Résumé : The complete integrability of the nonlinear Schrodinger equation (NLSE) via inverse scattering transform enables the decomposition of the initial conditions into elementary nonlinear modes : the spatially localized solitons/breathers and continuous spectrum waves and quasi-periodic finite gap solutions. Numerical simulations of the NLSE statistics always imply periodic boundary conditions. However, when the size of the periodic box is large the existence of localized solitons/breathters is possible. Such solitons/breathers under periodic boundary conditions correspond to the case of very narrow eigenvalue gaps. To study the role of solitons in the dynamics and statistics of a periodic wave field we create a gas of solitons of high density. We use a special stable numerical scheme to generate statistical ensembles of 128 strongly interacting solitons, i.e. solve the inverse scattering problem for the high number of discrete eigenvalues. Then we use this ensembles as initial conditions for numerical simulations in a large numerical box with periodic boundary conditions and study statistics of the obtained uniform strongly interacting soliton gas. We also study the role of breathers in the development of modulation instability from randomly perturbed quasimonochromatic plane wave. We solve the Zakharov-Shabat eigenvalue problem with high numerical accuracy and demonstrate typical eigenvalue portraits that reveal the coexistence of finite gap quasiperiodic waves and spatially localized breathers.
