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

Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons