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We study the formation of breathers in multi-dimensional lattices with nonlocal coupling that decays algebraically. By variational methods, the exact relationship between various parameters (dimension, nonlinearity, nonlocal parameter $\alpha$) that defines positive excitation thresholds is characterized. At the anti-continuum regime, there exists a family of unique ground states that determines excitation thresholds. We not only characterize the sharp spatial decay of ground states, which varies continuously in $\alpha$, but also identify the time decay of dispersive waves, which undergoes a discontinuous transition in $\alpha$.

The fractional discrete nonlinear Schr\"odinger equation (fDNLS) is studied on a periodic lattice from the analytic and dynamic perspective by varying the mesh size $h>0$ and the nonlocal L\'evy index $\alpha \in (0,2]$. We show that the discrete system converges to the fractional NLS as $h \rightarrow 0$ below the energy space by directly estimating the difference between the discrete and continuum solutions in $L^2(\mathbb{T})$ using the periodic Strichartz estimates. The sharp convergence rate via the finite-difference method is shown to be $O(h^{\frac{\alpha}{2+\alpha}})$ in the energy space. On the other hand for a fixed $h > 0$, the linear stability analysis on a family of continuous wave (CW) solutions reveals a rich dynamical structure of CW waves due to the interplay between nonlinearity, nonlocal dispersion, and discreteness. The gain spectrum is derived to understand the role of $h$ and $\alpha$ in triggering higher mode excitations. The transition from the quadratic dependence of maximum gain on the amplitude of CW solutions to the linear dependence, due to the lattice structure, is shown analytically and numerically.

The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations but not for nonlinear problems. We further show how to derive \emph{a priori} bounds on the size of the region of analyticity. This method is applied to the trypsin molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is sufficiently general enough to be applied to other nonlinear problems in uncertainty quantification., Comment: 28 pages,4 figures

Semi-linear elliptic Partial Differential Equations (PDEs) such as the non-linear Poisson Boltzmann Equation (nPBE) is highly relevant for non-linear electrostatics in computational biology and chemistry. It is of particular importance for modeling potential fields from molecules in solvents or plasmas with stochastic fluctuations. The extensive applications include ones in condensed matter and solid state physics, chemical physics, electrochemistry, biochemistry, thermodynamics, statistical mechanics, and materials science, among others. In this paper we study the complex analytic properties of semi-linear elliptic Partial Differential Equations with respect to random fluctuations on the domain. We first prove the existence and uniqueness of the nPBE on a bounded domain in $\mathbb{R}^3$. This proof relies on the application of a contraction mapping reasoning, as the standard convex optimization argument for the deterministic nPBE no longer applies. Using the existence and uniqueness result we subsequently show that solution to the nPBE admits an analytic extension onto a well defined region in the complex hyperplane with respect to the number of stochastic variables. Due to the analytic extension, stochastic collocation theory for sparse grids predict algebraic to sub-exponential convergence rates with respect to the number of knots. A series of numerical experiments with sparse grids is consistent with this prediction and the analyticity result. Finally, this approach readily extends to a wide class of semi-linear elliptic PDEs., Comment: arXiv admin note: substantial text overlap with arXiv:2106.05811

The continuum and discrete fractional nonlinear Schr\"odinger equations (fDNLS) represent new models in nonlinear wave phenomena with unique properties. In this paper, we focus on various aspects of localization associated to fDNLS featuring modulational instability, asymptotic construction of onsite and offsite solutions, and the role of Peierls-Nabarro barrier. In particular, the localized onsite and offsite solutions are constructed using the map approach. Under the long-range interaction characterized by the L\'{e}vy index $\alpha>0$, the phase space of solutions is infinite-dimensional unlike that of the well-studied nearest-neighbor interaction. We show that an orbit corresponding to this spatial dynamics translates to an approximate solution that decays algebraically. We also show as $\alpha \rightarrow \infty$, the discrepancy between local and nonlocal dynamics becomes negligible on a compact time interval, but persists on a global time scale. Moreover it is shown that data of small mass scatter to free solutions under a sufficiently high nonlinearity, which proves the existence of strictly positive excitation threshold for ground state solutions of fDNLS.

