Article
Materials Science, Multidisciplinary
M. M. M. Jaradat, Amna Batool, Asma Rashid Butt, Nauman Raza
Summary: The goal of this research is to find novel optical solutions to the Kundu-Eckhaus equation, which plays a crucial role in the field of nonlinear optics. A collective variable strategy is adopted to solve governing equation, and the results obtained from this approach can be applied to solve a variety of nonlinear problems.
RESULTS IN PHYSICS
(2022)
Article
Mathematics, Applied
Shou-Fu Tian, Jian-Min Tu, Tian-Tian Zhang, Yi-Ren Chen
Summary: This study investigates integrable discretizations of the Eckhaus-Kundu equation, presenting x-discrete, t-discrete, and fully discrete versions based on a bilinear form. Furthermore, soliton solutions of the derived discrete equations are successfully constructed using Hirota's bilinear method.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics
Karim K. Ahmed, Niveen M. Badra, Hamdy M. Ahmed, Wafaa B. Rabie
Summary: Our paper studies optical solitons for the Kundu-Eckhaus equation with quintic nonlinearity and Raman effect. By applying the improved modified extended tanh-function method, various soliton solutions and their physical nature are obtained and illustrated graphically.
Article
Mathematics, Applied
Yingmin Yang, Tiecheng Xia, Tongshuai Liu
Summary: In this paper, the n-component nonlocal Kundu-Eckhaus equation and its Darboux transformation are presented. The N-soliton solution and the one-exact solution to the equation are obtained. The difference between the solutions of the nonlocal and local Kundu-Eckhaus equations is that the former has symmetric constraints. Furthermore, specific parameters are used to draw the images of rogue wave solutions for a three-component nonlocal KE equation.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Optics
E. Parasuraman
Summary: This study investigates the modulational instability of nonlinear wave propagation in birefringent fiber of the Kundu-Eckhaus equation without four wave mixing term, and analyzes the effects of self phase modulation and cross phase modulation. The linear stability analysis is applied to the Kundu-Eckhaus equation in the absence of four wave mixing term, and modulational instability criteria are found from linearized equations for small perturbations. The study discusses the effects of self phase modulation and cross phase modulation on optical wave propagation in birefringent fiber using modulational instability criteria.
Article
Engineering, Electrical & Electronic
Karim K. K. Ahmed, Niveen M. M. Badra, Hamdy M. M. Ahmed, Wafaa B. B. Rabie
Summary: This paper introduces the generalized Kundu-Eckhaus equation (KEE) with extra-dispersion, which describes the propagation of ultra-short femtosecond pulses in an optical fiber. Many exact solutions are constructed using the improved modified extended tanh-function method and a new transformation. As a result, a variety of new families of exact traveling wave solutions are found, including bright solitons, dark solitons, singular solitons, Weierstrass elliptic doubly periodic solutions, Jacobi elliptic solutions, periodic solutions, and rational solutions. Physical interpretation for some of the obtained solutions are illustrated in Figures.
OPTICAL AND QUANTUM ELECTRONICS
(2023)
Article
Optics
Yu-Han Deng, Xiang-Hua Meng, Gui-Min Yue, Yu-Jia Shen
Summary: This paper investigates the propagation of ultrashort femtosecond pulses in optical fibers using the Kundu-Eckhaus (KE) equation. The nonlocal KE equation, also known as the nonlocal integrable nonlinear Schrodinger equation with cubic and quintic nonlinearities, is solved using the Hirota bilinear method. The N-soliton solution is derived using symbolic calculation, and the exact solution expressions for two-soliton and three-soliton are obtained. Various propagation situations, such as periodic solitary wave evolution and collision of two parallel, perpendicular, and periodic solitary waves, are demonstrated and discussed under different parameters.
Article
Mathematics, Applied
Shikun Cui, Zhen Wang
Summary: This paper develops the numerical inverse scattering transform (NIST) for the Kundu-Eckhaus equations, focusing on both the focusing and defocusing cases. The NIST consists of numerical direct scattering and numerical inverse scattering, which utilize the Chebyshev collocation method improved by tanh mapping and the Deift-Zhou nonlinear steepest descent method combined with Olver's numerical method, respectively. Unlike traditional methods, the NIST does not require advance time and is more effective for long-time evolution of solutions, making it of great significance.
PHYSICA D-NONLINEAR PHENOMENA
(2023)
Article
Physics, Multidisciplinary
Si-Yu Hong, Wei-Guo Zhang, Yu-Li Guo, Xing-Qian Ling
Summary: This paper mainly studies the orbital stability of the periodic traveling wave solution for the Eckhaus-Kundu equation with quintic nonlinearity. By constructing conserved quantities, conducting detailed spectral analysis, and employing appropriate techniques, the complexity of the equation is overcome, resulting in a conclusion on the orbital stability of the dn periodic wave solution.
