Article
Computer Science, Interdisciplinary Applications
Yunqing Huang, Jichun Li, Bin He
Summary: This paper introduces a finite element method for solving the time-dependent Maxwell's equations in nonlinear Kerr media. The fully-discrete scheme is shown to be conditionally stable in the spatial variable and second order in time. Numerical results support the theoretical analysis and demonstrate practical soliton propagation phenomena in Kerr media.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Meng Chen, Rong Gao, Yan He, Linghua Kong
Summary: Recently, researchers have developed a direct method for solving 2D Maxwell's equations in Kerr-type nonlinear media. This method eliminates iteration error and is more efficient than the classical iterative method. The stability of 3D Maxwell's equations has been theoretically investigated, and numerical results have confirmed a second-order convergence rate in both time and space.
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Dmitry V. Valovik
Summary: This paper considers a transmission eigenvalue problem for nonlinear Maxwell's equations, which describes the propagation of monochromatic transverse electric waves in a dielectric film with cubic inhomogeneous permittivity sandwiched between two half-spaces with constant permittivities. It is proved that guided waves exist with and without linear counterparts, even if the nonlinearity coefficient is small. The theoretical results suggest the existence of a new type of nonlinear guided waves. The paper proposes an original mathematical approach to determine the asymptotic distribution of the eigenvalues and zeros of the eigenfunctions. Numerical results are provided to illustrate the theoretical findings.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Li Zhang, Maohua Ran, Hanyue Zhang
Summary: In this research, a two-dimensional ADI-FDTD method for Maxwell's equations without sources and charges is proposed. The method shows fourth-order accuracy in time and is approximately energy-preserving. Numerical experiments confirm the theoretical results.
Article
Mathematics, Applied
Yernat M. Assylbekov, Ting Zhou
Summary: This paper explores an inverse boundary value problem of electromagnetism in a nonlinear Kerr medium, demonstrating the unique determination of the electromagnetic material parameters and nonlinear susceptibility parameters of the medium through electromagnetic measurements on the boundary. The study focuses on the case of the time-harmonic Maxwell equations.
JOURNAL OF SPECTRAL THEORY
(2021)
Article
Optics
Tiancheng Zhang, Yan Peng, Zhou Dai, Huaguang Bao, Zelong Xiao, Xuewen Chen, Dazhi Ding
Summary: In this paper, an efficient discontinuous Galerkin time-domain (DGTD) method is proposed for solving Maxwell's equations in nonlinear Kerr or Raman media. The method incorporates a high-order derivatives DGTD method with a local time-stepping scheme, allowing for independent time-stepping increments in the nonlinear and linear media regions. This greatly improves the efficiency of time-domain analysis. The proposed method preserves local features and offers an effective tool for numerical analysis of nonlinear optical phenomena.
Article
Mathematics, Applied
Nuodi Liu, Yanping Chen, Jianwei Zhou, Yunqing Huang
Summary: In this paper, a mixed finite element method (MFEM) is developed for solving the time-dependent Maxwell's equations in a Havriliak-Negami (H-N) dispersive medium. The unconditional stability of the fully discrete backward Euler scheme is proven, and global superconvergence results are obtained using interpolation postprocessing techniques. Numerical experiments in two dimensions are provided to demonstrate the validity of the theoretical analysis, which has not been discussed in any other publications.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics
Aleksandr Belov, Zhanna Dombrovskaya
Summary: This study proposes an approximate method of integrating the Maxwell equations along the beam propagation direction to reduce the computational costs of integrated photonics problems. By using one-dimensional bicompact schemes, the original two-dimensional problem is simplified to a one-dimensional one. The proposed method is validated through calculations of test and applied problems with known exact reflection spectra.
Article
Engineering, Multidisciplinary
Serbay Duran, Bayhan Karabulut
Summary: Trigonometric and hyperbolic type traveling wave solutions are produced by using the sub-equation analytical method considering the Kerr Law properties of the equation defining nematic liquid crystals. The resulting traveling wave solutions play an important role in energy transport and soliton molecules in liquid crystals. In addition, solitary wave behaviors for different values of constants in the produced traveling wave solutions are discussed.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Mathematics
Valeria Martynova, Dmitry Valovik
Summary: This paper investigates the propagation of monochromatic electromagnetic waves in a circular cylindrical waveguide filled with nonlinear dielectric medium. The waveguide's permittivity is described by the Kerr law, which is a significant nonlinear dependence in nonlinear optics. To identify guided waves, the authors develop an original mathematical approach that proves the existence of infinitely many guided waves. Moreover, they demonstrate that only a finite number of the discovered nonlinear guided waves have linear counterparts, and the rest are non-linearizable solutions. Numerical results are also presented.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Materials Science, Multidisciplinary
Jiahao Wang, Chunzhen Fan
Summary: A three-layer design of multi-functional terahertz metamaterials is proposed and investigated, utilizing the nonlinear Kerr-media sandwiched between a phase change material of vanadium dioxide. The design achieves high transmission before phase transformation and exhibits three band perfect absorption in the metallic phase of vanadium dioxide. Impedance matching and effective medium theory are used to extract high absorption. The physical mechanism is further studied through analysis of the normalized electric and magnetic field distribution. Additionally, a switcher is realized by irradiating the nonlinear Kerr-media with external pump light, and its sensing capability with different Kerr medium is explored.
