Role of 1D NLS anomalous-wave solutions in non-integrable multidimensional NLS models

Determine whether the anomalous-wave solutions of the 1+1-dimensional focusing nonlinear Schrödinger equation (such as the Akhmediev breather and the Peregrine instanton) play a significant dynamical role in the evolution of multidimensional, non-integrable NLS-type equations, including the elliptic and hyperbolic NLS equations in 2+1 and 3+1 dimensions.

Background

The paper studies the first nonlinear stage of modulation instability of quasi one-dimensional anomalous (rogue) waves in multidimensional generalizations of the focusing NLS equation. Many physically relevant multidimensional NLS-type models (e.g., elliptic and hyperbolic NLS in 2+1 and 3+1 dimensions) are non-integrable, which raises the question of whether exact anomalous-wave solutions from the integrable 1D NLS (like the Akhmediev breather and Peregrine instanton) are relevant outside the integrable setting.

The authors develop a universal leading-order description in the quasi one-dimensional regime via adiabatic deformations of the Akhmediev breather, showing processes such as growth, fission, and fusion. However, beyond this regime and in general non-integrable multidimensional settings, the fundamental issue of whether 1D NLS anomalous-wave solutions play a role remains explicitly uncertain.