Extreme Superposition: Rogue Waves of Infinite Order and the PainlevéIII Hierarchy
Abstract
We study the fundamental rogue wave solutions of the focusing nonlinear Schrödinger equation in the limit of large order. Using a recentlyproposed 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 formulæ\ with the exact solution with the help of numerical methods for solving RiemannHilbert problems. In a certain transitional region for the asymptotics the near field limit function is described by a specific globallydefined tritronquée solution of the PainlevéII equation. These properties lead us to regard the rogue wave of infinite order as a new special function.
 Publication:

arXiv eprints
 Pub Date:
 June 2018
 arXiv:
 arXiv:1806.00545
 Bibcode:
 2018arXiv180600545B
 Keywords:

 Nonlinear Sciences  Exactly Solvable and Integrable Systems;
 Mathematics  Analysis of PDEs;
 Mathematics  Classical Analysis and ODEs;
 Nonlinear Sciences  Pattern Formation and Solitons;
 35Q55;
 35Q15;
 35Q51;
 37K10;
 37K15;
 37K40;
 34M55
 EPrint:
 55 pages, 27 figures