Search Results (16)

Non-repetitive scanning LiDARs provide high coverage yet exhibit irregular sampling patterns, which destabilize local features and correspondences. To address this, we propose a novel spatiotemporal unified energy framework that integrates fractional calculus into rigid pose estimation. Spatially, we introduce a Riesz fractional regularization term to impose non-local smoothness constraints on the residual field, mitigating structural inconsistencies. Temporally, we design a Grünwald–Letnikov fractional dynamics solver that leverages long-memory effects of historical gradients to reduce the risk of being trapped in local minima. Comparative experiments on the Stanford 3D, MVTec ITODD, and HomebrewedDB (HB) datasets demonstrate that our method significantly outperforms state-of-the-art geometric and learning-based approaches. Specifically, it maintains a success rate exceeding 90% even under severe sampling perturbations where traditional methods fail. Ablation studies further validate that the introduction of non-local spatial constraints and historical gradient memory significantly reshapes the energy landscape, ensuring robust convergence. This work provides a rigorous theoretical foundation for applying fractional operators to point cloud processing. Full article

In this work, we consider a generalized form of the classical $(2+1)$-dimensional sine-Gordon system. The mathematical model considers a generalized reaction term, and the two-dimensional Laplacian includes the presence of space-fractional derivatives of the Riesz type with two different differentiation orders in general. The system is equipped with a conserved quantity that resembles the energy functional in the integer-order scenario. We propose a numerical model to approximate the solutions of the fractional sine-Gordon equation. A discretized form of the energy-like quantity is proposed, and we prove that it is conserved throughout the discrete time. Moreover, the analysis of consistency, stability, and convergence is rigorously carried out. The numerical model is implemented computationally, and some computer simulations are presented in this work. As a consequence of our simulations, we show that the discrete energy is approximately conserved throughout time, which coincides with the theoretical results. Full article

In this paper, we study a class of nonlocal Schrödinger–Poisson–Slater equations: $-\Delta u+u+\lambda \left(I_{\alpha }\ast {\left|u\right|}^q\right){\left|u\right|}^{q-2}u={\left|u\right|}^{p-2}u$, where $q,p>1$, $\lambda >0$, and $I_{\alpha }$ is the Riesz potential. We obtain the existence, stability, and symmetry-breaking of solutions for both radial and nonradial cases. In the radial case, we use variational methods to establish the coercivity and weak lower semicontinuity of the energy functional, ensuring the existence of a positive solution when p is below a critical threshold $\overline{p}={\displaystyle \frac{4q+2\alpha }{2+\alpha }}$. In addition, we prove that the energy functional attains a minimum, guaranteeing the existence of a ground-state solution under specific conditions on the parameters. We also apply the Pohozaev identity to identify parameter regimes where only the trivial solution is possible. In the nonradial case, we use the Nehari manifold method to prove the existence of ground-state solutions, analyze symmetry-breaking by studying the behavior of the energy functional and identifying the parameter regimes in the nonradial case, and apply concentration-compactness methods to prove the global well-posedness of the Cauchy problem and demonstrate the orbital stability of the ground state. Our results demonstrate the stability of solutions in both radial and nonradial cases, identifying critical parameter regimes for stability and instability. This work enhances our understanding of the role of nonlocal interactions in symmetry-breaking and stability, while extending existing theories to multiparameter and higher-dimensional settings in the Schrödinger–Poisson–Slater model. Full article

This paper introduces a highly accurate and efficient conservative scheme for solving the nonlocal damped Schrödinger system with Riesz fractional derivatives. The proposed approach combines the Fourier spectral method with the Crank–Nicolson time-stepping scheme. To begin, the original equation is reformulated into an equivalent system by introducing a new variable that modifies both energy and mass. The Fourier spectral method is employed to achieve high spatial accuracy in this semi-discrete formulation. For time discretization, the Crank–Nicolson scheme is applied, ensuring conservation of the modified energy and mass in the fully discrete system. Numerical experiments validate the scheme’s precision and its ability to preserve conservation properties. Full article

