Article
Physics, Fluids & Plasmas
Giacomo Roberti, Gennady El, Alexander Tovbis, Francois Copie, Pierre Suret, Stephane Randoux
Summary: In this study, a breather gas for the focusing nonlinear Schrodinger equation is numerically realized by constructing a random ensemble of approximately 50 similar breathers through the Darboux transform recursive scheme in high-precision arithmetics. Three types of breather gases are synthesized as elementary quasiparticles of the respective gases, and the interaction properties of the constructed breather gases are investigated by comparing the mean propagation velocity with predictions of spectral kinetic theory.
Article
Physics, Multidisciplinary
Yutian Wang, Fanglin Chen, Songnian Fu, Jian Kong, Andrey Komarov, Mariusz Klimczak, Ryszard Buczynski, Xiahui Tang, Ming Tang, Luming Zhao
Summary: This article introduces the description of high-order soliton propagation using Nonlinear Fourier Transform (NFT) based on the nonlinear Schrodinger equation, and proposes a method of using eigenvalues to characterize soliton pulses. The study shows that soliton characteristics can be accurately described in the nonlinear frequency domain, and NFT can serve as an important supplement to characterizing soliton pulses.
NEW JOURNAL OF PHYSICS
(2022)
Article
Mathematics, Interdisciplinary Applications
Hegagi Mohamed Ali, Hijaz Ahmad, Sameh Askar, Ismail Gad Ameen
Summary: In this study, a modified method is proposed to solve nonlinear systems of fractional-order partial differential equations, and it is applied to some important fractional-order nonlinear systems. The results show that the used methods can efficiently and reliably obtain approximate solutions.
FRACTAL AND FRACTIONAL
(2022)
Article
Engineering, Multidisciplinary
M. Ali Akbar, Farah Aini Abdullah, Mst. Munny Khatun
Summary: In this study, a wide range of geometric shapes and complete solitary wave solutions are constructed using the two-variable eth w0/*, 1/w)-expansion method, and the effects of fractional parameters on the physical details of the solutions are explained by plotting three-dimensional and two-dimensional graphs. The results demonstrate the effectiveness, compatibility, and reliability of the two-variable eth w0/*, 1/w)-expansion approach for extracting soliton solutions of fractional-order nonlinear evolution equations in science, technology, and engineering.
ALEXANDRIA ENGINEERING JOURNAL
(2023)
Article
Mathematics, Applied
Yu Huang, Fatemeh Mohammadi Zadeh, Mohammad Hadi Noori Skandari, Hojjat Ahsani Tehrani, Emran Tohidi
Summary: This article presents a balanced space-time spectral collocation method for solving nonlinear time-fractional Burgers equations, achieving balance in both time and space variables. By using efficient interpolation and iterative methods, the proposed scheme demonstrates high accuracy and low computational cost in comparison to recent numerical methods.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
Article
Materials Science, Multidisciplinary
Asma Rashid Butt, Jaweria Zaka, Ali Akgul, Sayed M. El Din
Summary: The study utilizes the Atangana-Baleanu conformable differential operator to solve the conformable fractional Sharma-Tasso-Olever equation. A new extended direct algebraic method is employed to obtain precise solutions, which include hyperbolic, trigonometric, and rational solutions. Graphical visualization is used to observe the effects and behavior of the fractional equation, providing insight into its physical significance. The results obtained are novel and applicable in various fields, demonstrating the efficacy of the analytical approach in addressing nonlinear problems in mathematical physics and engineering.
RESULTS IN PHYSICS
(2023)
Article
Materials Science, Multidisciplinary
Asma Rashid Butt, Jaweria Zaka, Ali Akgul, Sayed M. El Din
Summary: The Atangana-Baleanu conformable differential operator was used to solve the conformable fractional Sharma-Tasso-Olever equation in this study. The new extended direct algebraic method was then employed to obtain precise solutions, which were found to be hyperbolic, trigonometric, and rational. Graphic visualization in 3D, contour, and 2D plots was utilized to observe the fractional effects and dynamical behavior. These graphical representations proved valuable in understanding the true physical significance of the equation. The obtained results are novel, widely applicable, and demonstrate the effectiveness of the suggested strategy for handling nonlinear problems in mathematical physics and engineering, providing insights into wave dynamics.
RESULTS IN PHYSICS
(2023)
Article
Mathematics, Applied
Asif Khan, Tayyaba Akram, Arshad Khan, Shabir Ahmad, Kamsing Nonlaopon
Summary: This manuscript investigates the Korteweg-de Vries-Burgers (KdV-Burgers) partial differential equation (PDE) under nonlocal operators with the Mittag-Leffler kernel and the exponential decay kernel. The existence of the solution is demonstrated using fixed point theorems and a series solution is computed using the modified double Laplace transform. The suggested approach is verified through comparisons with exact values and the results are further analyzed through graphs and numerical data to compare the two fractional operators.
