EXTREME SUPERPOSITION: ROGUE WAVES OF INFINITE ORDER AND THE PAINLEVE-III HIERARCHY

We study the fundamental rogue wave solutions of the focusing nonlinear Schrodinger 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 Schrodinger 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 Painleve-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 tritronquee solution of the Painleve-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function.