2021/02/27 by Deniz Bilman, Bilman, Deniz, Peter D. Miller +1 · 2 citations
Physics and Astronomy · #35Q15 #35Q51 #35Q55 #37K10 #37K15 #37K40 #Advanced Fiber Laser Technologies #Analysis of PDEs (math.AP) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Pattern Formation and Solitons (nlin.PS)
paper · pdf · doi:10.48550/arxiv.2103.00337
openalex publication_date 2021/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables (x,t) are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as x→±∞. The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions C, S, and E contains the rescaled variables. On the regions C and S we show that the asymptotic behavior is the same for all continuous orders, while in the region E the discrete sequence of rogue-wave orders produces distinctive asymptotic behavior that is different from other cases.