|XXXIV Workshop on Geometric Methods in Physics||28.06-4.07.2015|
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Local inverse scattering
We develop a local version of the inverse scattering method for studying soliton equations of parabolic type (including KdV, NLS, Boussinesq, but not sine-Gordon, for example). The potentials are germs of holomorphic matrix-valued functions, without any boundary conditions. The scattering data are matrix-valued formal power series in the spectral parameter. We give a precise description of all possible scattering data and exact criteria for solubility of the local holomorphic Cauchy problem for a soliton equation of parabolic type in terms of the scattering data of the initial conditions. Applications include the strongest possible version of the Painleve property (global meromorphic extension in $x$ of any local holomorphic solution), the issues related to trivial monodromy, and a proof of divergence of the Kontsevich--Witten series with respect to all higher times.