2017/08/12 by P. G. Grinevich, P. M. Santini · 3 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Breather #Cauchy distribution #Cauchy problem #Initial value problem #Instability #Mathematical analysis #Mathematical physics #Mathematics #Modulational instability #Monochromatic color #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Nonlinear system #Optics #Perturbation (astronomy) #Physics #Quantum mechanics #math-ph #math.MP #nlin.SI #physics.ao-ph #physics.optics
paper · pdf · doi:10.1016/j.physleta.2018.02.014
published as Physics Letters A Volume 382, Issue 14, 12 April 2018, Pages 973-979 · 20 pages. arXiv admin note: text overlap with arXiv:1708.00762 and substantial text overlap with arXiv:1707.05659
arxiv created 2017/08/12 · openalex created_date 2017/08/31 · openalex publication_date 2018/02/12 · arxiv updated 2018/05/01 · openalex updated_date 2026/08/05
The focusing Nonlinear Schrödinger (NLS) equation is the simplest universal model describing the modulation instability (MI) of quasi monochromatic waves in weakly nonlinear media, the main physical mechanism for the generation of rogue (anomalous) waves (RWs) in Nature. In this paper we investigate the x-periodic Cauchy problem for NLS for a generic periodic initial perturbation of the unstable constant background solution, in the case of N=1,2 unstable modes. We use matched asymptotic expansion techniques to show that the solution of this problem describes an exact deterministic alternate recurrence of linear and nonlinear stages of MI, and that the nonlinear RW stages are described by the N-breather solution of Akhmediev type, whose parameters, different at each RW appearence, are always given in terms of the initial data through elementary functions. This paper is motivated by a preceeding work of the authors in which a different approach, the finite gap method, was used to investigate periodic Cauchy problems giving rise to RW recurrence.