Tau functions of solutions of soliton equations

Abstract

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.

Meromorphy of solutions for a wide class of ordinary differential equations of Painlevé type

“…But the converse is not true in general. A striking example of a hierarchy of such invariant manifolds was given by Bagderina [72] in the case of the Sawada-Kotera equation (7). The relation of these invariant manifolds to symmetries of ( 7) is highly non-trivial.…”

“…A direct verification shows that the system of ODE consisting of the equations ( 66) and the non-autonomous ODE U r = 0 (67) determines an invariant manifold (in the sense of Definition 2) of the evolutionary system consisting of the equation v t = u 4 − 30uu 2 + 60u 3 (68) and the SK equation (7).…”

“…In [75], the SK equation ( 7 By the main theorem in [5] (with an irrelevant inaccuracy corrected in [6]), all the coefficients of L (in our case, u(t, x)) for any local holomorphic solution extend to globally meromorphic functions of the spatial variable x. Since the system of ODE ( 66), ( 67) determines an invariant manifold of the system of evolution equations ( 68), (7), Theorem 1 yields a simultaneous solution of these two systems with initial conditions (33) in a neighborhood of the point t = t 0 , x = x 0 . The rest of the proof is clear.…”

“…As in [2], the main step in the proof of global meromorphy consists in using the meromorphic extendibility with respect to x of all local holomorphic solutions of the evolution equations ( 4) - (7). However, in contrast with [2], we consider not only ODE for self-similar solutions of IST-integrable evolution equations but also ODE that determine the so-called symmetries (and, more generally, invariant manifolds) of these evolution equations.…”

“…The main and almost the only obstacle is the necessity to establish that all local holomorphic solutions of the integrable partial differential equation admit a global meromorphic extension with respect to one of the independent variables. However, this was already done in [2], [4]- [9] for many equations including ( 4)- (7).…”