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 15 articles , 1 standard article )

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  1. Bilman, Deniz; Buckingham, Robert; Wang, Deng-Shan: Far-field asymptotics for multiple-pole solitons in the large-order limit (2021)
  2. Girotti, M.; Grava, T.; Jenkins, R.; McLaughlin, K. D. T.-R.: Rigorous asymptotics of a KdV soliton gas (2021)
  3. 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)
  4. Liu, Nan; Guo, Boling: Painlevé-type asymptotics of an extended modified KdV equation in transition regions (2021)
  5. Liu, Nan; Guo, Boling: Asymptotics of solutions to a fifth-order modified Korteweg-de Vries equation in the quarter plane (2021)
  6. Rao, Jiguang; Fokas, Athanassios S.; He, Jingsong: Doubly localized two-dimensional rogue waves in the Davey-Stewartson I equation (2021)
  7. Suleĭmanov, Bulat Irekovich; Shavlukov, Azamat Mavletovich: Integrable Abel equation and asymptotics of symmetry solutions of Korteweg-de Vries equation (2021)
  8. 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)
  9. Zhang, Guoqiang; Ling, Liming; Yan, Zhenya: Multi-component nonlinear Schrödinger equations with nonzero boundary conditions: higher-order vector Peregrine solitons and asymptotic estimates (2021)
  10. Bilman, Deniz; Ling, Liming; Miller, Peter D.: Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy (2020)
  11. Liu, Nan; Guo, Boling: Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann-Hilbert approach (2020)
  12. Wang, Yao; Xiong, Zhi-Jin; Ling, Liming: Fokas-Lenells equation: three types of Darboux transformation and multi-soliton solutions (2020)
  13. Bilman, Deniz; Buckingham, Robert: Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation (2019)
  14. Zhang, Xiaoen; Chen, Yong: Inverse scattering transformation for generalized nonlinear Schrödinger equation (2019)
  15. Miller, Peter D.: On the increasing tritronquée solutions of the Painlevé-II equation (2018)