VARIATIONAL PRINCIPLE FOR THE ZAKHAROV-SHABAT EQUATIONS

We put forth a variational principle for the Zakharov-Shabat (ZS) equa tions, which is the basis of the inverse scattering transform of a num ber of important nonlinear PDE's. Using the variational representation of the ZS equations, we develop an approximate analytical technique f or finding discrete eigenvalues of the complex spectral parameter in t he ZS equations for a given pulse-shaped potential, which is equivalen t to the physically important problem of finding the soliton content o f the given initial pulse. We apply the technique to several particula r shapes of the pulse and demonstrate that the simplest version of the variational approximation, based on trial functions with one or two f ree parameters, proves to be fully analytically tractable, and it yiel ds threshold conditions for the appearance of the first soliton, or of the first soliton pair, which are in a fairly good agreement with ava ilable numerical results. However, more free parameters are necessary to allow prediction of additional solitons produced by the given pulse .