Rogue Wave Formation and Asymptotics for Large-Order Coherent Structures

In the article A Robust IST for the Focusing NLS Equation, we introduced a robust inverse scattering transform (IST) that can handle arbitrary spectral singularities or poles of arbitrary order in a simple and unified way. With this development, we now for the first time have Riemann-Hilbert (RH) problem representations for solutions of the nonlinear Schrödinger equation that may involve coherent structures such as

High-order rogue wave

  • multi-pole solitons: also reflectionless solutions, which are associated with $N^{\text{th}}$-order ($N\geq 2$, $n\in\mathbb{Z}$) pole singularities in the complement of the continuous spectrum Multi-pole soliton

In Extreme Superposition: Rogue Waves of Infinite Order and the Painlevé-III Hierarchy we establish the existence of a limiting profile of rogue waves in the limit of large order 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 (the $\Psi^{\pm}$ function) 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.

Time evolution of the rogue wave of infinite order $\Psi(X,T)$ can be seen in the video below:

The rogue wave of infinite order

In Extreme Superposition: Rogue Waves of Infinite Order and the Painlevé-III Hierarchy we also compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formulæ 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 tritronqueé solution of the Painlevé-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function, the $\Psi^{\pm}$ function, characterized by its Riemann-Hilbert problem representation given in Extreme Superposition: Rogue Waves of Infinite Order and the Painlevé-III Hierarchy.

In Large-order asymptotics for multiple-pole solitons of the focusing nonlinear Schrödinger equation we establish that the very same $\Psi^{\pm}$ function describes the near-field asymptotics coalescing multi-pole solitons (on a zero-background) as the order of the pole becomes large under a certain choice of norming constants. In general, the near-field behavior is again given in terms of certain members of the Painlevé-III hierarchy, albeit with different boundary conditions, having $\Psi^\pm$ as a special case. In this work we also compute the boundary of quiescent regions exactly, and use the nonlinear steepest-descent method to prove the asymptotic limit of the solitons is zero in these regions.

Deniz Bilman
Deniz Bilman
Assistant Professor

Assistant Professor of Mathematics. Research interests include nonlinear waves, Riemann-Hilbert problems.