The nonlinear Schrödinger equation (NLSE) stands out as the dispersive nonlinear partial differential equation that plays prominent role in the modeling and understanding of features relevant to many fields of nonlinear physics. The question of random input problems in the one-dimensional and integrable NLSE enters within the framework of integrable turbulence [Agafontsev:15,Randoux:14,Walczak:15], and the specific question of formation of rogue waves has been recently extensively studied in this context [Toenger:15]. The determination of exact analytic solutions of the focusing 1D-NLSE prototyping rogue waves of statistical relevance is now considered as the problem of central importance. Here we address this question from the perspective of the inverse scattering transform (IST) method that relies on the integrable nature of the wave equation. We develop a conceptually new approach to the rogue wave identification in which appropriate, locally coherent structures are specifically isolated from a globally incoherent wave train to be subsequently analyzed by implementing a numerical IST procedure relying on a spatial periodization of the object under consideration. Using this approach we extend the existing classifications of the prototypes of statistically relevant rogue waves from standard solitons on finite background (SFBs) and their collisions to more general nonlinear modes characterized by the finite-band spectra.

References

[Agafontsev:15] Agafontsev, D.S. and Zakharov, V.E. Integrable turbulence and formation of rogue waves. Nonlinearity 28, 2791 (2015).

[Randoux:14] Randoux, S., Walczak, P., Onorato, M. and Suret, P. Intermittency in integrable turbulence. Phys. Rev. Lett. 113, 113902 (2014).

[Walczak:15] Walczak, P., Randoux, S. and Suret, P. Optical rogue waves in integrable turbulence. Phys. Rev. Lett. 114, 143903 (2015).

[Toenger:15] Toenger, S. et al. Emergent rogue wave structures and statistics in spontaneous modulation instability. Scientific Reports 5 (2015).