  It is shown that third-order 1D nonlinear dispersion equations admit single
point gradient catastrophe, described by blow-up-type similarity solutions.
After blow-up, the solutions admit shock wave-type self-similar extensions.
Snce such extensions are not unique, this implies the principle nonuniqueness
of shock-type solutions and also nonexistence of any entropy-type description
of proper unique solutions. A difficult free-boundary setting, with extra
conditions specified on shocks, are necessary to restore uniqueness in such
problems.
