Article
Engineering, Electrical & Electronic
Md. Tarikul Islam, Mst. Armina Akter, J. F. Gomez-Aguilar, Md. Ali Akbar, Eduardo Perez-Careta
Summary: Nonlinear models of fractional order play an important role in depicting the interior mechanisms of complicated phenomena in nature. This study presents accurate wave solutions of two arbitrary order nonlinear Schrodinger models using the rational (G'/G)-expansion scheme, combined with the Cole-Hopf transformation, and shows various wave structures in different profiles.
OPTICAL AND QUANTUM ELECTRONICS
(2022)
Article
Mathematics, Applied
Minghe Zhang, Weifang Weng, Zhenya Yan
Summary: This paper investigates the anomalous dispersive relations, inverse scattering transform, and fractional multi-solitons of the integrable combined fractional higher-order mKdV hierarchy. It explores the completeness of squared scalar eigenfunctions and constructs a matrix RH problem to represent three types of fractional N-solitons. The obtained results demonstrate anomalous dispersion and various interesting wave phenomena in fractional nonlinear media.
PHYSICA D-NONLINEAR PHENOMENA
(2023)
Article
Mathematics, Applied
Meryem Odabasi, Zehra Pinar, Huseyin Kocak
Summary: In this work, the exact solutions of fractional-order differential equations in mathematical physics were investigated, including three equations with important applications in science and engineering. Exact traveling wave solutions of these equations have been established by different efficient methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Optics
Jia-Dong Li, Ling-Zheng Meng, Li -Chen Zhao
Summary: The phase of rational solutions in nonlinear waves can undergo sudden phase inversion processes, which has implications for understanding the phase properties of these waves. By analyzing the topological vector potentials of related nonlinear waves, it is found that rational W-shaped solitons exhibit distinctive phases.
Article
Mathematics, Interdisciplinary Applications
H. G. Abdelwahed, A. F. Alsarhana, E. K. El-Shewy, Mahmoud A. E. Abdelrahman
Summary: The nonlinearity form of the Schrodinger equation (NLSE) provides a good explanation for energy and solitary transmission properties in modern communications with optical-fiber energy reinforcement actions. The coefficients of NLSE, such as nonlinear Kerr, evolutions, and dispersions, regulate the solitary representation during fiber transmissions which control the energy changes through the model. Higher-order nonlinear Schrodinger equations (HONLSEs) need to be explored to alleviate the implications in energy and wave features. The unified solver approach is employed in this work to evaluate the HONLSEs, taking into consideration steepness, HO dispersions, and nonlinearity self-frequency influences. The energy and solitary features in the investigated model were altered by the higher-order impacts. Furthermore, the new HONLSE solutions explain a wide range of important complex phenomena in wave energy and its applications.
FRACTAL AND FRACTIONAL
(2023)
Article
Engineering, Mechanical
Muwei Liu, Haotian Wang, Hujiang Yang, Wenjun Liu
Summary: This paper investigates a coupled higher-order variable-coefficient nonlinear Schrodinger equation, which has potential applications in optical fiber communications. The author presents a modified Kudryashov method to obtain soliton solutions for the fractional variable-coefficient equations. The interaction of solitons is studied through dynamical analysis, providing insights into the relationship between the dynamical structure of solitons and parameters in the fractional nonlinear optical system. The results of this paper contribute to the study of fractional nonlinear optical systems and provide theoretical guidance for optical communication in inhomogeneous optical fibers.
NONLINEAR DYNAMICS
(2023)
Article
Materials Science, Multidisciplinary
Qin Zhou, Yunzhou Sun, Houria Triki, Yu Zhong, Zhongliang Zeng, Mohammad Mirzazadeh
Summary: This paper investigates the propagation properties of optical soliton pulses with higher-order effects in a multimode fiber and proposes a method to control the physical properties of solitons by choosing different parameters.
RESULTS IN PHYSICS
(2022)
Article
Mathematics, Applied
Martin Bohner, Said R. Grace, Irena Jadlovska, Nurten Kilic
Summary: This paper deals with the asymptotic behavior of the nonoscillatory solutions of a certain forced fractional differential equation with positive and negative terms, involving the Caputo fractional derivative. The obtained results are new and generalize some known results in the literature. Two examples are also provided to illustrate the results.
