Behavior of solitons in random-function solutions of the periodic Korteweg–de Vries equation

I develop a general approach for computing random-function solutions of the periodic Korteweg–de Vries (KdV) equation using the inverse scattering transform (IST) in the hyperelliptic function representation. I exploit IST to construct realizations of KdV random processes which have power-law spectra, ${\mathit{k}}^{\mathrm{-}\gamma }$ (k the wave number, γ a constant), and uniformly distributed random IST phases on (-π,π). IST characterizes these realizations in terms of solitons moving in a sea of background radiation and is thus able to extract solitons, by nonlinear filtering techniques, from complex, random motions described by the KdV equation.