rogue-waves swMATH ID: 32953 Software Authors: Bilman, Deniz; Ling, Liming; Miller, Peter D. Description: 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. Homepage: https://arxiv.org/abs/1806.00545 Source Code: https://github.com/bilman/rogue-waves Keywords: rogue waves; equations and hierarchies of Painlevé type; nonlinear Schrödinger equation; Riemann-Hilbert problems; spectral singularities Related Software: DLMF; RHPackage; DiffSharp; L-BFGS; ISTPackage; GitHub Cited in: 24 Publications Standard Articles 1 Publication describing the Software, including 1 Publication in zbMATH Year Extreme superposition: rogue waves of infinite order and the Painlevé-III hierarchy. Zbl 1437.35617Bilman, Deniz; Ling, Liming; Miller, Peter D. 2020 all top 5 Cited by 32 Authors 7 Ling, Liming 4 Bilman, Deniz 4 He, Jingsong 4 Rao, Jiguang 3 Cheng, Yi 3 Guo, Boling 3 Liu, Nan 3 Miller, Peter D. 3 Zhang, Xiaoen 2 Buckingham, Robert J. 2 Chen, Yong 2 Feng, Baofeng 2 Mihalache, Dumitru 2 Yan, Zhenya 1 Chen, Jinbing 1 Chen, Yiren 1 Dong, Jieyang 1 Fokas, Athanassios S. 1 Girotti, Manuela 1 Grava, Tamara 1 Jiang, Ying 1 McLaughlin, Kenneth D. T.-R. 1 Pelinovsky, Dmitry Efimovich 1 Peng, Weiqi 1 Shavlukov, Azamat Mavletovich 1 Suleĭmanov, Bulat Irekovich 1 Wang, Dengshan 1 Wang, Yao 1 Weng, Weifang 1 White, Robert E. 1 Xiong, Zhi-Jin 1 Zhang, Guoqiang all top 5 Cited in 13 Serials 7 Physica D 3 Journal of Nonlinear Science 2 Journal of Differential Equations 2 Applied Mathematics Letters 2 Communications in Nonlinear Science and Numerical Simulation 1 Communications in Mathematical Physics 1 Letters in Mathematical Physics 1 Physics Letters. A 1 Duke Mathematical Journal 1 Nonlinear Dynamics 1 Analysis and Applications (Singapore) 1 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 1 Ufimskiĭ Matematicheskiĭ Zhurnal all top 5 Cited in 10 Fields 22 Partial differential equations (35-XX) 14 Dynamical systems and ergodic theory (37-XX) 4 Ordinary differential equations (34-XX) 2 Special functions (33-XX) 2 Quantum theory (81-XX) 1 Numerical analysis (65-XX) 1 Computer science (68-XX) 1 Fluid mechanics (76-XX) 1 Statistical mechanics, structure of matter (82-XX) 1 Geophysics (86-XX) Citations by Year