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The modified Camassa-Holm (also called FORQ) equation is one of numerous <i>cousins</i> of the Camassa-Holm equation possessing non-smoth solitons (<i>peakons</i>) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev <i>H</i>¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the <i>H</i>¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].

In this paper, we propose a direct method to evaluate Hankel determinants for some generating functions satisfying a certain type of quadratic equations, which cover generating functions of Catalan numbers, Motzkin numbers and Schrder numbers. Additionally, four recent conjectures proposed by Cigler (2011) [3] are proved.

In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $ LDU $ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schrder paths.

In this paper, we present a generalized Toeplitz determinant solution for the generalized Schur flow and propose a mixed form of the two known relativistic Toda chains together with its generalized Toeplitz determinant solution. In addition, we also give a Hankel type determinant solution for a nonisospectral Toda lattice. All these results are obtained by technical determinant operations. As a bonus, we finally obtain some new combinatorial numbers based on the moment relations with respect to these semi-discrete integrable systems and give the corresponding combinatorial interpretations by means of the lattice paths.

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.

In this paper, we show that the coupled modified KdV equations possess rich mathematical structures and some remarkable properties. The connections between the system and skew orthogonal polynomials, convergence acceleration algorithms and Laurent property are discussed in detail.

Using the tri-hamiltonian splitting method, the authors of [Anco and Mobasheramini, Physica D, 355: 1--23, 2017] derived two U (1) -invariant nonlinear PDEs that arise from the hierarchy of the nonlinear Schrdinger equation and admit peakons ($ non-smooth\solitons $). In the present paper, these two peakon PDEs are generalized to a family of U (1) -invariant peakon PDEs parametrized by the real projective line U (1) . All equations in this family are shown to posses $ conservative\peakon\solutions $(whose Sobolev U (1) norm is time invariant). The Hamiltonian structure for the sector of conservative peakons is identified and the peakon ODEs are shown to be Hamiltonian with respect to several Poisson structures. It is shown that the resulting Hamilonian peakon flows in the case of the two peakon equations derived in [Anco and Mobasheramini, Physica D, 355: 1--23, 2017] form orthogonal families, while in general the Hamiltonian peakon flows for two different equations in the general family intersect at a fixed angle equal to the angle between two lines in U (1) parametrizing those two equations. Moreover, it is shown that inverse spectral methods allow one to solve explicitly the dynamics of conservative peakons using explicit solutions to a certain interpolation problem. The graphs of multipeakon solutions confirm the existence of multipeakon breathers as well as asymptotic formation of pairs of two peakon bound states in the non-periodic time domain.