Firstly, a formal correspondence is established between the CamassaHolm (CH) equation and a two-component modified CH (or called SQQ) equation according to the method of <i>moment modification</i> for multipeakon formulae. Secondly, based on the generalized nonisospectral CH equation in Chang et al. (2014) [14] and the interlacing multipeakons of the two-component modified CH equation in Chang et al. (2016) [15], we propose a new generalized two-component modified CH equation with two parameters, which possesses a nonisospectral Lax pair. The proposed equation still admits multipeakon solutions of explicit and closed form. Sufficient conditions for global existence of solutions are given and two concrete examples with certain interesting phenomenon are presented. Last of all, as a by-product, a generalized nonisospectral modified CH equation is deduced, together with its Lax pair.

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by solving various problems in many fields, which can be hardly handled by alternative methods so far.

The chains studied in this paper generalize Chern–Moser chains for CR structures. They form a distinguished family of one dimensional submanifolds in manifolds endowed with a parabolic contact structure. Both the parabolic contact structure and the system of chains can be equivalently encoded as Cartan geometries (of different types). The aim of this paper is to study the relation between these two Cartan geometries for Lagrangean contact structures and partially integrable almost CR structures. We develop a general method for extending Cartan geometries which generalizes the Cartan geometry interpretation of Fefferman’s construction of a conformal structure associated to a CR structure. For the two structures in question, we show that the Cartan geometry associated to the family of chains can be obtained in that way if and only if the original parabolic contact structure is torsion free. In particular, the procedure works exactly on the subclass of (integrable) CR structures. This tight relation between the two Cartan geometries leads to an explicit description of the Cartan curvature associated to the family of chains. On the one hand, this shows that the homogeneous models for the two parabolic contact structures give rise to examples of non–flat path geometries with large automorphism groups. On the other hand, we show that one may (almost) reconstruct the underlying torsion free parabolic contact structure from the Cartan curvature associated to the chains. In particular, this leads to a very conceptual proof of the fact that chain preserving contact diffeomorphisms are either isomorphisms or antiisomorphisms of parabolic contact structures.

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