A new class of integrable systems and its relation to solitons

Systems of Calogero-Moser Type

“…More generally, (3.8) turns out to be valid for N > 3 when (3.9) holds [3]. (For N > 3 it is not known, however, whether other solutions to (3.8) exist.…”

“…First, specializing to the systems of type I, II, and VI, the repulsive character of the interparticle potentials j(xj -Xk) and fr(xj -xk), respectively, can be exploited to prove that the 8 1 dynamics leads to soliton scattering. Specifically, Moser's argument can be adapted to the 8 1 flow [3], yielding here (3.22) and, once more, (2.43). Therefore, the functions sk: t i---> Sk ( x(t), p(t)) reduce to symmetric functions of matrices L± = diag ( exp(f3pt), ... , exp(f3pt)) for t -> ±oo.…”

“…Integrable relativistic generalizations (corresponding once more to the root system AN -1) were first introduced in Ref. 3 at the classical and in Ref. 4 at the quantum level.…”

Sine-Gordon Solitons vs. Relativistic Calogero-Moser Particles

“…These equations were proved in Ref. (2], where the relativistic systems were pr!2'.sented for the first time.…”

“…The involutivity of these Hamiltonians can be reduced to a set of functional equations for the g.;-function (2]. These equations can be obtained as a corollary of a.…”