Role of NLS anomalous wave solutions in non‑integrable multidimensional NLS‑type models

Determine whether the anomalous (rogue) wave solutions of the 1+1 dimensional focusing nonlinear Schrödinger equation—specifically the Akhmediev breather and the Peregrine soliton—play any dynamical role in multidimensional generalizations of the nonlinear Schrödinger equation that are non‑integrable, including the elliptic and hyperbolic NLS equations in 2+1 and 3+1 dimensions relevant to water waves, nonlinear optics, and plasma physics.

Background

The paper studies quasi one‑dimensional anomalous (rogue) waves in multidimensional generalizations of the focusing nonlinear Schrödinger equation, such as the elliptic and hyperbolic NLS equations in 2+1 and 3+1 dimensions. These models are frequently non‑integrable in physically relevant contexts (e.g., ocean waves and nonlinear optics), raising foundational questions about the applicability of exact anomalous wave solutions known from the integrable 1+1 dimensional NLS.

The authors develop a universal leading‑order description of the first nonlinear stage of modulation instability in the quasi one‑dimensional regime using adiabatic deformations of the Akhmediev breather. This advances understanding of multidimensional dynamics, but the broader question of whether the canonical NLS anomalous wave solutions themselves have a dynamical role in non‑integrable multidimensional settings remains explicitly stated as unclear.