RogueWaveInfiniteNLS.jl

Julia package for computing the general rogue waves of infinite order for the focusing nonlinear Schrödinger equation. This is a family of special solutions of the focusing nonlinear equation denoted by $\Psi(X,T;\mathbf{G},\beta)$, whose exact and asymptotic properties (in different regimes where $(X,T)\in\mathbb{R}^2$ becomes large) were studied recently in the manuscript

General rogue waves of infinite order: exact properties, asymptotic behavior, and effective numerical computation [D. Bilman & P. D. Miller, 2024].

Importantly these solutions all lie in $L^2(\mathbb{R})\setminus L^1(\mathbb{R})$ and they exhibit quite slow temporal decay: $\Psi(X,T;\mathbf{G},\beta)\sim |T|^{-1/3}$ as $T\to\pm\infty$. In the sense that they decay slower than pure radiation generated by generic initial data in $L^1(\mathbb{R})$ ($\sim t^{-1/2}$).

The family of solutions is parametrized by a $2\times 2$ matrix $\mathbf{G}$ and a scalar $\beta>0$. The matrix $\mathbf{G}$ is given by

where $c^*$ denotes the complex conjugate of $c\in\mathbb{C}$.

The main routine is psi. The command

psi(0.8, 1.2, 1, 2im, 1)

computes $\Psi(X,T;\mathbf{G},\beta)$ at $(X,T)=(0.8,1.2)$ with $a=1$ and $b=2i$, and scalar $\beta=1$. So the syntax for the end user is psi(X, T, a, b, beta).

$\Psi(X,T;\mathbf{G},\beta)$ is defined in terms of the solution of a Riemann-Hilbert problem that depends parametrically on the arguments $(X,T;\mathbf{G},\beta)$. Depending on the location of $(X,T)$ in $\mathbb{R}^2$, a suitably regularized (deformed using the noncommutative steepest descent techniques) Riemann-Hilbert problem implemented as part of this package is picked automatically and it is solved numerically using OperatorApproximation.jl.

One can compute the timeslice at $T=0$ in a parallelized manner via:

using Distributed
addprocs()
using RogueWaveInfiniteNLS
@everywhere Xlist = collect(-16:0.05:16)
@everywhere atest = 0.8im
@everywhere btest = 1
@everywhere compute_psi_slice = X->psi(X,0,atest,btest,1)
outpsiX = pmap(compute_psi_slice, Xlist)

The variable outpsiX holds a vector of (complex) values of $\Psi(X,T=0,a=0.8i,b=1,\beta=1)$ for $-16\leq X \leq 16$ with a mesh-size of $dX=0.05$. Then one can plot, say, the absolute value of the computed solution via

using Plots
using LaTeXStrings
abs_outpsiX = abs.(outpsiX)
plot(Xlist, abs_outpsiX, label=L"|\Psi(X,0,a=0.8i,b=1,\beta=1)|", xlabel=L"X", ylabel=L"|\Psi|", legend=:topright, linewidth=2.5, linecolor="darkblue")

This work is supported by the National Science Foundation of the United States under the award NSF DMS-2108029.