Review
Engineering, Mechanical
Xuedong Chai, Yufeng Zhang
Summary: The boundary value problem for the focusing Kundu-Eckhaus equation with nonzero boundary conditions is studied using the Dbar dressing method in this work. The soliton solution is investigated by introducing a Dbar problem with non-canonical normalization condition at infinity. The eigenfunction of the Dbar problem is used to construct the Lax pair of the Kundu-Eckhaus equation, which is crucial for further searching for soliton solutions. Moreover, the N-soliton solutions of the focusing Kundu-Eckhaus equation with nonzero boundary conditions are discussed based on symmetries and distribution.
NONLINEAR DYNAMICS
(2023)
Article
Optics
E. Parasuraman
Summary: The role of inter modal dispersion on modulation instability of optical soliton in the presence of self phase modulation and cross phase modulation is investigated analytically. The modulational instability criterion for the Kundu-Eckhaus equation is obtained and used to analyze the effect of inter modal dispersion on the modulation instability of optical soliton in birefringent fiber. Through graphical illustration of modulation instability gain, the role of inter modal dispersion on modulation instability in optical fiber is discussed in the case of absence and presence of self phase modulation and cross phase modulation.
Article
Materials Science, Multidisciplinary
Gui-Min Yue, Xiang-Hua Meng
Summary: In this paper, the authors introduce a complex matrix into the differential relation satisfied by determinant elements based on the Hirota bilinear form of the Kundu-Eckhaus equation. They derive the solution in the generalized double Wronskian determinant form for the KE equation by constructing relations among the matrix elements. Soliton solutions and Jordan block solutions of the KE equation are obtained when the complex matrix introduced in the differential relation takes a diagonal matrix and Jordan block matrix respectively, and propagation situations are discussed via different parameters.
RESULTS IN PHYSICS
(2023)
Article
Mathematics, Applied
Jinghua Luo, Engui Fan
Summary: The spatial and time spectral problems associated with the Kundu-Eckhaus (KE) equation are derived via linear constraint equations, a KE hierarchy with source is proposed, N-solitons of the KE equation are constructed, and explicit one- and two-soliton solutions are obtained.
JOURNAL OF GEOMETRY AND PHYSICS
(2021)
Article
Engineering, Multidisciplinary
H. A. Eldidamony, Hamdy M. Ahmed, A. S. Zaghrout, Y. S. Ali, Ahmed H. Arnous
Summary: In this paper, the modified simple equation method and the extended simplest equation method are used to investigate the secure optical solitons and other solutions of the Radhakrishnan-Kundu Lakshmanan equation. Various types of solitons and solutions are obtained and presented graphically.
ALEXANDRIA ENGINEERING JOURNAL
(2022)
Article
Mathematics, Applied
Aly R. Seadawy, Mujahid Iqbal
Summary: In this research work, we obtained the optical soliton solutions of the nonlinear complex Kundu-Eckhaus (KE) equation using a modified mathematical method. These solutions include dark solitons, bright solitons, combined dark-bright solitons, travelling wave solutions, and periodic wave solutions with general coefficients. These solutions are useful in various fields such as optical fiber development, soliton dynamics, adiabatic parameter dynamics, fluid dynamics, biomedical problems, and industrial phenomena. The technique used in this research proves to be powerful, effective, and fruitful for studying other higher-order nonlinear complex PDEs in fields like mathematical physics, quantum physics, geophysics, fluid mechanics, mathematical biology, engineering, and other physical sciences.
APPLIED MATHEMATICS-A JOURNAL OF CHINESE UNIVERSITIES SERIES B
(2023)
Article
Engineering, Electrical & Electronic
Ali Murat Yalci, Mehmet Ekici
Summary: The paper focuses on the retrieval of stationary soliton solutions to the complex Ginzburg-Landau equation using Jacobi's elliptic function approach, leading to soliton solutions under certain conditions.
OPTICAL AND QUANTUM ELECTRONICS
(2022)
Article
Physics, Multidisciplinary
Anjan Biswas, Mehmet Ekici, Abdullah Sonmezoglu
Summary: This paper implements extended trial function algorithm to retrieve stationary optical soliton solutions to the governing nonlinear Schrodinger's equation with Kudryashov's lately proposed quintuple power law of refractive index, in the presence of nonlinear chromatic dispersion. Both, linear temporal evolution as well as generalized temporal evolution are taken into account.
Article
Mathematics, Interdisciplinary Applications
Mehmet Ekici
Summary: This paper discusses the interaction of multi-wave solutions of the model with Kudryashov's quintuple form coupled with the dual form of nonlocal law of refractive index, including kinky breathers, W-shaped and multi-peaked solitons. Analytical results are supported by numerical simulations of the obtained solutions.
CHAOS SOLITONS & FRACTALS
(2022)
Article
Optics
Mehmet Ekici
Summary: This paper investigates the recovery of stationary optical solitons using Kudryashov's recently proposed nonlinear refractive index structure with quintuple power laws. The extended Jacobi's elliptic function expansion is employed as the integration algorithm, considering both linear and generalized formats of the temporal evolution term.
