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A new class of integrable systems and its relation to solitons. (English) Zbl 0608.35071

A new integrable \(N\)-particle system on line is proposed with canonically conjugate coordinates \(q_ j,\vartheta_ j\), \(j=1,\dots,N\) and the Hamiltonian \[ H=const \sum^{N}_{j=1}ch\vartheta_ j\times \prod_{k\neq j}f(q_ j-q_ k),\quad f^ 2(q)=\lambda +\mu \wp (q), \] where \(\wp (q)\) equals the Weierstrass function. The integrals of the system \[ S_ k=\sum_{I\subset \{1,\dots,N\},| I| =k}\exp (\sum_{i\in I}\vartheta_ i)\prod_{i\in I,j\not\in I}f(q_ i-q_ j),\quad k=1,\dots,N \] are built. The Lax matrix is proposed such that the spectrum is conserved under each of the \(S_ k\) flows. The solutions to Hamiltonian equations for the flow generated by the \(S_ 1\) are found in explicit form with a method due to Perelomov and Olshanetsky. Degenerate limits of the Weierstrass functions and associated soliton solutions are studied. The system in question may be viewed as a relativistic generalization of the Calogero-Moser system with Hamiltonian \(H=\sum^{N}p^ 2_ j/2m-\sum^{N}_{i\leq j<k\leq N}\wp (x_ j-x_ k)\) and is intimately connected with algebraically integrable partial differential equations, e.g., Korteweg-de Vries, sine-Gordon, Toda and others.
Reviewer: E.Belokolos

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
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