Modulational instability analysis of the Peregrine soliton

We perform modulational instability analysis of the Peregrine soliton. The eigensystem of the linearized perturbations results in time-dependent gain curve. The instantaneous stability of the Peregrine soliton is studied at different times and in terms of modulated spacial width. A correlation between the most unstable eigenmode and the time evolution of the Peregrine soliton is established. Our analysis explains the bifurcation of the Peregrine soliton into Ma breathers and the generation of shock waves. The theoretical approach and numerical procedure followed here may be applied to any other localized solution with nontrivial time dependence.