Rogue waves generation through multiphase solutions degeneration for the derivative nonlinear Schrödinger equation

Abstract

The generation of rogue waves from the development of modulational instability of the plane waves due to small perturbations or interactions of solitons is usually modeled by some breather solutions, which are considered to as solitons on a finite background. The instability of small-amplitude perturbations is usually chaotic and may contain many frequencies in their spectra. Thus, a more general description of rogue wave generation can be achieved via the consideration of multiphase solutions and their interactions. The general N-order phase solutions of the derivative nonlinear Schrodinger (DNLS) equation are constructed from the trivial seed (zero solution) by using the determinant representation of the N-fold Darboux transformation. By adjusting the relative phases of the multiphase solutions in the interacting area, namely taking the limit $$\lambda _k \rightarrow \lambda _\mathrm{c}$$ for some of $$\lambda _k$$ (not for all of them), where $$\lambda _\mathrm{c}$$ is a critical eigenvalue associated with the synchronization of each phase in the multiphase nonlinear waves of the DNLS equation, it is possible to obtain different types of quasi-rational solutions from the multiphase solutions degeneration mechanism: quasi-rational traveling wave solutions, rogue waves, and quasi-rogue waves with periodic conditions. Hence, this multiphase solution degeneration represents a new mechanism of rogue wave generation.