[Paper Review] Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

Deniz Bilman, Peter D. Miller|arXiv (Cornell University)|Feb 27, 2021

Nonlinear Waves and Solitons7 citations

TL;DR

This paper studies high-order fundamental rogue waves in the focusing nonlinear Schrödinger equation in the far-field regime, where spatial and temporal variables scale linearly with the rogue wave order. Using a continuous Riemann-Hilbert problem formulation, it identifies three distinct asymptotic regions—C, S, and E—showing that large-order behavior is universal in C and S, but exhibits unique discrete-order-dependent features in E, with convergence to an infinite-order rogue wave solution in the limit of large order.

ABSTRACT

We study fundamental rogue-wave solutions of the focusing nonlinear Schrödinger equation in the limit that the order of the rogue wave is large and the independent variables $(x,t)$ are proportional to the order (the far-field limit). We first formulate a Riemann-Hilbert representation of these solutions that allows the order to vary continuously rather than by integer increments. The intermediate solutions in this continuous family include also soliton solutions for zero boundary conditions spectrally encoded by a single complex-conjugate pair of poles of arbitrary order, as well as other solutions having nonzero boundary conditions matching those of the rogue waves albeit with far slower decay as $x o\pm\infty$. The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions $\mathcal{C}$, $\mathcal{S}$, and $\mathcal{E}$ contains the rescaled variables. On the regions $\mathcal{C}$ and $\mathcal{S}$ we show that the asymptotic behavior is the same for all continuous orders, while in the region $\mathcal{E}$ the discrete sequence of rogue-wave orders produces distinctive asymptotic behavior that is different from other cases.

Motivation & Objective

• To analyze the large-order, far-field asymptotic behavior of fundamental rogue wave solutions in the focusing nonlinear Schrödinger equation.
• To extend the rogue wave solution framework from discrete integer orders to a continuous family of orders using Riemann-Hilbert problems.
• To identify and characterize three distinct asymptotic regions (C, S, E) in the rescaled space-time plane where the solution's behavior differs qualitatively.
• To establish the existence and properties of a limiting infinite-order rogue wave solution in the large-order far-field limit.

Proposed method

• Formulates a continuous Riemann-Hilbert problem for rogue wave solutions, allowing the order to vary continuously rather than in discrete integer steps.
• Introduces a spectral parameter transformation using the function $ \rho(\lambda) $, defined by $ \rho(\lambda)^2 = \lambda^2 + 1 $, with $ \rho(\lambda) \sim \lambda $ as $ \lambda \to \infty $.
• Uses a matrix-valued function $ \mathbf{E}(\lambda) $, constructed from $ f(\lambda) $ and $ B(\lambda) $, to encode the jump conditions on the contours $ \Sigma_c $ and $ \Sigma_\circ $.
• Applies the Riemann-Hilbert problem with jump conditions involving $ \mathrm{e}^{2i\rho_+(\lambda)(x+\lambda t)\sigma_3} $ on $ \Sigma_c $ and a more complex expression on $ \Sigma_\circ $ involving $ B(\lambda)^{sn\sigma_3} $.
• Defines the rogue wave via the limit $ \psi(x,t) = 2i \lim_{\lambda \to \infty} \lambda M_{12}^{(k)}(\lambda;x,t) $, ensuring normalization and rationality of the solution.
• Analyzes the large-order far-field limit by rescaling $ x = n^{-1}X $, $ t = n^{-2}T $, and studies the convergence of $ s n^{-1} \psi_k(n^{-1}X, n^{-2}T) $ to a limiting function $ \Psi(X,T) $.

Experimental results

Research questions

• RQ1 How does the asymptotic behavior of high-order rogue waves in the far-field regime depend on the rescaled space-time variables?
• RQ2 What is the nature of the limiting solution as the rogue wave order tends to infinity in the far-field scaling regime?
• RQ3 How do the asymptotic behaviors differ across the three disjoint regions $ \mathcal{C} $, $ \mathcal{S} $, and $ \mathcal{E} $ in the rescaled plane?
• RQ4 Can the discrete sequence of rogue wave orders be embedded into a continuous family of solutions via a Riemann-Hilbert formulation?

Key findings

• The large-order far-field asymptotic behavior of the rogue wave solution depends on the rescaled variables' location in one of three disjoint regions: $ \mathcal{C} $, $ \mathcal{S} $, or $ \mathcal{E} $.
• In regions $ \mathcal{C} $ and $ \mathcal{S} $, the asymptotic behavior is universal and independent of the continuous order, showing the same limiting form for all orders.
• In region $ \mathcal{E} $, the discrete sequence of integer orders produces distinct asymptotic behavior not seen in the continuous family, indicating a unique discrete effect.
• The solution converges to a limiting infinite-order rogue wave $ \Psi(X,T) $ in the large-order far-field limit, satisfying the focusing nonlinear Schrödinger equation with $ \mathrm{i}\Psi_T + \frac{1}{2}\Psi_{XX} + |\Psi|^2\Psi = 0 $.
• The continuous Riemann-Hilbert formulation includes soliton solutions with zero boundary conditions and other solutions with nonzero boundary conditions matching the rogue wave's asymptotic state but with slower decay.
• The analysis reveals that the solution's structure is governed by the interplay between spectral poles and the geometry of the jump contours $ \Sigma_c $ and $ \Sigma_\circ $, with the sign $ s = (-1)^k $ encoding parity of the order.