JOURNAL OF NONLINEAR OPTICAL PHYSICS & MATERIALS
(2023)
Article
Optics
Anjan Biswas, Abdullah Sonmezoglu, Mehmet Ekici
Summary: This work is a sequel to the previous study on stationary optical solitons with different forms of Kudryashov's law of self-phase modulation. The study explores generalized temporal evolution using an extended trial function approach, resulting in solutions expressed in terms of Jacobi's elliptic function. These solutions eventually lead to stationary solitons approaching the appropriate limit of ellipticity modulus.
JOURNAL OF NONLINEAR OPTICAL PHYSICS & MATERIALS
(2023)
Article
Engineering, Electrical & Electronic
Mehmet Ekici
Summary: The extended Jacobi's elliptic function approach is used in this paper to study optical solitons in the nonlinear Schrodinger's equation. This approach can recover various types of soliton solutions, including bright solitons, dark solitons, singular solitons, and dark-singular form of straddled solitons.
OPTICAL AND QUANTUM ELECTRONICS
(2022)
Article
Materials Science, Multidisciplinary
Abdullah Sonmezoglu, Mehmet Ekici, Anjan Biswas
Summary: This paper investigates solitons solutions by utilizing Kudryashov's law of refractive index with quadruple power-law, integrating using undetermined coefficients and Jacobi's elliptic functions. A wide range of soliton solutions are recovered as the ellipticity modulus approaches unity.
OPTOELECTRONICS AND ADVANCED MATERIALS-RAPID COMMUNICATIONS
(2022)
Article
Quantum Science & Technology
Yakup Yildirim, Anjan Biswas, Padmaja Guggilla, Salam Khan, Mehmet Ekici, Luminita Moraru, Houria Triki, E. M. E. Zayed, Abdullah K. Alzahrani, Milivoj R. Belic
Summary: This paper investigates the recovery of solitons using optical metamaterials in nonlinear directional couplers. Both twin-core and multiple-core couplers are considered, and six different nonlinear forms of refractive index are studied. The sine-Gordon equation approach is used for integration.
NONLINEAR OPTICS QUANTUM OPTICS-CONCEPTS IN MODERN OPTICS
(2022)
Article
Multidisciplinary Sciences
Elsayed M. E. Zayed, Mohamed E. M. Alngar, Anjan Biswas, Mehmet Ekici, Salam Khan, Abdullah K. Alzahrani, Milivoj R. Belic
Summary: This paper recovers cubic-quartic perturbed solitons in fiber Bragg gratings with quadratic-cubic law nonlinear refractive index using the unified Riccati equation expansion method and the modified Kudryashov's approach. The parameter constraints for the existence of such solitons are also presented.
PROCEEDINGS OF THE ESTONIAN ACADEMY OF SCIENCES
(2022)
Article
Optics
Elsayed M. E. Zayed, Mohamed E. M. Alngar, Anjan Biswas, Mehmet Ekici, Padmaja Guggilla, Salam Khan, Hashim M. Alshehri, Milivoj R. Belic
Summary: The paper discusses optical solitons generated by fiber Bragg gratings and the polynomial law of nonlinear refractive index. Soliton solutions to the model are identified utilizing the auxiliary equation approach and an addendum to Kudryashov's method, leading to the emergence of singular periodic solutions as a byproduct.
OPTICS AND SPECTROSCOPY
(2022)
Article
Materials Science, Multidisciplinary
Abdullahi Rashid Adem, Basetsana Pauline Ntsime, Anjan Biswas, Anelia Dakova, Mehmet Ekici, Yakup Yidirim, Hashim M. Alshehri
Summary: This paper examines the stationary optical soliton solutions to the nonlinear Schrodinger's equation that satisfy Kudryashov's law of refractive index. Both linear and generalized temporal evolutions are considered, and the results are expressed in terms of Appell's hypergeometric function.
OPTOELECTRONICS AND ADVANCED MATERIALS-RAPID COMMUNICATIONS
(2022)
Article
Optics
Jose Vega-Guzman, Anjan Biswas, Mir Asma, Aly R. Seadawy, Mehmet Ekici, Abdullah Khamis Alzahrani, Milivoj R. Belic
Summary: This paper explores exact bright, dark, and singular soliton solutions with parabolic-nonlocal combo nonlinearity in polarization-preserving optical fibers, where perturbation terms are of Hamiltonian type. Subsequently, an analytical bright 1-soliton solution is obtained in the presence of higher-order dispersion effects through the semi-inverse variational principle.
JOURNAL OF OPTICS-INDIA
(2022)
Article
Optics
O. Gonzalez-Gaxiola, Anjan Biswas, Mehmet Ekici, Salam Khan
Summary: This paper numerically studies highly dispersive bright and dark optical soliton solutions using the variational iteration method. The model considers a quadratic-cubic nonlinear form of refractive index and recovers soliton solutions with an impressive error measure.
JOURNAL OF OPTICS-INDIA
(2022)
