Article
Mathematics, Applied
Chao-Jiang Xu, Yan Xu
Summary: This study investigates the Cauchy problem of the spatially homogeneous fractional Kramers-Fokker-Planck equation and demonstrates that the solution exhibits Gevrey regularity and decay estimation with an L2 initial datum for positive time.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2023)
Article
Mathematics, Applied
Qifeng Zhang, Lu Zhang, Hai-wei Sun
Summary: This paper studies two-dimensional nonlinear spatial fractional complex Ginzburg-Landau equations, proposing a method with second order accuracy on both time and space, and proving its uniqueness and unconditional stability. Numerical experiments confirm the computational advantage of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Haotian Qian, Minjie Shan
Summary: This article investigates the initial value problem for the generalized Zakharov-Kuznetsov equation on R2 and proves that it has a locally well-posed solution within a strip of analytic functions without shrinking the width of the strip in time. The proof relies on the local smoothing effect, a multi-dimensional maximal function estimate, and Bourgain-type spaces, which are significant for derivative nonlinear dispersive equations in low regularity spaces. Additionally, under the assumption of a bounded Sobolev norm, the generalized Zakharov-Kuznetsov equation is shown to be of Gevrey-class analysis. Furthermore, an explicit lower bound on the possible decreasing rate of the uniform radius of analyticity is obtained for a solution starting from analytic initial data.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2023)
Article
Mathematics, Interdisciplinary Applications
Li Peng, Yong Zhou
Summary: This paper discusses the well-posedness and regularity results of weak solution for a fractional wave equation allowing that the coefficients may have low regularity. The analysis relies on mollification arguments, Galerkin methods, and energy arguments.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics
Jishan Fan, Yuxi Hu, Gen Nakamura
Summary: This work proves the local well-posedness of local strong solutions to an isentropic compressible Ginzburg-Landau-Navier-Stokes system with vacuum in a bounded domain omega subset of R3.
MATHEMATISCHE NACHRICHTEN
(2021)
Article
Mathematics, Applied
Nguyen Duc Phuong, Le Dinh Long, Devender Kumar, Ho Duy Binh
Summary: In this paper, we studied a pseudo-parabolic equation with the Atangana-Baleanu Caputo fractional derivative. We used the Fractional Tikhonov method and the generalized Tikhonov method as our main tools and provided numerical experiments to validate our theory.
Article
Mathematics, Applied
Hamed Mohebalizadeh, Hojatollah Adibi, Mehdi Dehghan
Summary: This study investigates theoretical results, numerical study, and real-world application of the SFGLE with the fractional Laplacian. The study explores the application of SFGLE in Bose-Einstein condensate for energy transportation. It also analyzes the existence, uniqueness, and regularity of local weak solutions. Moreover, the study introduces new Green reproducing kernels for numerical solutions and evaluates the accuracy and efficiency of the method.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics
V. J. Ervin
Summary: This article investigates the regularity of solutions to the fractional diffusion, advection, reaction equation on a bounded domain in R-1. The regularity of the solution is determined by the endpoint behavior of the solution, and it is lower for a sufficiently smooth right hand side function. The regularity of the solution to the fractional diffusion advection reaction equation is two orders lower than that of the fractional diffusion reaction equation.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics
Qiaoyuan Cheng, Engui Fan
Summary: In this paper, the existence of global solutions in H3(R) boolean AND H2,1(R) to the Fokas-Lenells (FL) equation on the line is proven when the initial data includes solitons. A modified Darboux transformation is used as a key tool to add or subtract a soliton with given spectral and scattering parameters. The global well-posedness of the initial value problem with a finite number of solitons is established using the inverse scattering transform technique based on previous results on the global well-posedness of the FL equation. (c) 2023 Elsevier Inc. All rights reserved.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Optics
Salim B. Ivars, Muriel Botey, Ramon Herrero, Kestutis Staliunas
Summary: We propose a method to control turbulence by modifying the excitation cascade. The method is based on the asymmetric coupling between spatiotemporal excitation modes using non-Hermitian potentials. We demonstrate that unidirectional coupling towards larger or smaller wave numbers can increase or reduce the energy flow in turbulent states, thereby influencing the character of turbulence. The study uses the complex Ginzburg-Landau equation, a universal model for pattern formation and turbulence in various systems.
Article
Mathematics, Applied
Lu Zhang, Qifeng Zhang, Hai-wei Sun
Summary: This paper presents a fast and high-order finite difference method for two-dimensional space-fractional complex Ginzburg-Landau equations, utilizing innovative time discretization scheme and compact spatial method. The rigorous theoretical analysis using energy argument is conducted, and numerical results demonstrate the performance of the proposed method.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2021)
Article
Operations Research & Management Science
Xiaoju Zhang, Yao Lu, Dong Liu
Summary: This paper investigates the initial boundary value problems for time-space fractional Ginzburg-Landau equations with Caputo time fractional derivatives and fractional Laplacian operators. The study establishes the global existence and decay estimates of weak solutions using the Galerkin method. The dependence of weak solutions on the given data is examined, leading to the stability of the system. Additionally, an optimal control for the time-space fractional Ginzburg-Landau system is obtained.
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
(2023)
Article
Mathematics, Interdisciplinary Applications
Orazio Descalzi, Carlos Cartes
Summary: This article investigates the formation of localized spatiotemporal chaos in the complex cubic Ginzburg-Landau equation with nonlinear gradient terms and reviews the influence of multiplicative noise on stationary pulses stabilized by nonlinear gradients. Surprising results are obtained through numerical simulations and explained analytically, including the induction of velocity change in propagating dissipative solitons.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Mathematics, Interdisciplinary Applications
Shao-Wen Yao, Esin Ilhan, P. Veeresha, Haci Mehmet Baskonus
Summary: This paper aims to find the iterative solution for the generalized quintic complex Ginzburg-Landau equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus, demonstrating its efficiency and applicability. The obtained results' nature is presented in three distinct cases, illustrated with surfaces and contour plots for a particular fractional order, and plots with different fractional orders are presented to show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The investigation confirms the efficiency and applicability of the considered method and fractional operators in analyzing phenomena in science and technology.
FRACTALS-COMPLEX GEOMETRY PATTERNS AND SCALING IN NATURE AND SOCIETY
(2021)
Article
Mathematics, Applied
Priscila L. da Silva
Summary: In this work, we studied the Cauchy problem in Gevrey spaces for a generalized class of equations. We proved that for the generalized equation, the local initial value problem is well-posed in Gevrey spaces. Additionally, by choosing a specific parameter in the equation, we showed that the local solution is globally analytic in both time and spatial variables.
PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A-MATHEMATICS
(2023)
