Asymptotics of a Class of Solutions to the Cylindrical Toda Equations
نویسندگان
چکیده
The small t asymptotics of a class of solutions to the 2D cylindrical Toda equations is computed. The solutions, qk(t), have the representation qk(t) = log det (I − λ Kk) − log det (I − λ Kk−1), where Kk 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 qk(t) are derived by computing the small t asymptotics det (I − λ Kk) ∼ bk ( t n )ak , where explicit formulas are given for the quantities ak and bk. The method consists of showing that the resolvent operator of Kk has an approximation in terms of resolvents of certain Wiener-Hopf operators, for which there are explicit integral formulas.
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