Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy. 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.

References in zbMATH (referenced in 24 articles , 1 standard article )

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  1. Bilman, Deniz; Miller, Peter D.: Broader universality of rogue waves of infinite order (2022)
  2. Dong, Jieyang; Ling, Liming; Zhang, Xiaoen: Kadomtsev-Petviashvili equation: one-constraint method and lump pattern (2022)
  3. Feng, Bao-Feng; Ling, Liming: Darboux transformation and solitonic solution to the coupled complex short pulse equation (2022)
  4. Peng, Wei-Qi; Chen, Yong: (N)-double poles solutions for nonlocal Hirota equation with nonzero boundary conditions using Riemann-Hilbert method and PINN algorithm (2022)
  5. Rao, Jiguang; He, Jingsong; Cheng, Yi: The Davey-Stewartson I equation: doubly localized two-dimensional rogue lumps on the background of homoclinic orbits or constant (2022)
  6. Rao, Jiguang; Mihalache, Dumitru; He, Jingsong; Cheng, Yi: Dynamics of general higher-order rogue waves in the two-component nonlinear Schrödinger equation coupled to the Boussinesq equation (2022)
  7. Bilman, Deniz; Buckingham, Robert; Wang, Deng-Shan: Far-field asymptotics for multiple-pole solitons in the large-order limit (2021)
  8. Chen, Yiren; Feng, Bao-Feng; Ling, Liming: The robust inverse scattering method for focusing Ablowitz-Ladik equation on the non-vanishing background (2021)
  9. Girotti, M.; Grava, T.; Jenkins, R.; McLaughlin, K. D. T.-R.: Rigorous asymptotics of a KdV soliton gas (2021)
  10. Jiang, Ying; Rao, Jiguang; Mihalache, Dumitru; He, Jingsong; Cheng, Yi: Rogue breathers and rogue lumps on a background of dark line solitons for the Maccari system (2021)
  11. Liu, Nan; Guo, Boling: Painlevé-type asymptotics of an extended modified KdV equation in transition regions (2021)
  12. Liu, Nan; Guo, Boling: Asymptotics of solutions to a fifth-order modified Korteweg-de Vries equation in the quarter plane (2021)
  13. Rao, Jiguang; Fokas, Athanassios S.; He, Jingsong: Doubly localized two-dimensional rogue waves in the Davey-Stewartson I equation (2021)
  14. Suleĭmanov, Bulat Irekovich; Shavlukov, Azamat Mavletovich: Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation (2021)
  15. Weng, Weifang; Yan, Zhenya: Inverse scattering and (N)-triple-pole soliton and breather solutions of the focusing nonlinear Schrödinger hierarchy with nonzero boundary conditions (2021)
  16. Zhang, Guoqiang; Ling, Liming; Yan, Zhenya: Multi-component nonlinear Schrödinger equations with nonzero boundary conditions: higher-order vector Peregrine solitons and asymptotic estimates (2021)
  17. Zhang, Xiaoen; Ling, Liming: Asymptotic analysis of high-order solitons for the Hirota equation (2021)
  18. Bilman, Deniz; Ling, Liming; Miller, Peter D.: Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy (2020)
  19. Chen, Jinbing; Pelinovsky, Dmitry E.; White, Robert E.: Periodic standing waves in the focusing nonlinear Schrödinger equation: rogue waves and modulation instability (2020)
  20. Liu, Nan; Guo, Boling: Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann-Hilbert approach (2020)

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