The Hirota {\tau}-function and well-posedness of the KdV equation with an arbitrary step like initial profile decaying on the right half line

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.