Article
Computer Science, Interdisciplinary Applications
Lingxiao Li, Maohui Lyu, Weiying Zheng
Summary: In this paper, we propose a second order accurate finite element method for the three-dimensional nonlinear Maxwell's equations in optical media. The method is shown to be unconditionally stable through energy stability analysis. We also develop an effective preconditioner to efficiently solve the linearized algebraic system in each Newton sub-iteration. The proposed scheme is validated through numerical experiments and applied to simulate soliton propagation and wave scattering in nonlinear optical glass.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Physics, Fluids & Plasmas
A. Demiquel, V. Achilleos, G. Theocharis, V. Tournat
Summary: In this paper, the modulation instabilities (MI) in a one-dimensional chain configuration of a flexible mechanical metamaterial (flexMM) are studied. The flexMMs can be modeled using the lumped element approach, which represents a coupled system of discrete equations for the longitudinal displacements and rotations of the rigid mass units. A nonlinear Schrodinger equation for slowly varying envelope rotational waves is derived using the multiple-scales method in the long wavelength regime. The occurrence of MI is mapped to the parameters of the metamaterials and the wave numbers, highlighting the importance of the rotation-displacement coupling in the manifestation of MI. The analytical findings are confirmed through numerical simulations of the full discrete and nonlinear lump problem. These results provide design guidelines for nonlinear metamaterials with either stability to high amplitude waves or potential for observing instabilities.
Article
Mathematics, Applied
Xixian Bai, Shuang Wang, Hongxing Rui
Summary: Two efficient numerical schemes based on L1 formula and Finite-Difference Time-Domain (FDTD) method are constructed for Maxwell's equations in a Cole-Cole dispersive medium. By rigorously carrying out energy stability and error analysis using the energy method, it is proven that both schemes converge with a certain order. Numerical experiments are performed to confirm the theoretical analysis.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Environmental Sciences
Diana Semenikhina, Nikolay N. Gorbatenko
Summary: A research on nonlinear cylindrical metastructure was conducted, which developed an algorithm for finding the total field of a nonlinearly loaded perfectly conducting cylinder covered with a metamaterial layer based on Maxwell's equations with nonlinear boundary conditions. The algorithm was implemented in a software package, and numerical calculations and simulations were carried out to study the scattering diagrams and dependencies of the scattered field harmonics on the parameters of nonlinear loads. The experimental study confirmed the results of numerical simulations, showing the potential of the nonlinear cylindrical marker with metamaterial for applications in both military and civilian areas.
Article
Mathematics, Applied
Ming Sun, Jichun Li, Peizhen Wang, Zhimin Zhang
JOURNAL OF SCIENTIFIC COMPUTING
(2018)
Article
Mathematics, Applied
Jichun Li, Meng Chen, Min Chen
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2019)
Article
Engineering, Multidisciplinary
Yunqing Huang, Jichun Li, Chao Wu
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2018)
Article
Engineering, Multidisciplinary
Jichun Li, Zhiwei Fang, Guang Lin
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2018)
Article
Mathematics, Applied
Yunqing Huang, Meng Chen, Jichun Li, Yanping Lin
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2018)
Article
Mathematics, Applied
Yunqing Huang, Hongen Jia, Jichun Li
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2018)
Article
Mathematics, Applied
Cengke Shi, Jichun Li, Chi-Wang Shu
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2018)
Article
Computer Science, Interdisciplinary Applications
Jichun Li, Xiu Ye, Shangyou Zhang
JOURNAL OF COMPUTATIONAL PHYSICS
(2018)
Article
Mathematics, Applied
Xiaofeng Jia, Jichun Li, Hongen Jia
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2019)
Article
Mathematics, Applied
Jichun Li, Chen Meng, Yunqing Huang
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2019)
Article
Mathematics, Applied
Zhiwei Fang, Jichun Li, Tao Tang, Tao Zhou
JOURNAL OF SCIENTIFIC COMPUTING
(2019)
Article
Mathematics, Applied
Yunqing Huang, Jichun Li, Shangyou Zhang
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2019)
Proceedings Paper
Engineering, Electrical & Electronic
Meng Chen, Yunqing Huang, Jichun Li
2018 PROGRESS IN ELECTROMAGNETICS RESEARCH SYMPOSIUM (PIERS-TOYAMA)
(2018)
Article
Mathematics, Applied
Jichun Li, Zhiwei Fang
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2018)
