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

TL;DR

This paper proves the well-posedness of the KdV equation with initial profiles that decay on one side and are arbitrary on the other, using the Hirota { au}-function and spectral data analysis.

Contribution

It introduces a novel integrability framework for the KdV equation with arbitrary step-like initial data via the Hirota { au}-function representation.

Findings

Solution V(x,t) is real meromorphic in x for t>0.

The approach generalizes previous results by Cohen, Kappeler, Khruslov, Kotlyarov, Venakides, Zhang.

Under mild conditions, the initial value problem is strongly well-posed.

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.