This paper studies how spatially periodic rogue waves—large, sudden peaks in wave amplitude—appear again and again in a class of mathematical wave models that extend the familiar nonlinear Schrödinger equation to more than one space dimension. The authors work in a regime they call quasi one dimensional, where the waves vary rapidly along a main direction x but more slowly across the transverse directions. Their central finding is that the very first nonlinear growth of small disturbances is almost the same for many models, but the later recurrence of the rogue waves can differ significantly between models and become increasingly complex.

The models considered are multidimensional nonlinear Schrödinger (MNLS) equations that are physically relevant in two and three spatial dimensions. These include both elliptic and hyperbolic generalizations of the standard focusing nonlinear Schrödinger equation. “Rogue” or anomalous waves (AWs) in these models are x-periodic patterns that can undergo modulation instability—the tendency of a steady pattern to break up when small perturbations grow. Modulation instability is the mechanism that leads to large amplitude peaks from small initial ripples.

In earlier work the authors showed that the first nonlinear stage of modulation instability is universal across these MNLS models: it is well described by slow changes (adiabatic deformations) of a special exact waveform of the integrable one-dimensional NLS equation called the Akhmediev breather. The Akhmediev breather is a known solution that captures the formation of a large periodic peak from a small modulation. In the present paper they concentrate on what happens after that first stage—how the pattern returns or “recurs” over longer times.

They find that recurrence dynamics are not universal. Different MNLS models show order-one (i.e., substantial) differences in how and when waves come back. As the system evolves through successive nonlinear stages, peaks can split (fission) and merge (fusion) in various combinations, producing richer and more complicated sequences of events. To explain and predict this behaviour the authors use a mathematical tool called finite gap perturbation theory. In plain terms, this is an analytic method that treats the multidimensional models as small perturbations of the integrable one-dimensional NLS and tracks how the known exact waves deform under those perturbations.