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We are concerned with the Cauchy problem for the KdV equation on the wholeline with an initial profile V_0 which is decaying sufficiently fast at +\inftyand arbitrarily enough (i.e., no decay or pattern of behavior) at -\infty. Weshow 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 certainscattering and spectral data suitably defined in terms of the Titchmarsh-Weylm-functions associated with the two half-line Schr\"odinger operatorscorresponding to V_0. We show that V(x,t) is real meromorphic with respect to xfor any t>0. We also show that under a very mild additional condition on V_0representation implies a strong well-posedness of the KdV equation with suchV_0's. Among others, our approach yields some relevant results due to Cohen,Kappeler, Khruslov, Kotlyarov, Venakides, Zhang and others.

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