Global Convergence of a Sticky Particle Method for the Modified Camassa-Holm Equation

In this paper, we prove convergence of a sticky particle method for the modified Camassa–Holm equation (mCH) with cubic nonlinearity in one dimension. As a byproduct, we prove global existence of weak solutions u with regularity: u and ux are space-time BV functions. The total variation of m(·, t) = u(·, t)− uxx(·, t) is bounded by the total variation of the initial data m0. We also obtain W 1,1(R)-stability of weak solutions when solutions are in L∞(0,∞;W 2,1(R)). (Notice that peakon weak solutions are not in W 2,1(R).) Finally, we provide some examples of nonuniqueness of peakon weak solutions to the mCH equation.