In this paper, our primary objective is to develop a robust and efficient higher-order structure-preserving algorithm for the numerical solution of the two-dimensional nonlinear spatial fractional Schrödinger equation. This equation, which incorporates fractional derivatives, poses significant challenges due to its non-local nature and nonlinearity, making it essential to design numerical methods that not only achieve high accuracy but also preserve the intrinsic physical and mathematical properties of the system. To address these challenges, we employ the scalar auxiliary variable (SAV) method, a powerful technique known for its ability to maintain energy stability and simplify the treatment of nonlinear terms. Combined with the composite Simpson’s formula for numerical integration, which ensures high precision in approximating integrals, and a fourth-order numerical differential formula for discretizing the Riesz derivative, we construct a highly effective finite difference scheme. This scheme is designed to balance computational efficiency with numerical accuracy, making it suitable for long-time simulations. Furthermore, we rigorously analyze the conserving properties of the numerical solution, including mass and energy conservation, which are critical for ensuring the physical relevance and stability of the results. Full article

Significant new progress has been made in nonlinear integrable systems with Riesz fractional-order derivative, and it is impressive that such nonlocal fractional-order integrable systems exhibit inverse scattering integrability. The focus of this article is on extending this progress to nonlocal fractional-order Schrödinger-type equations with variable coefficients. Specifically, based on the analysis of anomalous dispersion relation (ADR), a novel variable-coefficient Riesz fractional-order generalized NLS (vcRfgNLS) equation is derived. By utilizing the relevant matrix spectral problems (MSPs), the vcRfgNLS equation is solved through the inverse scattering transform (IST), and analytical solutions including n-soliton solution as a special case are obtained. In addition, an explicit form of the vcRfgNLS equation depending on the completeness of squared eigenfunctions (SEFs) is presented. In particular, the 1-soliton solution and 2-soliton solution are taken as examples to simulate their spatial structures and analyze their structural properties by selecting different variable coefficients and fractional orders. It turns out that both the variable coefficients and fractional order can influence the velocity of soliton propagation, but there is no energy dissipation throughout the entire motion process. Such soliton solutions may not only have important value for studying the super-dispersion transport of nonlinear waves in non-uniform media, but also for realizing a new generation of ultra-high-speed optical communication engineering. Full article

In this article, a generalized form of the Davey–Stewartson system, consisting of three nonlinear coupled partial differential equations, will be studied. The system considers the presence of fractional spatial partial derivatives of the Riesz type, and extensions of the classical mass, energy, and momentum operators will be proposed in the fractional-case scenario. In this work, we will prove rigorously that these functionals are conserved throughout time using some functional properties of the Riesz fractional operators. This study is intended to serve as a stepping stone for further exploration of the generalized Davey–Stewartson system and its wide-ranging applications. Full article

The present paper investigates the following inhomogeneous generalized Hartree equation $i\dot{u}+\Delta u=\pm {\left|u\right|}^{p-2}{\left|x\right|}^b(I_{\alpha }{\ast \left|u\right|}^p{|·|}^b)u$, where the wave function is $u:=u(t,x):\mathbb{R}\times {\mathbb{R}}^N\to \mathbb{C}$, with $N\ge 2$. In addition, the exponent $b>0$ gives an unbounded inhomogeneous term ${\left|x\right|}^b$ and $I_{\alpha }\approx {|·|}^{-(N-\alpha )}$ denotes the Riesz-potential for certain $0<\alpha <N$. In this work, our aim is to establish the local existence of solutions in some radial Sobolev spaces, as well as the global existence for small data and the decay of energy sub-critical defocusing global solutions. Our results complement the recent work (Sharp threshold of global well-posedness versus finite time blow-up for a class of inhomogeneous Choquard equations, J. Math. Phys. 60 (2019), 081514). The main challenge in this work is to overcome the singularity of the unbounded inhomogeneous term ${\left|x\right|}^b$ for certain $b>0$. Full article

The Riesz space-fractional derivative is discretized by the Fourier pseudo-spectral (FPS) method. The Riesz space-fractional nonlinear Klein–Gordon–Zakharov (KGZ) and Klein–Gordon–Schrödinger (KGS) equations are transformed into two infinite-dimensional Hamiltonian systems, which are discretized by the FPS method. Two finite-dimensional Hamiltonian systems are thus obtained and solved by the second-order average vector field (AVF) method. The energy conservation property of these new discrete schemes of the fractional KGZ and KGS equations is proven. These schemes are applied to simulate the evolution of two fractional differential equations. Numerical results show that these schemes can simulate the evolution of these fractional differential equations well and maintain the energy-preserving property. Full article

