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The WKB approximation of semiclassical eigenvalues of the Zakharov-Shabat problem

We numerically compute eigenvalues of the non-self-adjoint Zakharov--Shabat problem in the semiclassical regime. In particular, we compute the eigenvalues for a Gaussian potential and compare the results to the corresponding (formal) WKB approximations used in the approach to the semiclassical or zero-dispersion limit of the focusing nonlinear Schroedinger equation via semiclassical soliton ensembles. This numerical experiment, taken together with recent numerical experiments [17,18], speaks directly to the viability of this approach; in particular, our experiment suggests a value for the rate of convergence of the WKB eigenvalues to the true eigenvalues in the semiclassical limit. This information provides some hint as to how these approximations might be rigorously incorporated into the asymptotic analysis of the singular limit for the associated nonlinear partial differential equation.