MEDITERRANEAN JOURNAL OF MATHEMATICS
(2022)
Article
Multidisciplinary Sciences
Sheng Zhang, Feng Zhu, Bo Xu
Summary: This article illustrates the feasibility of extending the Darboux transformation (DT) and generalized DT (GDT) methods to construct solitary wave solutions for fractional integrable systems using the coupled nonlinear Schrodinger (CNLS) equations as an example. The study found that the symmetric solitary wave solutions of the integer-order CNLS equations exhibit asymmetry in the fractional order case.
Article
Mathematics
Rafail K. Gazizov, Stanislav Yu. Lukashchuk
Summary: The study proves that linear anomalous diffusion equation has infinite sequences of nontrivial higher-order symmetries and some of these can be rewritten as fractional-order symmetries. This approach is applicable for a wide class of linear fractional differential equations.
Article
Mathematics
Shou-Ting Chen, Wen-Xiu Ma
Summary: We generate integrable Hamiltonian hierarchies from a type of higher-order matrix spectral problems using the zero-curvature formulation. The Liouville integrability of the obtained hierarchies is ensured by establishing their Hamiltonian structures through the trace identity. Illustrative examples of coupled nonlinear Schrodinger equations and coupled modified Korteweg-de Vries equations are presented.
Article
Physics, Multidisciplinary
Yue-Jin Cai, Jian-Wen Wu, Lang-Tao Hu, Ji Lin
Summary: This paper investigates femtosecond nondegenerate solitons in optical fibers, described by coupled higher-order nonlinear Schrodinger equations. Analytical solutions for nondegenerate solitons are constructed using the Hirota method, with constraints for stable structures. In addition, soliton molecules and asymmetric solitons are identified as new types of nondegenerate solitons, with their interactions during propagation studied.
Article
Engineering, Mechanical
Han-Dong Guo, Tie-Cheng Xia
Summary: The study focuses on multi-soliton solutions for a higher-order coupled nonlinear Schrodinger system in an optical fiber. The analysis reveals the impact of higher-order linear and nonlinear terms on wave dynamics, and shows different collision behaviors for second-order and third-order breathers and soliton solutions.
NONLINEAR DYNAMICS
(2021)
Article
Physics, Mathematical
Wen-Xiu Ma
Summary: The paper aims to generate nonlocal integrable nonlinear Schrodinger hierarchies of type (-lambda, lambda) by imposing two nonlocal matrix restrictions of the AKNS matrix characteristic-value problems of arbitrary order. Exact soliton solutions are formulated by applying the associated reflectionless generalized Riemann-Hilbert problems based on the explored outspreading of characteristic-values and adjoint characteristic-values, in which characteristic-values and adjoint characteristic-values could have a nonempty intersection. Illustrative models of the resultant mixed-type nonlocal integrable nonlinear Schrodinger equations are presented.
INTERNATIONAL JOURNAL OF GEOMETRIC METHODS IN MODERN PHYSICS
(2023)
Article
Mathematics, Interdisciplinary Applications
Vladimir I. Kruglov, Houria Triki
Summary: We have discovered two types of multiple-hump soliton modes in a highly dispersive optical fiber with a Kerr nonlinearity. These multi-hump solitons of quartic or dipole types can exist in the fiber system in the presence of higher-order dispersion. The third-and fourth-order dispersion effects in the fiber material can lead to the coupling of quartic or dipole solitons into double-, triple-, and multi-humped solitons. These newly found multi-soliton pulses are potentially stable to small noise perturbation and can be utilized for transmission in optical fibers medium with higher-order dispersions.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Physics, Multidisciplinary
Zijian Zhou, Zhenya Yan
Summary: This paper investigates the logarithmic nonlinear Schrodinger equation with PT-symmetric harmonic potential using physics-informed neural networks (PINNs) deep learning method. Different initial and boundary conditions are considered and compared with results from the Fourier spectral method. The effectiveness of PINNs deep learning for this equation is also examined by varying space widths and optimization steps.
