Eur. Phys. J. Plus (2026) 141: 99
https://doi.org/10.1140/epjp/s13360-025-07276-y

Regular Article

Quasi one dimensional anomalous (rogue) waves in multidimensional nonlinear Schrödinger equations: fission and fusion

¹ Dipartimento di Fisica, Università di Roma “La Sapienza” and Istituto Nazionale di Fisica Nucleare (INFN), Piazz.le Aldo Moro 2, 00185, Rome, Italy
² Department of Mathematics, University at Buffalo, Mathematics Building UB North Campus, Buffalo, NY, USA

^a This email address is being protected from spambots. You need JavaScript enabled to view it. , This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 2 November 2025
Accepted: 30 December 2025
Published online: 30 January 2026

Abstract

In this paper we study the first nonlinear stage of modulation instability (NLSMI) of x-periodic anomalous waves (AWs) in multidimensional generalizations of the focusing nonlinear Schrödinger (NLS) equation, like the non-integrable elliptic and hyperbolic NLS equations in $2+1$ and $3+1$ dimensions. In the quasi one-dimensional (Q1D) regime, where the wavelength in the x direction of propagation is significantly smaller than in the transversal directions, the behavior at leading order is universal, independent of the particular model, and described by adiabatic deformations of the Akhmediev breather solution of NLS. Varying the initial data, the first NLSMI shows various combinations of basic processes, like AW growth from the unstable background, followed by fission in the slowly varying transversal directions, and the inverse process of fusion, followed by AW decay to the background. Fission and fusion are critical processes showing similarities with multidimensional wave breaking, and with phase transitions of second kind and critical exponent 1/2. In $3+1$ dimensions with radial symmetry in the transversal plane, fission consists in the formation of an opening smoke ring, while if the symmetry is hyperbolic in the transversal plane, the growing Q1D AW is an X-wave undergoing fission into branches of hyperbolas. In the long wave limit, the Q1D Akhmediev breather reduces to the Q1D analogue of the Peregrine instanton, rationally localized in space. Numerical experiments on the hyperbolic NLS equation show that the process of “AW growth + fission” is not restricted to the Q1D regime, extending to a finite region of the modulation instability domain. At last, we pose and solve the “inverse time-scattering problem of AWs”: the reconstruction of the $O(ϵ)$ initial perturbation of the background, from the knowledge of the first nonlinear stages of modulation instability for positive and negative times. The universality of these processes suggests their observability in natural phenomena related to AWs in contexts such as water waves, nonlinear optics, plasma physics, and Bose–Einstein condensates.