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1. Extreme Superposition: High-Order Fundamental Rogue Waves in the Far-Field Regime

   https://doi.org/10.1090/memo/1505  

   Bilman, Deniz; Miller, Peter (August 2024, Memoirs of the American Mathematical Society)

   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 $$\mathcal{E}$$. 

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2. A focusing and defocusing semi-discrete complex short-pulse equation and its various soliton solutions

   https://doi.org/10.1098/rspa.2020.0853  

   Feng, Bao-Feng; Ling, Liming; Zhu, Zuonong (March 2021, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   In this paper, we are concerned with a semi-discrete complex short-pulse (sdCSP) equation of both focusing and defocusing types, which can be viewed as an analogue to the Ablowitz–Ladik lattice in the ultra-short-pulse regime. By using a generalized Darboux transformation method, various soliton solutions to this newly integrable semi-discrete equation are studied with both zero and non-zero boundary conditions. To be specific, for the focusing sdCSP equation, the multi-bright solution (zero boundary conditions), multi-breather and high-order rogue wave solutions (non-zero boundary conditions) are derived, while for the defocusing sdCSP equation with non-zero boundary conditions, the multi-dark soliton solution is constructed. We further show that, in the continuous limit, all the solutions obtained converge to the ones for its original CSP equation (Ling et al . 2016 Physica D 327 , 13–29 ( doi:10.1016/j.physd.2016.03.012 ); Feng et al . 2016 Phys. Rev. E 93 , 052227 ( doi:10.1103/PhysRevE.93.052227 )). 

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3. Long‐time asymptotics and the radiation condition with time‐periodic boundary conditions for linear evolution equations on the half‐line and experiment

   https://doi.org/10.1111/sapm.12665  

   Mao, Yifeng; Mantzavinos, Dionyssios; Hoefer, Mark A. (December 2023, Studies in Applied Mathematics)

   Abstract The asymptotic Dirichlet‐to‐Neumann (D‐N) map is constructed for a class of scalar, constant coefficient, linear, third‐order, dispersive equations with asymptotically time/periodic Dirichlet boundary data and zero initial data on the half‐line, modeling a wavemaker acting upon an initially quiescent medium. The large timetasymptotics for the special cases of the linear Korteweg‐de Vries and linear Benjamin–Bona–Mahony (BBM) equations are obtained. The D‐N map is proven to be unique if and only if the radiation condition that selects the unique wave number branch of the dispersion relation for a sinusoidal, time‐dependent boundary condition holds: (i) for frequencies in a finite interval, the wave number is real and corresponds to positive group velocity, and (ii) for frequencies outside the interval, the wave number is complex with positive imaginary part. For fixed spatial locationx, the corresponding asymptotic solution is (i) a traveling wave or (ii) a spatially decaying, time‐periodic wave. The linearized BBM asymptotics are found to quantitatively agree with viscous core‐annular fluid experiments. 

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4. Holder estimates for kinetic Fokker-Planck equations up to the boundary

   Luis Silvestre (December 2022, Ars inveniendi analytica)

   We obtain local Holder continuity estimates up to the boundary for a kinetic Fokker-Planck equation with rough coefficients, with the prescribed influx boundary condition. Our result extends some recent developments that incorporate De Giorgi methods to kinetic Fokker-Planck equations. We also obtain higher order asymptotic estimates near the incoming part of the boundary. In particular, when the equation has a zero boundary conditions and no source term, we prove that the solution vanishes at infinite order on the incoming part of the boundary. 

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5. Inference for Gaussian Processes with Matern Covariogram on Compact Riemannian Manifolds

   Li, D; Tang, W; Banerjee, S (March 2023, Journal of machine learning research)

   Gaussian processes are widely employed as versatile modelling and predictive tools in spa- tial statistics, functional data analysis, computer modelling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on Gaussian processes over Riemannian manifolds in order to develop richer and more flexible inferential frameworks for non-Euclidean data. While numerical approximations through graph representations have been well studied for the Mat´ern covariogram and heat kernel, the behaviour of asymptotic inference on the param- eters of the covariogram has received relatively scant attention. We focus on asymptotic behaviour for Gaussian processes constructed over compact Riemannian manifolds. Build- ing upon a recently introduced Mat´ern covariogram on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Mat´ern Gaussian random measures on compact manifolds to derive the parameter that is identifiable, also known as the microergodic parameter, and formally establish the consistency of the maximum like- lihood estimate and the asymptotic optimality of the best linear unbiased predictor. The circle is studied as a specific example of compact Riemannian manifolds with numerical experiments to illustrate and corroborate the theory 

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