<i>Peakons</i> are special weak solutions of a class of nonlinear partial differential equations modelling non-linear phenomena such as the breakdown of regularity and the onset of shocks. We show that the natural concept of weak solutions in the case of the modified CamassaHolm equation studied in this paper is dictated by the distributional compatibility of its Lax pair and, as a result, it differs from the one proposed and used in the literature based on the concept of weak solutions used for equations of the Burgers type. Subsequently, we give a complete construction of peakon solutions satisfying the modified CamassaHolm equation in the sense of distributions; our approach is based on solving certain inverse boundary value problem, the solution of which hinges on a combination of classical techniques of analysis involving Stieltjes continued fractions and multi-point Pad approximations. We propose

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 <i>peakon</i> 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 <i>H</i>¹ norm.

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There is much research on the dynamical complexity on irregular sets andlevel sets of ergodic average from the perspective of density in base space, theHausdorff dimension, Lebesgue positive measure, positive or full topological entropy (andtopological pressure), etc. However, this is not the case from the viewpoint of chaos.There are many results on the relationship of positive topological entropy and variouschaos. However, positive topological entropy does not imply a strong version of chaos,called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. Inthis paper, we will show that, for dynamical systems with specification properties, thereexist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, weprove that several recurrent level sets of points with different recurrent frequency haveuncountable DC1-scrambled subsets. The major argument in proving the above results isthat there exists uncountable DC1-scrambled subsets in saturated sets.

We prove we can build (transitive or nontransitive) Anosov flows on closed three-dimensional manifolds by gluing together filtrating neighborhoods of hyperbolic sets. We give several applications of this result; for example: 1.We build a closed three-dimensional manifold supporting both a transitive Anosov vector field and a nontransitive Anosov vector field. 2.For any n, we build a closed three-dimensional manifold M supporting at least n pairwise different Anosov vector fields. 3.We build transitive hyperbolic attractors with prescribed entrance foliation; in particular, we construct some incoherent transitive hyperbolic attractors. 4.We build a transitive Anosov vector field admitting infinitely many pairwise nonisotopic transverse tori.