Thursday, January 31, 2013

1301.7211 (Boris Dubrovin et al.)

On an isomonodromy deformation equation without the Painlevé property    [PDF]

Boris Dubrovin, Andrei Kapaev
We show that the fourth order nonlinear ODE which controls the pole dynamics in the general solution of equation $P_I^2$ compatible with the KdV equation exhibits two remarkable properties: 1)~it governs the isomonodromy deformations of a $2\times2$ matrix linear ODE with polynomial coefficients, and 2)~it does not possesses the Painlev\'e property. We also study the properties of the Riemann--Hilbert problem associated to this ODE and find its large $t$ asymptotic solution for the physically interesting initial data.
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