dorsal/arxiv
View SchemaThe complex Toda chains and the simple Lie algebras - solutions and large time asymptotics -- II
| Authors | V. S. Gerdjikov, E. G. Evstatiev, R. I. Ivanov |
|---|---|
| Categories | |
| ArXiv ID | solv-int/9909020 |
| URL | https://arxiv.org/abs/solv-int/9909020 |
| DOI | 10.1088/0305-4470/33/5/312 |
| Journal | J. Phys. A: Math. Gen. 33 (2000) 975 |
Abstract
We propose a compact and explicit expression for the solutions of the complex Toda chains related to the classical series of simple Lie algebras g. The solutions are parametrized by a minimal set of scattering data for the corresponding Lax matrix. They are expressed as sums over the weight systems of the fundamental representations of g and are explicitly covariant under the corresponding Weyl group action. In deriving these results we start from the Moser formula for the A_r series and obtain the results for the other classical series of Lie algebras by imposing appropriate involutions on the scattering data. Thus we also show how Moser's solution goes into the one of Olshanetsky and Perelomov. The results for the large-time asymptotics of the A_r -CTC solutions are extended to the other classical series B_r - D_r. We exhibit also some `irregular' solutions for the D_{2n+1} algebras whose asymptotic regimes at t ->\pm\infty are qualitatively different. Interesting examples of bounded and periodic solutions are presented and the relations between the solutions for the algebras D_4, B_3 and G_2 $ are analyzed.
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"abstract": "We propose a compact and explicit expression for the solutions of the complex\nToda chains related to the classical series of simple Lie algebras g. The\nsolutions are parametrized by a minimal set of scattering data for the\ncorresponding Lax matrix. They are expressed as sums over the weight systems of\nthe fundamental representations of g and are explicitly covariant under the\ncorresponding Weyl group action. In deriving these results we start from the\nMoser formula for the A_r series and obtain the results for the other classical\nseries of Lie algebras by imposing appropriate involutions on the scattering\ndata. Thus we also show how Moser\u0027s solution goes into the one of Olshanetsky\nand Perelomov. The results for the large-time asymptotics of the A_r -CTC\nsolutions are extended to the other classical series B_r - D_r. We exhibit also\nsome `irregular\u0027 solutions for the D_{2n+1} algebras whose asymptotic regimes\nat t -\u003e\\pm\\infty are qualitatively different. Interesting examples of bounded\nand periodic solutions are presented and the relations between the solutions\nfor the algebras D_4, B_3 and G_2 $ are analyzed.",
"arxiv_id": "solv-int/9909020",
"authors": [
"V. S. Gerdjikov",
"E. G. Evstatiev",
"R. I. Ivanov"
],
"categories": [
"solv-int",
"nlin.SI"
],
"doi": "10.1088/0305-4470/33/5/312",
"journal_ref": "J. Phys. A: Math. Gen. 33 (2000) 975",
"title": "The complex Toda chains and the simple Lie algebras - solutions and large time asymptotics -- II",
"url": "https://arxiv.org/abs/solv-int/9909020"
},
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