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We propose a modification of the standard inverse scattering transform forthe focusing nonlinear Schr\"odinger equation (also other equations by naturalgeneralization) formulated with nonzero boundary conditions at infinity. Thepurpose is to deal with arbitrary-order poles and potentially severe spectralsingularities in a simple and unified way. As an application, we use themodified transform to place the Peregrine solution and related higher-order"rogue wave" solutions in an inverse-scattering context for the first time.This allows one to directly study properties of these solutions such as theirdynamical or structural stability, or their asymptotic behavior in the limit ofhigh order. The modified transform method also allows rogue waves to begenerated on top of other structures by elementary Darboux transformations,rather than the generalized Darboux transformations in the literature or otherrelated limit processes.

A robust inverse scattering transform for the focusing nonlinear Schr\"odinger equation