A rigorous trace-type representation for the KdV solution is obtained via left reflection coefficients and Hankel operators on the Hardy space for L1 and L2 summable half-line supported initial data.
Rybkin, The Hirota τ-function and well-posedness of the KdV equation with an arb itrary step-like initial profile decaying on the right half-line , Nonlinearity 24 (2011), 2953–2990
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On the inverse scattering transform for the KdV equation with summable initial data
A rigorous trace-type representation for the KdV solution is obtained via left reflection coefficients and Hankel operators on the Hardy space for L1 and L2 summable half-line supported initial data.