An inverse problem for the modified Camassa-Holm equation and multi-point Pad{e} approximants

In this Letter the main steps in the inverse spectral construction of a family of non-smooth solitons (peakons) to the modified Camassa-Holm equation are oulined. It is shown that the inverse problem is solvable in terms of multi-point Pad'{e} approximations.

💡 Research Summary

The paper addresses the inverse spectral problem for the modified Camassa‑Holm (mCH) equation
(m_t+((u^2-u_x^2)m)x=0,; m=u-u{xx}),
focusing on the class of non‑smooth “peakon” solutions. A peakon ansatz
(u(x,t)=\sum_{j=1}^n m_j(t),e^{-|x-x_j(t)|})
produces a discrete measure (m=2\sum_{j=1}^n m_j\delta_{x_j}). Because the nonlinear term ((u^2-u_x^2)m) is singular, the authors define it via the arithmetic average (\langle\cdot\rangle) of left‑ and right‑hand limits, which preserves the integrable structure and yields a well‑defined distributional Lax pair.

A gauge transformation (\Phi=\operatorname{diag}(e^{x/2},e^{-x/2})\Psi) simplifies the spatial part of the Lax pair to
(\Phi_x=\begin{pmatrix}0&\lambda h\ -\lambda g&0\end{pmatrix}\Phi),
with (g_j=m_j e^{-x_j}) and (h_j=m_j e^{x_j}). Imposing the natural boundary conditions (\Phi_1(-\infty)=0) and (\Phi_2(+\infty)=0) forces (\Phi) to be piecewise constant, and the problem reduces to a finite difference system:

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