Laplacian Research Articles

Modulation instability in nonidentical metamaterial waveguide arrays by graph Laplacian approach

We theoretically investigate modulation instability (MI) in a nonidentical waveguide array, which is made up of positive and negative index metamaterial waveguides. The unit cell of the optical waveguide array consists of three waveguides arranged in a triangular manner. Waveguides 1 and 3 are made up of positive index material (PIM) channels and waveguide 2 is by negative index material (NIM) channels, as a result, they show different light propagation characteristics. We model this array of waveguides using a generalized nonlinear Schro¨dinger equation, replacing the Laplacian operator with the graph Laplacian. Following linear stability analysis, we will discuss MI for different values of transverse wave number, as it determines the order of the Brillouin zone. We also discuss the effect of input power on periodic MI in normal and anomalous dispersion regimes. Thus we report a comprehensive study on the MI and hence the better ways to generate and manipulate the solitons or ultra-short pulses in NIM PIM waveguide arrays.

Nonlinear Eigenvalue Problems for (p, 2)-Equations and Laplace Equations with Perturbations

We consider two types of nonlinear eigenvalue problems involving Laplace and p-Laplace operators (p>2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p>2)$$\\end{document}. The main result establishes the existence of at least two nontrivial weak solutions in the case of the perturbed equation and the existence of a continuous spectrum in the case of the (p, 2)-equation. In both cases, variational methods play a central role in our arguments.

Symmetrization for Mixed Operators

Abstract In this paper, we prove Talenti’s comparison theorem for mixed local/nonlocal elliptic operators and derive the Faber–Krahn inequality for the first eigenvalue of the Dirichlet mixed local/nonlocal problem. Our findings are relevant to the fractional p&q−Laplacian operator.

On the geometric trace of a generalized Selberg trace formula

Abstract A certain generalization of the Selberg trace formula was proved by the first named author in 1999. In this generalization instead of considering the integral of K ⁢ ( z , z ) {K(z,z)} (where K ⁢ ( z , w ) {K(z,w)} is an automorphic kernel function) over the fundamental domain, one considers the integral of K ⁢ ( z , z ) ⁢ u ⁢ ( z ) {K(z,z)u(z)} , where u ⁢ ( z ) {u(z)} is a fixed automorphic eigenfunction of the Laplace operator. This formula was proved for discrete subgroups of PSL ⁢ ( 2 , ℝ ) {\mathrm{PSL}(2,\mathbb{R})} , and just as in the case of the classical Selberg trace formula it was obtained by evaluating in two different ways (“geometrically” and “spectrally”) the integral of K ⁢ ( z , z ) ⁢ u ⁢ ( z ) {K(z,z)u(z)} . In the present paper we work out the geometric side of a further generalization of this generalized trace formula: we consider the case of discrete subgroups of PSL ⁢ ( 2 , ℝ ) n {\mathrm{PSL}(2,\mathbb{R})^{n}} where n > 1 {n>1} . Many new difficulties arise in the case of these groups due to the fact that the classification of conjugacy classes is much more complicated for n > 1 {n>1} than in the case n = 1 {n=1} .

Hypergraph p-Laplacians and Scale Spaces

AbstractThe aim of this paper is to revisit the definition of differential operators on hypergraphs, which are a natural extension of graphs in systems based on interactions beyond pairs. In particular, we focus on the definition of Laplacian and p-Laplace operators for oriented and unoriented hypergraphs, their basic properties, variational structure, and their scale spaces. We illustrate that diffusion equations on hypergraphs are possible models for different applications such as information flow on social networks or image processing. Moreover, the spectral analysis and scale spaces induced by these operators provide a potential method to further analyze complex data and their multiscale structure. The quest for spectral analysis and suitable scale spaces on hypergraphs motivates in particular a definition of differential operators with trivial first eigenfunction and thus more interpretable second eigenfunctions. This property is not automatically satisfied in existing definitions of hypergraph p-Laplacians, and we hence provide a novel axiomatic approach that extends previous definitions and can be specialized to satisfy such (or other) desired properties.

