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.
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References in zbMATH (referenced in 5 articles , 1 standard article )
Showing results 1 to 5 of 5.
- Bilman, Deniz; Ling, Liming; Miller, Peter D.: Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy (2020)
- Liu, Nan; Guo, Boling: Solitons and rogue waves of the quartic nonlinear Schrödinger equation by Riemann-Hilbert approach (2020)
- Bilman, Deniz; Buckingham, Robert: Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation (2019)
- Zhang, Xiaoen; Chen, Yong: Inverse scattering transformation for generalized nonlinear Schrödinger equation (2019)
- Miller, Peter D.: On the increasing tritronquée solutions of the Painlevé-II equation (2018)