Multipeakons of a two-component modified Camassa–Holm equation and the relation with the finite Kac–van Moerbeke lattice

Abstract

A spectral and the inverse spectral problem are studied for the two-component modified Camassa-Holm type for measures associated to interlacing peaks. It is shown that the spectral problem is equivalent to an inhomogenous string problem with Dirichlet/Neumann boundary conditions. The inverse problem is solved by Stieltjes's continued fraction expansion, leading to an explicit construction of peakon solutions. Sufficient conditions for global existence of solutions are given. The large time asymptotics reveals that, asymptotically, peakons pair up to form bound states moving with constant speeds. The peakon flow is shown to project to one of the isospectral flows of the finite Kac-van Moerbeke lattice. XIANG-KE CHANG, XING-BIAO HU, AND JACEK SZMIGIELSKI Appendix A. Lax pair for peakon ODEs 24 References 26

Isospectral Flows Related to Frobenius–Stickelberger–Thiele Polynomials

“…We point out that the map between the Kac-van Moerbeke lattice and the 2-mCH interlacing peakon lattice was established in [17]. As we show above the degenerate FST lattice system (7.1) can also be mapped to the 2-mCH interlacing peakon lattice which clearly suggests a close connection between the degenerate FST lattice and the Kac-Moerbeke lattice which merits further studies.…”

“…in the sense of distributions as explained in [17]. The above degeneration seems counter intuitive since the Toda-type lattices are usually viewed as positive flows in the spectral variable while peakon flows are negative flows based on previous works [11,14,16,17].…”

“…The finite truncation of such an FST lattice is introduced in Section 4 while its relation with the modified Camassa-Holm (mCH) peakon lattice [18] is established in Section 6. Finally, in Section 7, we discuss a degenerate case of the FST lattice and associate it with the interlacing peakon ODE system of a two-component mCH (2-mCH) equation studied in [17]. It should be mentioned that the FST lattice is an isospectral flow connected with a generalized eigenvalue problem.…”

An application of Pfaffians to multipeakons of the Novikov equation and the finite Toda lattice of BKP type

“…The CH peakons can be expressed by ordinary orthogonal polynomials [5]. Symmetric orthogonal polynomials play a substantial role in the study of interlacing peakons of a two component modified CH equation [12] . The DP peakons [40] are associated with Cauchy bimoment determinants, which induce the research on Cauchy biorthogonal polynomials and subsequent applications to random matrix models [7,8].…”

The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type

“…Remark We recall that the peakons for the two‐component modified Camassa‐Holm (2mCH) equation satisfy an identically looking ODE system , but with an important difference that , where are the two (real) components. One natural reduction of the 2mCH equation is the modified Camassa‐Holm (1mCH) equation obtained by putting .…”