Lipschitz continuity of the solutions to the Dirichlet problems for the invariant Laplacians

This short note is motivated by an attempt to understand the distinction between the Laplace operator and the hyperbolic Laplacian on the unit ball of Rn, regarding the Lipschitz continuity of the solutions to the corresponding Dirichlet problems.We investigate the Dirichlet problem{Δϑu=0, in Bn,u=ϕ, on Sn−1, whereΔϑ:=(1−|x|2){1−|x|24Δ+ϑ∑j=1nxj∂∂xj+ϑ(n2−1−ϑ)I}.We show that: (i) when ϑ>0, the Lipschitz continuity of boundary data always implies the Lipschitz continuity of the solutions; (ii) when ϑ=0, there exists a Lipschitz continuous function ϕ:Sn−1→R such that the solution is not Lipschitz continuous; (iii) when ϑ<0, the solution is definitely not Lipschitz continuous unless ϕ≡0.

Asymptotics for a parabolic problem of Kirchhoff type with singular critical exponential nonlinearity

AbstractThe main objective of this paper is to characterize stable sets based on the asymptotic behavior of solutions as goes to infinity for the following class of parabolic Kirchhoff equations: where is a bounded domain with a Lipschitz boundary, , , , , , and is the fractional ‐Laplacian operator, .

A linearized L2-1[formula omitted] Galerkin FEM for Kirchhoff type quasilinear subdiffusion equation with memory

This article aims to construct a novel linearized Galerkin FEM based on Alikhanov’s L2-1σ scheme to solve a Kirchhoff type quasilinear subdiffusion equation with memory Dα. Near time t=0, the weak singularity of the solution to the equation Dα is taken into account by using the graded mesh in the time direction. The nonlocal behavior of Kirchhoff term causes huge computer storage thereby increasing computational cost. We reduce these costs by linearizing the Kirchhoff term. We derive a priori bounds and global rates of convergence for the developed numerical scheme. We prove that the convergence rate of the proposed numerical scheme is first-order in space and second-order in time in L∞(0,T;L2(Ω)). In addition, we prove the same convergence rate in L∞(0,T;H01(Ω)) by making use of a discrete Laplacian operator. A numerical experiment is presented to verify the theoretical results.

Three weak solutions for a class of quasilinear choquard equations involving the fractional p(x, .)-Laplacian operator with weight

In this paper, we establish the existence of at least three weak solutions to a problem involving the the fractional p(x, .)- Laplacian operator with weight. Our method used for obtaining the existence of three solutions for a class of Choquard equations is based on the variational method concerned a type of version of Ricceri.

Superdiffusion of energetic particles at shocks: A fractional diffusion and Lévy flight model of spatial transport

The observed power laws in space and time profiles of energetic particles in the heliosphere can be the result of an underlying superdiffusive transport behavior. Such anomalous, non-Gaussian transport regimes can arise, for example, as a consequence of intermittent structures in the solar wind. Non-diffusive transport regimes may also play a critical role in other astrophysical environments such as supernova remnant shocks. To clarify the role of superdiffusion in the transport of particles near shocks, we study the solutions of a fractional diffusion-advection equation to investigate this issue. A fractional generalization of the Laplace operator, the Riesz derivative, provides a model of superdiffusive propagation. e vy motion. The expected power law profiles of particles upstream of the plasma shock, where particles are injected, can be reproduced with this approach. The method provides a full, time-dependent solution of the fractional diffusion-advection equation. The developed models enable a quantitative comparison to energetic particle properties based on a comprehensive, superdiffusive transport equation and allow for an application in a number of scenarios in astrophysics and space science.

Laplacian atom search optimization algorithm: development and application for harmonic estimator design

Harmonics, are the major source of contaminations in fundamental voltage and current signals. For consumer satisfaction and for better equipment performance, these contaminations shall be identified and mitigated. Although, for accurate identification of harmonics, several methods have been proposed, yet metaheuristic based approaches have been used and in practice, since last two decades. The work reported here is based on a newly proposed optimization algorithm named as Atom Search Optimisation (ASO). In the first phase, a variant of ASO is proposed, based on Laplacian operator based position update mechanism named as Laplacian-ASO (L-ASO) for enhancing the performance of ASO and in the next phase, application of a newly developed L-ASO is carried out on harmonic estimator design problems. The improvisation of our proposed L-ASO is validated through conducted analyses and results showcased in the discussion section.

