On the self-similar character of rogue waves

Abstract: Rogue waves are unexpectedly high waves which occur seemingly without a reason on a background of waves of the moderate amplitude. Most frequently, they are associated with the action of the nonlinear modulational instability of uniform waves with respect to weak long perturbations, and are considered within the frameworks of the nonlinear Schrödinger equation (NLSE). So-called Peregrine breathers, which are exact solutions of the NLSE, are considered to be the simplest mathematical prototypes of rogue waves. Other types of NLSE breather solutions are also known (named after E.A. Kuznetsov and N.N. Akhmediev). It should be noted that breather solutions have always been obtained either within the framework of the Inverse Scattering Technique or as a result of abstract mathematical constructions.

We discuss that from the general viewpoint, the shape of the most amplified due to the modulational instability envelope should possess a general form. Even more, the breather solutions are shown to be represented by fully coherent perturbations with self-similar shapes. The evolving modulations are characterized by constant values of the similarity parameter of the equation (i.e., the nonlinearity to dispersion ratio), just like classic solitons. Thus, breather solutions acquire a clear physical interpretation that is not based on the integrability property of the model. Approximate analytic breather-type solutions are obtained for non-integrable versions of the NLSE with different orders of nonlinearity. They are verified by the direct numerical simulation of the modulational instability.

Publications:

R.M. Rozental, A.V. Slunyaev, N.S. Ginzburg, A.S. Sergeev, I.V. Zotova, Self-similarity of rogue wave generation in gyrotrons: Beyond the Peregrine breather. Chaos, Solitons & Fractals 183, 114884 (2024).

A.V. Slunyaev, Breathers of the nonlinear Schrödinger equation are coherent self-similar solutions. Physica D 474, 134575 (2025).

C. Ward, P. Kevrekidis, Rogue waves as self-similar solutions on a background: a direct calculation. Romanian J. Phys. 64, 112 (2019).

Russianmathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods