Application of the Darboux transform to spectral problems in ocean acoustics

We propose a generalization of the n-order Darboux transform, which can be applied to solving spectral problems in ocean acoustics in cases when the medium can be considered stratified with respect to density. For a second-order Darboux transform with the boundary conditions characteristic of a single-layer shallow sea model, conditions are determined under which the spectra of the initial and transformed spectral problems coincide. It is demonstrated that, within the framework of this model, it is possible to use the n-order Darboux transform for constructing sound velocity profiles admitting exact solutions.