The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step like initial profile decaying on the right half line (1012.0052v1)

Abstract: We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V_0 which is decaying sufficiently fast at +\infty and arbitrarily enough (i.e., no decay or pattern of behavior) at -\infty. We show that this system is completely integrable in a very strong sense. Namely, the solution V(x,t) admits the Hirota {\tau}-function representation V(x,t)=-2\partial_{x}^2 logdet(I+M_{x,t}) where M_{x,t} is a Hankel integral operator constucted from certain scattering and spectral data suitably defined in terms of the Titchmarsh-Weyl m-functions associated with the two half-line Schr\"odinger operators corresponding to V_0. We show that V(x,t) is real meromorphic with respect to x for any t>0. We also show that under a very mild additional condition on V_0 representation implies a strong well-posedness of the KdV equation with such V_0's. Among others, our approach yields some relevant results due to Cohen, Kappeler, Khruslov, Kotlyarov, Venakides, Zhang and others.