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Asymptotics of a Class of Solutions to the Cylindrical Toda Equations

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t

asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, , have the representation

where K k $ are integral operators. This class includes the n-periodic cylindrical Toda equations. For n=2 our results reduce to the previously computed asymptotics of the 2D radial sinh-Gordon equation and for n=3 (and with an additional symmetry constraint) they reduce to earlier results for the radial Bullough-Dodd equation. Both of these special cases are examples of Painlevé III and have arisen in various applications. The asymptotics of are derived by computing the small t asymptotics

where explicit formulas are given for the quantities a k and b k . The method consists of showing that the resolvent operator of K k has an approximation in terms of resolvents of certain Wiener-Hopf operators, for which there are explicit integral formulas. Abstract:

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Authors and Affiliations

  1. Department of Mathematics and Institute of Theoretical Dynamics, University of California, Davis, CA 95616, USA. E-mail: tracy@itd.ucdavis.edu, , , , , , US

    Craig A. Tracy

  2. Department of Mathematics, University of California, Santa Cruz, CA 95064, USA.¶E-mail: widom@math.ucsc.edu, , , , , , US

    Harold Widom

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  1. Craig A. Tracy
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  2. Harold Widom
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Received: 23 January 1997 / Accepted: 8 May 1997

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Tracy, C., Widom, H. Asymptotics of a Class of Solutions to the Cylindrical Toda Equations . Comm Math Phys 190, 697–721 (1998). https://doi.org/10.1007/s002200050257

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  • DOI: https://doi.org/10.1007/s002200050257

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