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

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’ continued fraction expansion, leading to an explicit construction of peakon solutions. Sufficient conditions for the global existence in $t$ 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.

💡 Research Summary

The paper investigates the spectral and inverse‑spectral problems associated with the two‑component modified Camassa‑Holm (2‑mCH) equation when the solution is a superposition of interlacing peakons. The authors start by writing the fields (u(x,t)) and (v(x,t)) as finite sums of exponential “peak” functions whose amplitudes (m_k(t)) and (n_k(t)) evolve in time and whose positions (x_k(t)) are ordered so that the supports of the two measures interlace. Under this ansatz the Lax pair of the 2‑mCH equation reduces to a 2 × 2 linear system whose eigenvalue problem can be interpreted as an inhomogeneous vibrating string with mixed Dirichlet–Neumann boundary conditions. The string’s mass density and tension are piecewise constant and are directly determined by the discrete measures representing the peakon amplitudes.

The inverse problem is solved by exploiting Stieltjes’ continued‑fraction expansion. Given the spectral data (the eigenvalues and the associated norming constants), the continued fraction coefficients are recovered recursively; these coefficients are precisely the ratios of tension to mass on each sub‑interval of the string. Consequently the original measures—and hence the peakon amplitudes and positions—are reconstructed explicitly. The authors prove convergence of the continued fraction and establish uniqueness of the reconstruction as long as the interlacing condition holds, which guarantees positivity of all coefficients.

Global existence in time is addressed by deriving sufficient conditions: the product (m_k n_k) must remain positive and the distances between neighboring peakons must stay bounded away from zero. Under these hypotheses the solution exists for all (t) and no wave‑breaking occurs.

Long‑time asymptotics are then analyzed. As (t\to\infty) the peakons tend to form bound pairs. Within each pair the two peaks lock together, moving with a constant speed equal to the reciprocal of a corresponding eigenvalue. This pairing phenomenon is a direct manifestation of the spectral symmetry: eigenvalues appear in conjugate pairs, and the associated norming constants enforce the formation of a stable “soliton‑like” compound. Numerical simulations included in the paper illustrate the emergence of these travelling bound states.

Finally, the authors demonstrate that the finite‑dimensional peakon dynamics can be projected onto one of the isospectral flows of the finite Kac‑van Moerbeke lattice. By mapping the peakon positions and amplitudes to the lattice variables ((a_i,b_i)), they show that the Lax pair governing the peakon ODEs coincides with the Lax representation of the Kac‑van Moerbeke system. Hence the peakon flow is an exact reduction of the lattice flow, establishing a deep link between a continuous integrable PDE and a discrete integrable lattice.

In summary, the work provides a complete spectral characterization of interlacing 2‑mCH peakons, an explicit inverse‑spectral reconstruction via Stieltjes fractions, rigorous global‑existence criteria, a clear description of the asymptotic pairing mechanism, and a novel identification of the peakon dynamics with the finite Kac‑van Moerbeke lattice. These results enrich the theory of multi‑component integrable wave equations and open new avenues for transferring techniques between continuous and discrete integrable systems.