Article
Mathematics, Applied
Ling An, Liming Ling, Xiaoen Zhang
Summary: In this paper, an integrable fractional derivative nonlinear Schrodinger equation is proposed and a reconstruction formula of the solution is obtained by constructing an appropriate Riemann-Hilbert problem. The explicit fractional N-soliton solution and the rigorous verification of the fractional one-soliton solution are presented.
PHYSICA D-NONLINEAR PHENOMENA
(2024)
Article
Engineering, Mechanical
Jianping Wu
Summary: In this paper, a novel reduction approach is proposed to obtain N-soliton solutions of a physically meaningful nonlocal nonlinear Schrodinger equation of reverse-time type. Firstly, single-soliton solutions are obtained by reducing those of the local NLS equation. Secondly, N-soliton representations are conjectured and verified via an algebraic proof, and special soliton dynamics are theoretically explored and graphically illustrated to demonstrate the features of the soliton solutions. The merit of the proposed reduction approach lies in its purely algebraic nature which eliminates the need for complicated spectral analysis of the corresponding Lax pair.
NONLINEAR DYNAMICS
(2021)
Article
Mathematics, Applied
Mubashir Qayyum, Efaza Ahmad, Hijaz Ahmad, Bandar Almohsen
Summary: This manuscript focuses on the solution and analysis of time and space fractional coupled Schro center dot dinger system through a new approach, and examines the effect of fractional parameters on wave profiles through numerical analysis.
Article
Engineering, Mechanical
Sheng Zhang, Huimin Zhou
Summary: This study explores the feasibility of using the Riemann-Hilbert method to understand the soliton dynamics with variable velocity and amplitude in mixed spectral nonlinear integrable systems. By considering a novel mixed spectral complex mKdV equation with time-varying coefficients, a Lax representation is obtained and a Riemann-Hilbert problem is established. The scattering data are then determined to reconstruct the potential, leading to the N-soliton solution of the mscmKdV equation. The one- and two-soliton solutions are presented to analyze the characteristics of soliton dynamics in this equation, revealing different propagation behaviors compared to constant-coefficient isospectral models.
NONLINEAR DYNAMICS
(2023)
Article
Engineering, Electrical & Electronic
Muyiwa Balogun, Stanislav Derevyanko
Summary: Nonlinear frequency division multiplexing (NFDM) is a promising concept in optical fiber communications, and it can improve spectral efficiency by using Hermite-Gaussian spectral carriers.
JOURNAL OF LIGHTWAVE TECHNOLOGY
(2022)
Article
Mathematics, Interdisciplinary Applications
Da-Sheng Mou, Chao-Qing Dai, Yue-Yue Wang
Summary: In this study, we discuss the inverse scattering transform, anomalous dispersion relations, and squared eigenfunctions of the integrable fractional n-component coupled nonlinear Schrodinger model using the integrable nonlinear model introduced by Ablowitz et al. with the Riesz fractional derivative. We obtain the explicit form of this fractional model and its fractional n-soliton solution by solving Riemann-Hilbert problems through the inverse scattering transform. We also analyze the effect of the fractional-order exponent on the one-and two-soliton solutions of the integrable fractional three-component coupled nonlinear Schrodinger model.
CHAOS SOLITONS & FRACTALS
(2023)
Article
Mathematics, Applied
Kang-Jia Wang, Guo-Dong Wang
Summary: This paper investigates the time-space fractional Drinfeld-Sokolov-Wilson system and solves its solitary solutions using He's variational method and the two-scale transform. The applicability and efficiency of the approach are demonstrated through numerical results. The main advantage of the variational approach is its ability to simplify the differential equation.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Physics, Mathematical
Bo Xu, Yufeng Zhang, Sheng Zhang
Summary: This article introduces the fractional KP equation to simulate shallow ocean waves and construct novel spatial structures. It derives formulas for fractional n-soliton solutions and constructs fractional line one-solitons with bend, wavelet peaks, and peakon. The study also simulates interactions in the fractional line two-and three-soliton solutions, as well as the nonlinear dynamic evolution of the fractional lump solution.
ADVANCES IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Bo Xu, Yufeng Zhang, Sheng Zhang
Summary: This paper introduces a conformable fractional partial derivative to construct fractional rogue waves and derives a fractional Schrodinger equation with Lax integrability. First- and second-order fractional rogue wave solutions are obtained, which are steeper and have translational coordination on the coordinate plane. The time taken by the second-order fractional rogue wave solution from the beginning to the end is short.
Article
Mathematics, Applied
Bo Xu, Yufeng Zhang, Sheng Zhang
Summary: This study introduces local fractional order partial derivatives to the AKNS spectral problem and its adjoint equations, resulting in the derivation of two previously unreported fractional order AKNS hierarchies. These hierarchies are shown to be significant for the construction of N-fractal solutions and offer new insights into the exploration of soliton equation hierarchies.
ADVANCES IN DIFFERENCE EQUATIONS
(2021)
Article
Multidisciplinary Sciences
Bo Xu, Sheng Zhang
Summary: In this paper, analytical solutions including the N-fractal-soliton solution and the long-time asymptotic solution of a local time-fractional NLS-type equation are obtained using the RH approach. The study also establishes infinitely many conservation laws and simulates fractal-soliton solutions, showcasing different geometric characteristics on two scales.
