The modulational instability of the focusing nonlinear Schrödinger equation has been proposed as a possible mechanism in the formation of rogue waves on the ocean and other nonlinear dispersive systems. In this paper, the focus is on the pole dynamics associated with the modulational instability. A range of pole behaviours is identified, ranging from simple periodic motion along the imaginary axis when the nonlinear parameter is relatively small, to motions that trace complicated trajectories in the complex plane when the parameter is larger. Possibly of relevance to rogue wave theories is the situation where several poles have a near-collision close to the real axis, leading to a disproportionately high peak in the modulus of the solution. In the absence of explicit solution formulas, this study is based on computational pole tracking using Fourier–Padé methods.
