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Abstract. We prove existence of a global conservative solution of the Cauchy problem for the two-component Camassa-Holm (2CH) system on the line, allowing for nonvanishing and distinct asymptotics at plus and minus infinity. The solution is proven to be smooth as long as the density is bounded away from zero. Furthermore, we show that by taking the limit of vanishing density in the 2CH system, we obtain the global conservative solution of the (scalar) Camassa-Holm equation, which provides a novel way to define and obtain these solutions. Finally, it is shown that while solutions of the 2CH system have infinite speed of propagation, singularities travel with finite speed.

We study stability of solutions of the Cauchy problem for the Hunter-Saxton equationIn particular, we derive a new Lipschitz metric dD with the property that for two solutions u and v of the equation we have dD(u(t), v(t)) ≤ e Ct dD(u0, v0).

We show how to construct globally defined multipeakon solutions of the Camassa–Holm equation. The construction includes in particular the case with peakon-antipeakon collisions. The solutions are conservative in the sense that the associated energy is constant for almost all times. Furthermore, we construct a new set of ordinary differential equations that determines the multipeakons globally. The system remains globally well-defined.