Section: New Results

Stability analysis for rogue waves

Participant : Guillaume James.

The study of rogue waves (large amplitude waves localized both in space and time) has gained importance in various fields, such as the mathematical modeling of water waves and nonlinear optics. The analysis of their stability is delicate because of their transient nature. In the work [46], we introduce a new method to tackle this problem. Our approach relies on the approximation of rogue waves by large amplitude breathers (localized in space and time-periodic) having a large period, and the use of Floquet theory to analyze breather stability. This problem is examined for the nonlinear Schrödinger equation, which describes the envelope of nonlinear waves in a large class of systems, for example granular chains [15]. This model admits a family of breather solutions (Kuznetsov-Ma breathers) which converge to a rogue-wave profile (Peregrine soliton) when their period tends to infinity. We show numerically that the Floquet exponents of the breathers approach a finite limit for large periods, and observe that a motion of the localized wave can be induced by a dynamical instability. This work suggests an analytical way to define the spectral stability of the (transient) Peregrine soliton, but this remains an open problem to prove analytically the convergence of Floquet exponents in the limit of infinite period.