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1985•Journal of Mathematical PhysicsRequires access

On the one- and two-dimensional Toda lattices and the Painlevé property

John Gibbon, M. Tabor

Open publisher page 16 citations

Abstract

The Toda lattice and the two-dimensional Toda lattice (2-DTL) are shown to possess a type of ‘‘Painlevé property’’ that is based on the use of separate ‘‘singular manifolds’’ for each dependent variable. The isospectral problem for the 2-DTL found by both Mikhailov and by Fordy and Gibbons can be simply and logically derived from this analysis. Some remarks are made about the connection between our work and independent work of Kametaka and Airhault on the relationship between the Toda lattice and the second Painlevé transcendent.

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What this paper is about

The Toda lattice and the two-dimensional Toda lattice (2-DTL) are shown to possess a type of ‘‘Painlevé property’’ that is based on the use of separate ‘‘singular manifolds’’ for each dependent variable. The isospectral problem for the 2-DTL found by both Mikhailov and by Fordy and Gibbons can be simply and logically derived from this analysis. Some remarks are made about the connection between our work and independent work of Kametaka and Airhault on the relationship between the Toda lattice and the second Painlevé transcendent.

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OpenAlex reports 16 citations for this work. Citation counts describe recorded attention and do not establish research quality.

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Available abstract

The Toda lattice and the two-dimensional Toda lattice (2-DTL) are shown to possess a type of ‘‘Painlevé property’’ that is based on the use of separate ‘‘singular manifolds’’ for each dependent variable. The isospectral problem for the 2-DTL found by both Mikhailov and by Fordy and Gibbons can be simply and logically derived from this analysis. Some remarks are made about the connection between our work and independent work of Kametaka and Airhault on the relationship between the Toda lattice and the second Painlevé transcendent.

Key concepts: Toda lattice, Isospectral, Lattice (music), Property (philosophy), Mathematics, Pure mathematics, Connection (principal bundle), Mathematical physics

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