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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.

<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

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The Novikov equation is an integrable analogue of the CamassaHolm equation with a cubic (rather than quadratic) nonlinear term. Both these equations support a special family of weak solutions called multipeakon solutions. In this paper, an approach involving Pfaffians is applied to study multipeakons of the Novikov equation. First, we show that the Novikov peakon ODEs describe an isospectral flow on the manifold cut out by certain Pfaffian identities. Then, a link between the Novikov peakons and the finite Toda lattice of BKP type (B-Toda lattice) is established based on the use of Pfaffians. Finally, certain generalizations of the Novikov equation and the finite B-Toda lattice are proposed together with special solutions. To our knowledge, it is the first time that the peakon problem is interpreted in terms of Pfaffians.