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The Hirota τ-function and well-posedness of the KdV equation with an arbitrary step-like initial profile decaying on the right half line
We are concerned with the Cauchy problem for the KdV equation on the whole line with an initial profile V0 which is decaying sufficiently fast at +∞ and arbitrarily enough (i.e. no decay or pattern of behaviour) at −∞. We show that this system is completely integrable in a very …