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15 March 2020 Extreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchy
Deniz Bilman, Liming Ling, Peter D. Miller
Duke Math. J. 169(4): 671-760 (15 March 2020). DOI: 10.1215/00127094-2019-0066


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 formulas with the exact solution using 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.


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Deniz Bilman. Liming Ling. Peter D. Miller. "Extreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchy." Duke Math. J. 169 (4) 671 - 760, 15 March 2020.


Received: 5 June 2018; Revised: 2 August 2019; Published: 15 March 2020
First available in Project Euclid: 28 January 2020

zbMATH: 07198464
MathSciNet: MR4072637
Digital Object Identifier: 10.1215/00127094-2019-0066

Primary: 35Q55
Secondary: 34M55, 35Q15, 35Q51, 37K10, 37K15, 37K40

Rights: Copyright © 2020 Duke University Press


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Vol.169 • No. 4 • 15 March 2020
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