Article
Mathematics, Applied
Shuyan Chen, Zhenya Yan, Boling Guo
Summary: In this study, the long-time asymptotic behavior of the solution for the focusing Hirota equation with symmetric, non-zero boundary conditions at infinity is investigated. The oscillatory Riemann-Hilbert problem with distinct jump curves is established using the Lax pair with NZBCs, and long-time asymptotic solutions are proposed in two distinct domains of space-time plane through the Deift-Zhou nonlinear steepest-descent method. Additionally, the modulation instability of the Hirota equation is also explored.
MATHEMATICAL PHYSICS ANALYSIS AND GEOMETRY
(2021)
Article
Mathematics, Applied
Guoqiang Zhang, Liming Ling, Zhenya Yan
Summary: The study focuses on multi-component nonlinear Schrodinger equations with nonzero boundary conditions, exploring fundamental and higher-order vector Peregrine solitons using loop group theory. It also examines vector rational rogue waves, their symmetry under certain parameter constraints, and proposes a systematic approach to study their asymptotic behaviors and decompositions. Additionally, the research highlights the determination of vector rogue waves with maximal amplitudes using parameter vectors for multi-component nonlinear physical systems.
JOURNAL OF NONLINEAR SCIENCE
(2021)
Article
Physics, Multidisciplinary
Li Wang, Zhenya Yan
Summary: This paper uses the multi-layer PINN deep learning method to study data-driven rogue wave solutions of the defocusing nonlinear Schrodinger (NLS) equation with time-dependent potential. By considering various initial conditions and periodic boundary conditions, the study discusses learning parameters in the context of rogue wave solutions.
Article
Mathematics, Applied
Yong Chen, Jin Song, Xin Li, Zhenya Yan
Summary: In this paper, a new class of potential functions composed of the Dirac delta (x) and hyperbolic functions, called PT-delta-hyperbolic-function potentials, is introduced. These potentials support fully real energy spectra in non-Hermitian Hamiltonians, and the threshold curves of PT symmetry breaking are numerically presented. In addition, in self-focusing and defocusing Kerr-nonlinear media, the PT-symmetric potentials also support stable peakons, preserving total power and quasi-power conservation.
Article
Mathematics, Applied
Li Wang, Zhenya Yan
Summary: This paper successfully applies multi-layer physics-informed neural networks (PINNs) deep learning to study the data-driven peakon and periodic peakon solutions of various nonlinear dispersive equations, as well as the data-driven parameter discovery of the CH equation. These results will be useful for further research on peakon solutions and experimental designs of nonlinear dispersive equations.
PHYSICA D-NONLINEAR PHENOMENA
(2021)
Article
Mathematics, Applied
Yong Chen, Zhenya Yan, Boris A. Malomed
Summary: We studied a class of physically intriguing PT-symmetric generalized Scarf-II (GS-II) potentials that can support exact solitons in one- and multi-dimensional nonlinear Schrodinger equation. In the 1D and multi-D settings, we found that a properly adjusted localization parameter may support fully real energy spectra. Continuous families of fundamental and higher-order solitons were produced. While the fundamental states were stable, the higher-order ones were unstable. The stable solitons were capable of robust propagation and remained trapped in slowly moving potential wells, which offers possibilities for manipulating optical solitons. Adiabatic variation of potential parameters could transform solitons into stable forms.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Weifang Weng, Guoqiang Zhang, Minghe Zhang, Zijian Zhou, Zhenya Yan
Summary: In this paper, the wave structures of an n-component nonlinear Schrodinger equation with mixed nonzero and zero boundary conditions are investigated. Semi-rational vector rogon-soliton solutions and soliton-like solutions are found, and their amplitude characteristics are analyzed. These results are important for understanding the physical phenomena in the n-NLS equation and other related physical models.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Ming Zhong, Shibo Gong, Shou-Fu Tian, Zhenya Yan
Summary: In this paper, we investigate the forward and inverse problems of the generalized Gross-Pitaevskii (GP) equation with complex PT-symmetric potentials using deep physics-informed neural networks (PINNs). Data-driven rogue waves (RWs) are studied in the forward problem, and the accuracy of the PINNs solution is demonstrated by comparing the data-driven RWs with numerical results. Additionally, we focus on the impact of critical factors such as the depths of neural networks and numbers of training points on the performance of the PINNs algorithm. Furthermore, the inverse problem is explored to identify system parameters from training data.
