Rogue waves formation by solitons synchronization and resonance: Gerdjikov-Ivanov equation

Rogue waves usually possess localized large-amplitude waves and coming from nowhere and disappearing without any trace. The Peregrine solitons with the above characteristics are regarded as rogue waves prototype. The paper mainly studies the effects of multiple solitons in the formation of rogue waves for the Gerdjikov-Ivanov equation on vanishing boundary and nonvanishing boundary conditions by the N -fold Darboux transformation with the pure imaginary spectral parameters: (i) under the vanishing boundary conditions, all spectral parameters close to one value, and combined with the number of Darboux transformations, the solitons synchronization interactions can generate Peregrine-like solitons. (ii) under the nonvanishing boundary conditions, the spectral parameters λ k → λ c 1 ( λ c 2 ) , where λ c 1 and λ c 2 are a pair of key eigenvalues associated with the phase resonance of the solitons, the solitons resonance and degradation interactions enable to obtain Peregrine-like solitons, which can be considered as the generalization of Peregrine solitons. These results also well illustrate the relationship between the generation of rogue waves and spectral parameters and boundary conditions and provide a new idea for rogue waves excitation.