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In the absence of solitons, the nonlinear Schrödinger equation has an asymptotic solution which decays in time as t−1/2, and contains two arbitrary functions (in the amplitude and phase, respectively). For appropriate initial data, the amplitude function is uniquely determined in terms of the initial data by the conservation laws; the other function is undetermined. This method determines the leading two terms in each of the asymptotic expansions for the amplitude and phase, but no more. The method makes no direct use of th Marchenko integral equations.
Segur et al. (Sat,) studied this question.