Geodesic
Tau functions of solutions of soliton equations
Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 367-387.

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In the holomorphic version of the inverse scattering method, we prove that the determinant of a Toeplitz-type Fredholm operator arising in the solution of the inverse problem is an entire function of the spatial variable for all potentials whose scattering data belong to a Gevrey class strictly less than 1. As a corollary, we establish that, up to a constant factor, every local holomorphic solution of the Korteweg–de Vries equation is the second logarithmic derivative of an entire function of the spatial variable. We discuss the possible order of growth of this entire function. Analogous results are given for all soliton equations of parabolic type.
Keywords: holomorphic solution, analytic continuation.
Mots-clés : soliton equation 
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A. V. Domrin. Tau functions of  solutions of  soliton equations. Izvestiya. Mathematics , Tome 85 (2021) no. 3, pp. 367-387. http://geodesic.mathdoc.fr/item/IM2_2021_85_3_a3/

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