Providing rich and useful information regarding spectrum activities and propagation channels, radiomaps characterize the detailed distribution of power spectral density (PSD) and are important tools for network planning in modern wireless systems. Generally, radiomaps are constructed from radio strength measurements by deployed sensors and user devices. However, not all areas are accessible for radio measurements due to physical constraints and security consideration, leading to non-uniformly spaced measurements and blanks on a radiomap. In this work, we explore distribution of radio spectrum strengths in view of surrounding environments, and propose two radiomap inpainting approaches for the reconstruction of radiomaps that cover missing areas. Specifically, we first define a propagation-based priority and integrate exemplar-based inpainting with radio propagation model for fine-resolution small-size missing area reconstruction on a radiomap. Then, we introduce a novel radio depth map and propose a two-step template-perturbation approach for large-size restricted region inpainting. Our experimental results demonstrate the power of the proposed propagation priority and radio depth map in capturing the PSD distribution, as well as the efficacy of the proposed methods for radiomap reconstruction., Comment: submitted to IEEE journal for possible publication

In this paper, it is shown that all non-trivial solutions of a wide class of dispersive equations, defined by a dispersion relation with minimal hypotheses on its regularity, are compactly supported for at most one time element. With plane waves of different frequencies traveling at different speeds, the finite speed of propagation is not expected for dispersive phenomena, and this intuition is made rigorous by complex-analytic arguments. Furthermore the method developed here applies to time fractional-order systems given by the Caputo derivative where the generalized space-time fractional Schr\"odinger equation is considered for concreteness. When the energy operator $i\partial_t$ and the Riesz derivative $(-\Delta)^{\frac{\beta}{2}}$ are fractionalized to the same order, it is shown that the sharp frequency-localized dispersive estimates yield the time decay of solutions that depends on both the spatial dimension and the long-memory effect. Asymptotically in time, the solution operator strongly, but not uniformly, converges in $L^2(\mathbb{R}^d)$ to the half-wave operator.

Radio map describes network coverage and is a practically important tool for network planning in modern wireless systems. Generally, radio strength measurements are collected to construct fine-resolution radio maps for analysis. However, certain protected areas are not accessible for measurement due to physical constraints and security considerations, leading to blanked spaces on a radio map. Non-uniformly spaced measurement and uneven observation resolution make it more difficult for radio map estimation and spectrum planning in protected areas. This work explores the distribution of radio spectrum strengths and proposes an exemplar-based approach to reconstruct missing areas on a radio map. Instead of taking generic image processing approaches, we leverage radio propagation models to determine directions of region filling and develop two different schemes to estimate the missing radio signal power. Our test results based on high-fidelity simulation demonstrate efficacy of the proposed methods for radio map reconstruction., Comment: To appear in 2022 IEEE Global Communications Conference (Globecom)

We prove that the solutions to the discrete Nonlinear Schr\"odinger Equation (DNLSE) with non-local algebraically-decaying coupling converge strongly in $L^2(\mathbb{R}^2)$ to those of the continuum fractional Nonlinear Schr\"odinger Equation (FNLSE), as the discretization parameter tends to zero. The proof relies on sharp dispersive estimates that yield the Strichartz estimates that are uniform in the discretization parameter. An explicit computation of the leading term of the oscillatory integral asymptotics is used to show that the best constants of a family of dispersive estimates blow up as the non-locality parameter $\alpha \in (1,2)$ approaches the boundaries., Comment: Revised Article

