Extreme superposition: Rogue waves of infinite order and the Painlevé-III hierarchy

Abstract

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 formulae with the exact solution with the help of 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.

Large-Order Asymptotics for Multiple-Pole Solitons of the Focusing Nonlinear Schrödinger Equation

“…This regime was analyzed by two of the authors in [1], the first large-n analysis of nth-order pole solitons. The asymptotic solution seems to be a type of universal behavior, also appearing in the study of high-order Peregrine breathers for the NLS equation with constant, non-zero boundary conditions [2].…”

“…Define First, set τ = 0 and 0 < χ < 2 β . Then ϕ(λ) has two real distinct critical points λ (1) and λ (2) , where we choose λ (1) < λ (2) (the third critical point is at infinity). See Figure 8.…”

“…The algebraic-decay region (with χ > 0) consists of those χ and τ values that can be reached by continuously varying χ and τ with no two critical points coinciding. In this region if τ = 0 then ϕ(λ) has three distinct real critical points, which we label λ (0) < λ (1) < λ (2) if τ > 0 and λ (1) < λ (2) < λ (0) if τ < 0. The region is bounded by the locus of points in the (χ,τ )-plane satisfying…”

“…The arc with endpoints P − and P + passes through the point 2 β , 0 on the χ-axis and is denoted by L AE . This arc is a boundary between the algebraic-decay and the exponential-decay regions and corresponds to λ (1) = λ (2) . The arc from P 0 to P + is denoted by L + AN (and corresponds to λ (1) = λ (0) ), while that from P 0 to P − is denoted by L − AN (and corresponds to λ (2) = λ (0) ).…”

On the optimal focusing of solitons and breathers in long‐wave models

“…The description of the limiting case of an "extreme superposition" when N → ∞ was provided in Ref. 37. Obviously, such analogies can contribute to the studies of rogue-wave-type solutions of the NLS and mKdV equations.…”