PHYSICA D-NONLINEAR PHENOMENA
(2022)
Article
Mathematics, Applied
Ming Zhong, Li Wang, Pengfei Li, Zhenya Yan
Summary: We report a novel spontaneous symmetry breaking phenomenon and the existence of ghost states in the framework of the fractional nonlinear Schrodinger equation. The symmetry of fundamental solitons is broken into two branches of asymmetry solitons (ghost states) with complex conjugate propagation constants, exclusively in fractional media. The influences of fractional Levy index (alpha) and saturable nonlinear parameters (S) on the symmetry breaking of solitons are analyzed in detail. Stability analysis, direct propagations, and collision phenomena between symmetric and asymmetric solitons are explored. The results provide a theoretical basis for studying spontaneous symmetry breaking phenomena and related physical experiments in fractional media with PT-symmetric potentials.
Article
Mathematics, Applied
Minghe Zhang, Weifang Weng, Zhenya Yan
Summary: This paper investigates the anomalous dispersive relations, inverse scattering transform, and fractional multi-solitons of the integrable combined fractional higher-order mKdV hierarchy. It explores the completeness of squared scalar eigenfunctions and constructs a matrix RH problem to represent three types of fractional N-solitons. The obtained results demonstrate anomalous dispersion and various interesting wave phenomena in fractional nonlinear media.
PHYSICA D-NONLINEAR PHENOMENA
(2023)
Article
Mathematics, Applied
Jin Song, Zhenya Yan, Boris A. Malomed
Summary: In this paper, vortex solitons in various 2D spinning quantum droplets (QDs) are investigated in a PT-symmetric potential using the amended Gross-Pitaevskii equation with Lee-Huang-Yang corrections. Exact QD states are obtained under specific parameter constraints, which serve as a guide for finding the corresponding generic family. The stability of different families of QDs originating from linear modes is explored in the unbroken PT symmetry region. The effect of PT-symmetric potential on spinning and nonspinning QDs is studied by varying the strength of the gain-loss distribution. Interactions between spinning or nonspinning QDs are also examined, demonstrating elastic collisions under certain conditions.
Article
Mathematics, Applied
Ming Zhong, Zhenya Yan, Shou-Fu Tian
Summary: We investigate the effects of PT-symmetric and non-PT-symmetric potentials on spinor F = 1 Bose-Einstein condensates. By numerically analyzing the linear matrix non-Hermitian Hamiltonian, we determine the parameter regions where the spectra are real for PT-symmetric Scarf-II and harmonic-Hermitian-Gaussian potentials. Despite broken PT phases, we observe stable solitons with various system parameters, and find that solitons with different shapes exhibit anti-interference in the exact nonlinear models. Adiabatic changes of parameters allow for the stable excitation of bright, single-hump, and triple-humps solitons, and we present stable numerical solitons as well. Theoretical support for related physical experiments and extensions to other nonlinear multi-wave models in PT-symmetric structures can be derived from this study.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Physics, Multidisciplinary
Ming Zhong, Zhenya Yan
Summary: The interaction between fractional diffraction and parity-time (PT) symmetry brings unique properties to certain physical systems. We report a spontaneous symmetry breaking (SSB) phenomenon and ghost states of solitons in a two-dimensional (2D) fractional nonlinear Schrodinger equation with focusing and defocusing Kerr media under a 2D non-Hermitian PT-symmetric potential. Asymmetric solitons bifurcate out in an SSB way, destabilizing symmetric solitons. We also investigate the dependence of the symmetry breaking of solitons on the fractional Levy index and explore the stabilities of different solitons. Furthermore, the management of optical field propagation and stable excitations of solitons are achieved through modulation of the external potential.
COMMUNICATIONS PHYSICS
(2023)
