Ruijsenaars, S. N. M.; Schneider, H. A new class of integrable systems and its relation to solitons. (English) Zbl 0608.35071 Ann. Phys. 170, 370-405 (1986). 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 Cited in 7 ReviewsCited in 166 Documents MSC: 35Q99 Partial differential equations of mathematical physics and other areas of application Keywords:integrable \(N\)-particle system; conjugate coordinates; Hamiltonian; Weierstrass function; Lax matrix; soliton; Calogero-Moser system; Korteweg-de Vries PDFBibTeX XMLCite \textit{S. N. M. Ruijsenaars} and \textit{H. Schneider}, Ann. Phys. 170, 370--405 (1986; Zbl 0608.35071) Full Text: DOI References: [1] Ruijsenaars, S. N.M., Ann. Phys. (N.Y.), 126, 399-449 (1980) [2] Zamolodchikov, A. B., Comm. Math. Phys., 55, 183-186 (1977) [3] Karowski, M.; Thun, H. J., Nucl. Phys. 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