The Wheeler–DeWitt equation for a flat and compact Friedmann–Lemaître–Robertson–Walker cosmology at the pre-inflation epoch is studied in the contexts of the standard and fractional quantum cosmology. Working within the semiclassical regime and applying the Wentzel-Kramers-Brillouin (WKB) approximation, we show that some fascinating consequences are obtained for our simple fractional scenario that are completely different from their corresponding standard counterparts: (i) The conventional de Sitter behavior of the inflationary universe for constant potential is replaced by a power-law inflation. (ii) The non-locality of the Riesz’s fractional derivative produces a power-law inflation that depends on the fractal dimension of the compact spatial section of space-time, independent of the energy scale of the inflaton. Full article

For the first time, a new dissipation-preserving scheme is proposed and analyzed to solve a Caputo–Riesz time-space-fractional multidimensional nonlinear wave equation with generalized potential. We consider initial conditions and impose homogeneous Dirichlet data on the boundary of a bounded hyper cube. We introduce an energy-type functional and prove that the new mathematical model obeys a conservation law. Motivated by these facts, we propose a finite-difference scheme to approximate the solutions of the continuous model. A discrete form of the continuous energy is proposed and the discrete operator is shown to satisfy a conservation law, in agreement with its continuous counterpart. We employ a fixed-point theorem to establish theoretically the existence of solutions and study analytically the numerical properties of consistency, stability and convergence. We carry out a number of numerical simulations to verify the validity of our theoretical results. Full article

In this paper, we construct and analyze a class of high-order and dissipation-preserving schemes for the nonlinear space fractional generalized wave equations by the newly introduced scalar auxiliary variable (SAV) technique. The system is discretized by a fourth-order Riesz fractional difference operator in spatial discretization and the collocation methods in the temporal direction. Not only can the present method achieve fourth-order accuracy in the spatial direction and arbitrarily high-order accuracy in the temporal direction, but it also has long-time computing stability. Then, the unconditional discrete energy dissipation law of the present numerical schemes is proved. Finally, some numerical experiments are provided to certify the efficiency and the structure-preserving properties of the proposed schemes. Full article

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results. Full article

We investigated the energy of N points on an infinite compact metric space $(A,d)$ of a diameter less than 1 that interact through the potential $(1/d^s){(log1/d)}^t,$ where $s,t\ge 0$ and d is the metric distance. With ${\mathcal{E}}_{{log}^t}^s(A,N)$ denoting the minimal energy for such N-point configurations, we studied certain continuity and differentiability properties of ${\mathcal{E}}_{{log}^t}^s(A,N)$ in the variable $s.$ Then, we showed that in the limits, as $s\to \infty$ and as $s\to s_0>0,$N-point configurations that minimize the $s,{log}^t$-energy tends to an N-point best-packing configuration and an N-point configuration that minimizes the $s_0,{log}^t$-energy, respectively. Furthermore, we considered when A are circles in the Euclidean space ${\mathbb{R}}^2.$ In particular, we proved the minimality of N distinct equally spaced points on circles in ${\mathbb{R}}^2$ for some certain s and t. The study on circles shows a possibility for the utilization of N points generated through such new potential to uniformly discretize on objects with very high symmetry. Full article

A finite difference/Galerkin spectral discretization for the temporal and spatial fractional coupled Ginzburg–Landau system is proposed and analyzed. The Alikhanov $L2$-$1_{\sigma }$ difference formula is utilized to discretize the time Caputo fractional derivative, while the Legendre-Galerkin spectral approximation is used to approximate the Riesz spatial fractional operator. The scheme is shown efficiently applicable with spectral accuracy in space and second-order in time. A discrete form of the fractional Grönwall inequality is applied to establish the error estimates of the approximate solution based on the discrete energy estimates technique. The key aspects of the implementation of the numerical continuation are complemented with some numerical experiments to confirm the theoretical claims. Full article