We investigate the well-posedness theory of the 2-D fractional nonlinear Schr\"odinger equation (NLSE) with a mixed degree of derivatives. Motivated by models in optics and photonics where the light propagation is governed by non-quadratic, fractional, and anisotropic dispersion profile, this paper presents first results in this direction. Dispersive estimates are developed in the context of anisotropic Sobolev spaces defined by inhomogeneous symbols. The main model is shown to exhibit scattering for small data in the scaling-critical space. Furthermore the continuity of solution with respect to the dispersion parameter is shown on a compact time interval., Comment: Revised Article

We prove the existence and uniqueness of the complexified Nonlinear Poisson-Boltzmann Equation (nPBE) in a bounded domain in $\mathbb{R}^3$. The nPBE is a model equation in nonlinear electrostatics. The standard convex optimization argument to the complexified nPBE no longer applies, but instead, a contraction mapping argument is developed. Furthermore, we show that uniqueness can be lost if the hypotheses given are not satisfied. The complixified nPBE is highly relevant to regularity analysis of the solution of the real nPBE with respect to the dielectric (diffusion) and Debye-H\"uckel coefficients. This approach is also well-suited to investigate the existence and uniqueness problem for a wide class of semi-linear elliptic Partial Differential Equations (PDEs).

We consider the compact case of one-dimensional quantum Zakharov system, as an initial-value problem with periodic boundary conditions. We apply the Bourgain norm method to show low regularity local well-posedness for a certain class of Sobolev exponents that are sharp up to the boundary, under the condition that Schr\"odinger Sobolev regularity is non-negative. Using the conservation law and energy method, we show global well-posedness for sufficiently regular initial data, without any smallness assumption. Lastly we show the semi-classical limit as $\epsilon \to 0$ on a compact time interval, whereas the quantum perturbation proves to be singular on an infinite time interval., Comment: 18 pages

Systems that require high-throughput and fault tolerance, such as key-value stores and databases, are looking to persistent memory to combine the performance of in-memory systems with the data-consistent fault-tolerance of nonvolatile stores. Persistent memory devices provide fast bytea-ddressable access to non-volatile memory. We analyze the design space when integrating persistent memory into in-memory key value stores and quantify performance tradeoffs between throughput, latency, and and recovery time. Previous works have explored many design choices, but did not quantify the tradeoffs. We implement persistent memory support in Redis and Memcached, adapting the data structures of each to work in two modes: (1) with all data in persistent memory and (2) a hybrid mode that uses persistent memory for key/value data and non-volatile memory for indexing and metadata. Our experience reveals three actionable design principles that hold in Redis and Memcached, despite their very different implementations. We conclude that the hybrid design increases throughput and decreases latency at a minor cost in recovery time and code complexity

We prove modified Strichartz estimates on the one-dimensional torus, that are adapted to a fourth-order dispersion relation, and use them to show global well-posedness of nonlinear fourth-order Schr\"odinger equations. This extends the (low regularity) existence theory of the adiabatic transition of the Quantum Zakharov system to NLS. We show that the solutions behave continuously with respect to the quantum parameter in every compact time interval. Globally in time, however, we also show that such continuous dependence is generally not uniform., Comment: This article is absorbed into "Remark on the Adiabatic Limit of Quantum Zakharov System" arXiv:1906.10807

We consider the Carleson's problem regarding small time almost everywhere convergence to initial data for the Schr\"odinger equation, both linear and nonlinear on $\mathbb{R}$. It is shown, via the smoothing effect of the Schr\"odinger flow, that the (sharp) result proved by Dahlberg and Kenig for initial data in Sobolev spaces still holds when one considers the full Schr\"odinger equation with a certain class of potentials. As for $s<\frac{1}{4}$, the failure of $L^p$-boundedness of the (localized) maximal operator is investigated., Comment: Revised article

This paper is concerned with the low regularity well-posedness of the adiabatic limit of the quantum Zakharov system, which is a modified nonlinear Schr\"odinger equation (NLSE). As the quantum parameter tends to zero, the modified NLSE formally converges to the standard NLSE, and we show this rigorously., Comment: Revised article