  In this Letter we propose that for Lax integrable nonlinear partial
differential equations the natural concept of weak solutions is implied by the
compatibility condition for the respective distributional Lax pairs. We
illustrate our proposal by comparing two concepts of weak solutions of the
modified Camassa-Holm equation pointing out that in the peakon sector (a family
of non-smooth solitons) only one of them, namely the one obtained from the
distributional compatibility condition, supports the time invariance of the
Sobolev $H^1$ norm.
