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One-dimensional Toda molecule. I: General solution. (English) Zbl 1171.53341

Summary: A mechanism is presented for producing explicitly the general solutions to the one-dimensional Toda molecule equations associated with the classical complex Lie algebras of rank \(N\): \(sl (N+1,\mathbb{C})\), \(so(2N+1,\mathbb{C})\), \(sp(N,\mathbb{C})\) and \(so (2N,\mathbb{C})\). The method itself is applicable, however, to any semi-simple Lie algebra. A particular feature of our solutions is their description in terms of the “initial data”, which is consistent with the fact that the one-dimensional Toda molecule equation is a Newtonian equation of motion. The solutions for the algebras associated with \(SU(2)\), \(SU(3)\) and \(Sp(2)\) are given explicitly.

MSC:

53C80 Applications of global differential geometry to the sciences
17B20 Simple, semisimple, reductive (super)algebras
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