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Maslov indices and singularities of Integrable systems. JM Robbins, University of Bristol J Foxman S Creagh, H Dullen, G Tanner, H Waalkens. Outline. Singularities of integrable systems Maslov index Monodromy and Maslov index Singularity formula for Maslov index
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Maslov indices and singularities of Integrable systems JM Robbins, University of Bristol J Foxman S Creagh, H Dullen, G Tanner, H Waalkens
Outline • Singularities of integrable systems • Maslov index • Monodromy and Maslov index • Singularity formula for Maslov index • Examples, including Toda chain
n=1 x x
n=2 X X
V y x Monodromy - Example
Monodromy and Maslov indices (Creagh, Dullen, JMR, Tanner Waalkens (2005))
L Butler, C Viterbo (1990)
r p pj rj p(j-1) r(j-1) r(j) p(j) Singularities
3 1 2 4 The Periodic Toda Chain Henon,Flaschka Kac & Moerbeke Gutzwiller, Sklyanin
rth Inequivalent Lax pair
j ( ) * 0 * Integrability
Questions • Other Lax pairs • Higher Maslov classes 2 H4n-3 (Trofimov, Suzuki) • Higher-order semiclassical quantization of Toda chain (Littlejohn, Colin de Verdiere)
V y x Quantum Monodromy- Example from Child et al (1999)