Some remarks on a WKB method for the nonselfadjoint Zakharov–Shabat eigenvalue problem with analytic potentials and fast phase

A formal method for approximating eigenvalues of the nonselfadjoint Zakharov–Shabat eigenvalue problem in the semiclassical scaling is described. Analyticity of the potential is assumed and appears to be crucial. The method involves finding appropriate paths between pairs of complex turning points, and reproduces the Y-shaped spectra observed by Bronski [Physica D 97 (1996) 376]. An application to all-optical ultrashort pulse generation is briefly described, and the kind of tools that are required to make the results rigorous are indicated.