# 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

**rogue waves**: Peregrine breather and its higher-order generalizations, which are (pseudo-rational) solutions associated with higher-order singularities at the branch points of the continuous spectrum of the associated Zakharov-Shabat operator - see the description of the preceding project A Robust Inverse Scattering Transform for Arbitrary Spectral Singularities

**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

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:

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.