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This paper is devoted to a comprehensive analysis of a family of solutions of the focusing nonlinear Schrödinger equation called general rogue waves of infinite order. These solutions have recently been shown to describe various limit processes involving large-amplitude waves, and they have also appeared in some physical models not directly connected with nonlinear Schrödinger equations. We establish the following key property of these solutions: they are all in $$L^2(\mathbb{R})$$ with respect to the spatial variable but they exhibit anomalously slow temporal decay. In this paper, we define general rogue waves of infinite order, establish their basic exact and asymptotic properties, and provide computational tools for calculating them accurately. 

We consider the problem of computing a class of soliton gas primitive potentials for the Korteweg-de Vries equation that arise from the accumulation of solitons on an infinite interval in the physical domain, extending to $$-\infty$$. This accumulation results in an associated Riemann-Hilbert Problem (RHP) on a number of disjoint intervals. In the case where the jump matrices have specific square-root behaviour, we describe an efficient and accurate numerical method to solve this RHP and extract the potential. The keys to the method are, first, the deformation of the RHP, making numerical use of the so-called $$g$$-function, and, second, the incorporation of endpoint singularities into the chosen basis to discretize and solve the associated singular integral equation. 

We study fundamental rogue-wave solutions of the focusing nonlinear Schr\"odinger 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 $$x\to\pm\infty$$. The large-order far-field asymptotic behavior of the solution depends on which of three disjoint regions $$\mathcal{C}$$ (the ``channels''), $$\mathcal{S}$$ (the ``shelves''), and $$\mathcal{E}$$(the ``exterior domain'') contains the rescaled variables. On the region \mathcal{C}, the amplitude is small and the solution is highly oscillatory, while on the region \mathcal{S}, the solution is approximated by a modulated plane wave with a highly oscillatory correction term. The asymptotic behavior on these two domains is the same for all continuous orders. Assuming that the order belongs to the discrete sequence characteristic of rogue-wave solutions, the asymptotic behavior of the solution on the region $$\exterior$$ resembles that on \mathcal{S} but without the oscillatory correction term. Solutions of other continuous orders behave quite differently on $$\mathcal{E}$$. 

RogueWaveInfiniteNLS v0.1.0 Initial release with the fixes.