On axially rational regular functions and Schur analysis in the Clifford-Appell setting

In this paper we start the study of Schur analysis for Cauchy–Fueter regular quaternionic-valued functions, i.e. null solutions of the Cauchy–Fueter operator in R4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^4$$\\end{document}. The novelty of the approach developed in this paper is that we consider axially regular functions, i.e. functions spanned by the so-called Clifford-Appell polynomials. This type of functions arises naturally from two well-known extension results in hypercomplex analysis: the Fueter mapping theorem and the generalized Cauchy–Kovalevskaya (GCK) extension. These results allow one to obtain axially regular functions starting from analytic functions of one real or complex variable. Precisely, in the Fueter theorem two operators play a role. The first one is the so-called slice operator, which extends holomorphic functions of one complex variable to slice hyperholomorphic functions of a quaternionic variable. The second operator is the Laplace operator in four real variables, that maps slice hyperholomorphic functions to axially regular functions. On the other hand, the generalized CK-extension gives a characterization of axially regular functions in terms of their restriction to the real line. In this paper we use these two extensions to define two notions of rational function in the regular setting. For our purposes, the notion coming from the generalized CK-extension is the most suitable. Our results allow to consider the Hardy space, Schur multipliers and their relation with realizations in the framework of Clifford-Appell polynomials. We also introduce two notions of regular Blaschke factors, through the Fueter theorem and the generalized CK-extension.

Nonlinear Fokker–Planck equations with fractional Laplacian and McKean–Vlasov SDEs with Lévy noise

AbstractThis work is concerned with the existence of mild solutions to nonlinear Fokker–Planck equations with fractional Laplace operator $$(- \Delta )^s$$ ( - Δ ) s for $$s\in \left( \frac{1}{2},1\right) $$ s ∈ 1 2 , 1 . The uniqueness of Schwartz distributional solutions is also proved under suitable assumptions on diffusion and drift terms. As applications, weak existence and uniqueness of solutions to McKean–Vlasov equations with Lévy noise, as well as the Markov property for their laws are proved.

Global regularity and decay behavior for Leray equations with critical-dissipation and its application to self-similar solutions

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our approach hinges on the maximal smoothing effect, L p L^{p} -type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations. Most notably, we prove the global regularity and the optimal decay without the need for additional requirements found in existing literatures.

On the existence of positive solutions of some nonlocal elliptic problems involving fractional p–Laplacian operator

Our aim in this paper is to analyze the existence of solutions to a nonlocal elliptic problem involving the fractional Laplacian operator. These operators are utilized to solve an equation defined within a bounded domain  in . The operator  is a fractional Laplacian , where  with the condition  and  is a non-negative function almost everywhere with respect to the variable , and  are real numbers. The article establishes two existence theorems for weak solutions using Tychonoff and Schauder fixed-point theorems. These theorems are formulated based on different hypotheses regarding the parameters and .

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

Characterization of Tubular Surfaces in Terms of Finite III-type

In this paper, we first define relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E 3 . Then, we will characterize the tubular surfaces in terms of their coordinate finite type.

On a class of fractional Γ(.)-Kirchhoff-Schrödinger system type

This paper focuses on the investigation of a Kirchhoff-Schrödinger type elliptic system involving a fractional \(\gamma(.)\)-Laplacian operator. The primary objective is to establish the existence of weak solutions for this system within the framework of fractional Orlicz-Sobolev Spaces. To achieve this, we employ the critical point approach and direct variational principle, which allow us to demonstrate the existence of such solutions. The utilization of fractional Orlicz-Sobolev spaces is essential for handling the nonlinearity of the problem, making it a powerful tool for the analysis. The results presented herein contribute to a deeper understanding of the behavior of this type of elliptic system and provide a foundation for further research in related areas.

On the Pohozaev identity for the fractional p$p$‐Laplacian operator in RN$\mathbb {R}^N$

AbstractIn this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional ‐Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.

Spectral Graph Convolutional Neural Networks in the Context of Regularization Theory.

Graph convolutional neural networks (GCNN) have been widely used in graph learning and related applications. It has been identified that the filters in the state-of-the-art spectral graph convolutional networks (SGCNs) are essentially low-pass filters that enforce smoothness across the graph and use the functions of graph Laplacian as a tool that injects graph structure into the learning algorithm. There had been research findings that connect the smoothness functional in graphs, graph Laplacian, and regularization operators. We review the existing SGCNs in this context and propose a framework where the state-of-the-art filter designs can be deduced as special cases. We designed new filters that are associated with well-defined low-pass behavior and tested their performance on semisupervised node classification tasks. Their performance was found to be superior to that of the other state-of-the-art techniques. We further investigate the representation capability of low-pass features and make useful observations. In this context, we discuss a few points to further optimize the network, new strategies for designing SGCNs, their challenges, and some latest related developments. Based on our framework, we also deduce the connection between support vector kernels and SGCN filters.