Abstract
A previous treatment that holds for the Korteweg--de Vries equation is extended to cover the case of nonlinear integrable equations associated with the standard Zakharov-Shabat eigenvalue problem that has a complex discrete spectrum. Particularly, an analytical expression for the distribution function of solitons as a functional of the initial conditions is found. This distribution function gives the correct values of the infinite set of constants of motion and leads to a large number of conclusions that agree with previous numerical and analytical results. Special emphasis is given to a comparison of these results in the case of the derivative nonlinear Schr\"odinger equation. The distribution function is particularly useful for the statistical description of the nonlinear equations involved in the formalism (such as the nonlinear Schr\"odinger or the derivative nonlinear Schr\"odinger equations) when an ensemble of initial conditions is considered.
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