roguewaves
swMATH ID:  32953 
Software Authors:  Bilman, Deniz; Ling, Liming; Miller, Peter D. 
Description:  Extreme superposition: rogue waves of infinite order and the PainlevéIII hierarchy. We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recently proposed RiemannHilbert 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 nearfield 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 PainlevéIII hierarchy. We compute the farfield asymptotic behavior of the nearfield limit solution and compare the asymptotic formulas with the exact solution using numerical methods for solving RiemannHilbert problems. In a certain transitional region for the asymptotics, the nearfield limit function is described by a specific globally defined tritronquée solution of the PainlevéII 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/roguewaves 
Keywords:  rogue waves; equations and hierarchies of Painlevé type; nonlinear Schrödinger equation; RiemannHilbert problems; spectral singularities 
Related Software:  DLMF; RHPackage; DiffSharp; LBFGS; 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 PainlevéIII hierarchy. Zbl 1437.35617 Bilman, Deniz; Ling, Liming; Miller, Peter D. 
2020

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Cited by 32 Authors
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Cited in 13 Serials
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