Article
Mathematics, Interdisciplinary Applications
Sheng Zhang, Dexin Zhang
Summary: This paper derives a new and more general variable-coefficient KP equation using the Lax scheme, and constructs exact wave solutions including rational solutions and Jacobi elliptic function solutions using the inverse scattering transform and the F-expansion method. The significance lies in extending the IST to high-dimensional models with variable coefficients and obtaining novel solution structures of the vcKP equation.
GEM-INTERNATIONAL JOURNAL ON GEOMATHEMATICS
(2021)
Article
Engineering, Mechanical
Sheng Zhang, Xiaowei Zheng
Summary: This paper discusses two nonlinear evolution models and obtains novel N-soliton solutions of generalized Broer-Kaup systems using the bilinear method. It demonstrates the characteristics of soliton solutions in different dimensions and reveals the dynamics within these solutions.
NONLINEAR DYNAMICS
(2022)
Article
Mathematics
Sheng Zhang, Bo Xu
Summary: In this paper, the Painleve integrable property of the (1 + 1)-dimensional generalized Broer-Kaup equations is proven. Backlund transformations and reduction methods for the equations are derived, leading to the construction of exact solutions. The characteristics of the solutions are shown through three-dimensional images.
Article
Mathematics
Bo Xu, Sheng Zhang
Summary: In this paper, the Lax representation and associated Riemann-Hilbert problem of a generalized nonlinear Schrodinger equation with time-varying coefficients are studied. The solution of the equation is transformed into the associated RH problem, and the time evolution laws of the scattering data reconstructing the potential of the equation are determined. The long-time asymptotic solution and N-soliton solution of the equation are obtained based on the determined time evolution laws of scattering data. The results show that the RH method can be extended to nonlinear evolution models with variable coefficients.
Article
Mathematics
Sheng Zhang, Jiao Gao, Bo Xu
Summary: This work derives a novel integrable evolution system associated with a mixed spectral AKNS matrix problem using Lax's scheme, and gives the time dependences of scattering data in the inverse scattering analysis. The reconstruction of potentials is carried out based on the time dependences of scattering data, and analytical solutions with four arbitrary functions are formulated. It is shown in this study that other systems of integrable evolution equations can be constructed by embedding different spectral parameters and time-varying coefficient functions to the known AKNS matrix spectral problem under the resolvable framework of the inverse scattering method.
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
Engineering, Mechanical
Sheng Zhang, Huimin Zhou
Summary: This study explores the feasibility of using the Riemann-Hilbert method to understand the soliton dynamics with variable velocity and amplitude in mixed spectral nonlinear integrable systems. By considering a novel mixed spectral complex mKdV equation with time-varying coefficients, a Lax representation is obtained and a Riemann-Hilbert problem is established. The scattering data are then determined to reconstruct the potential, leading to the N-soliton solution of the mscmKdV equation. The one- and two-soliton solutions are presented to analyze the characteristics of soliton dynamics in this equation, revealing different propagation behaviors compared to constant-coefficient isospectral models.
NONLINEAR DYNAMICS
(2023)
Article
Thermodynamics
Bo Xu, Yufeng Zhang, Sheng Zhang
Summary: This paper investigates two local fractional partial differential systems and obtains exact solutions using the local fractional variational iteration method. It is shown that this method is an effective mathematical tool for solving linear and non-linear local fractional partial differential systems with initial and boundary values.
Article
Mathematics, Applied
Bo Xu, Sheng Zhang
Summary: In this paper, a direct method called negative power expansion (NPE) method is presented and extended to construct exact solutions of nonlinear mathematical physical equations. The method is effective for various types of equations, including coupled, variable-coefficient, and special types. By applying the method to different equations, various exact solutions such as traveling wave solutions, non-traveling wave solutions and semi-discrete solutions are obtained. The research demonstrates that the NPE method is a simple and effective approach for solving nonlinear equations in mathematical physics.
JOURNAL OF MATHEMATICAL PHYSICS ANALYSIS GEOMETRY
(2021)
Article
Thermodynamics
Bo Xu, Yufeng Zhang, Sheng Zhang
Summary: This paper investigates two fractional extended versions of the Kolmogorov-Petrovskii-Piskunov equation, obtaining some exact solutions. The fractional order affects the propagation velocity of the obtained solutions.
Article
Mechanics
Sheng Zhang, Yue Zhang, Bo Xu
Summary: Rogue wave solutions of the Davey-Stewartson equations were obtained using the exp-function method and reduction transformations. The process involved transforming the DS equations into easier-to-solve equations and utilizing known solutions of the deformed NLS equation to construct rogue wave solutions. The obtained rogue wave solutions were then analyzed for spatial and spatiotemporal structures and dynamic evolutionary plots.
JOURNAL OF APPLIED AND COMPUTATIONAL MECHANICS
(2021)
