2018/06/01 by Deniz Bilman, Liming Ling, Peter D. Miller · 1 citation
Physics and Astronomy · Mathematics · #nlin.SI #math.AP #math.CA #nlin.PS #msc:35Q55 #msc:35Q15 #msc:35Q51 #msc:37K10 #msc:37K15 #msc:37K40 #msc:34M55
paper · pdf · doi:10.1215/00127094-2019-0066
published as Duke Math. J. 169, no. 4 (2020), 671-760 · 55 pages, 27 figures
arxiv created 2018/06/01 · arxiv updated 2020/03/18
We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently-proposed Riemann-Hilbert representation of the rogue wave solution of arbitrary order k, we establish the existence of a limiting profile of the rogue wave in the large-k limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schrödinger equation in the rescaled variables --- the rogue wave of infinite order --- which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlevé-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulæ with the exact solution with the help of numerical methods for solving Riemann-Hilbert problems. In a certain transitional region for the asymptotics the near field limit function is described by a specific globally-defined tritronquée solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.