Eckhaus Equation Research Articles

On the dispersive shock waves of the defocusing Kundu–Eckhaus equation in an optical fiber

On the dispersive shock waves of the defocusing Kundu–Eckhaus equation in an optical fiber

On the analysis and deeper properties of the fractional complex physical models pertaining to nonsingular kernels.

This study solves the coupled fractional differential equations defining the massive Thirring model and the Kundu Eckhaus equation using the Natural transform decomposition method. The massive Thirring model is a dynamic component of quantum field theory, consisting of a coupled nonlinear complex differential equations. Initially, we study the suggested equations under the fractional derivative of Caputo-Fabrizio. The Atangana-Baleanu derivative is then used to evaluate the comparable equations. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approach and the fractional operators in this situation to develop satisfactory approximations to the offered problems. The proposed approach combines the natural transform technique with the efficient Adomian decomposition scheme. Obtaining numerical findings in the form of a fast-converge series significantly improves the scheme's accuracy. Some graphical plot distributions are presented to show that the present approach is very simple and straightforward. We performed a fractional order analysis of assumed phenomena to demonstrate and validate the effectiveness of the future technique. The behaviour of the approximate series solution for several fractional orders is shown visually. Additionally, the nature of the derived outcome has been observed for various fractional orders. The derived results demonstrate how simple and efficient the proposed method is to apply for analysing the behaviour of fractionally-order complex nonlinear differential equations that arise in related fields of engineering and science.

Whitham modulation theory and Riemann problem for the Kundu–Eckhaus equation

In this paper, the Riemann problem for the defocusing Kundu–Eckhaus equation is investigated by Whitham modulation theory. First, we study the dispersion relation for linear waves. Then, the zero-phase and one-phase periodic solutions of the Kundu–Eckhaus equation along with the corresponding Whitham modulation equations are derived by the finite-gap integration method. Further, employing the Whitham equations parametrized by the Riemann invariants, the main fundamental wave structures induced by the discontinuous initial data are found. Analytical and graphic methods are utilized to provide the wave structures of rarefaction waves and dispersive shock waves, and thus for a complete classification of solutions under general step-like conditions of initial discontinuity.

Stability and instability nature of solitons in an optical fiber with four wave mixing effect

The investigation into modulational instability (MI) within the Kundu-Eckhaus (KE) equation, governing optical solitons, involves a thorough examination of the effects of self-phase modulation, cross-phase modulation, and intermodal dispersion. Special attention is given to understanding the influence of the four-wave mixing effect. The KE equation, which models birefringent fiber and includes terms related to intermodal dispersion, cross-phase modulation, and self-phase modulation, serves as the fundamental framework for this analytical study. Employing conventional linear stability analysis, the gain within the KE equation is determined. To shed light on the role of four-wave mixing in various scenarios, the gain spectrum is utilized as a tool to analyze the behavior of the KE equation under different conditions. This methodology seeks to provide insightful information about the intricate interactions that impact the modulational instability of solitonic pulses in an optical systems. After that, we have investigated the soliton solution by implementing the Jacobian elliptical function approach. Finally, our focus here is on linear stability analysis, which employs eigenvalue spectra to study solitons’ stability via direct numerical simulation.

Analytical solutions of the space–time fractional Kundu–Eckhaus equation by using modified extended direct algebraic method

The study of soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) has gained prominence recently because of its ability to realistically recreate complex physical processes. Numerous mathematical techniques have been devised to handle the problem of NFPDEs where soliton solutions are difficult to obtain. Due to their accuracy in reproducing complex physical phenomena, soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) have recently attracted interest. Several mathematical techniques have been devised to tackle the difficult task of solving non-finite partial differential equations (NFPDEs) soliton. Studies of soliton solutions for Nonlinear Fractional Partial Differential Equations (NFPDEs) have garnered increased attention recently due to its capacity to accurately represent complex physical processes. Due to the difficulty of obtaining soliton solutions, NFPDEs can be solved using a wide variety of mathematical methods. In this way, it facilitates the extraction of the recently found abundance of optical soliton solutions. To further understanding of the results, the study also includes contour and three-dimensional images that visually depict particular optical soliton solutions for particular parameter selections, suggesting the existence of different soliton structures in the nonlinear fractional Kundu–Eckhaus equation (NFKEE) region. It is shown that the proposed technique is quite powerful and effective in solving several nonlinear FDEs.

Binary Darboux transformation and N-dark solitons for the defocusing Kundu-Eckhaus equation in an optical fiber

Binary Darboux transformation and N-dark solitons for the defocusing Kundu-Eckhaus equation in an optical fiber

Asymptotic analysis of high‐order solitons of an equivalent Kundu–Eckhaus equation

In this work, we study the asymptotic characteristics of high‐order solitons for the focusing Kundu–Eckhaus (KE) equation. Based on the loop group theory, we construct the general Darboux transformation within the framework of Riemann–Hilbert problems to derive the general high‐order soliton solution. Using high‐order Bäcklund transformation, we derive the leading order term of the determinant solution to obtain the asymptotic representation for the high‐order soliton solution. Furthermore, this method is also extended to the construction of more general high‐order cases with multiple poles. We further find that if a soliton propagates along the logarithm characteristic curve, the high‐order soliton can be decomposed into individual solitons with the same amplitude and velocity. Finally, these solutions are theoretically and graphically analyzed in detail.

Highly Dispersive Optical Gap Solitons with Kundu-Eckhaus Equation Having Multiplicative White Noise

This paper explores the dynamics of highly dispersive gap solitons within the framework of the Kundu-Eckhaus equation, augmented by the inclusion of multiplicative white noise in the Itô sense, a novel addition to the model. Two integration schemes, the extended simplest equation approach and the generalized Riccati equation mapping scheme, are employed to analyze and integrate the model. Despite yielding bright, singular, and dark-singular straddled solitons, both methods independently fail to capture dark optical solitons. Additionally, our investigation highlights that the influence of white noise predominantly affects the phase aspect of the solitons, with negligible impact on their amplitude. Further details regarding the specific physical system or optical medium under study would provide readers with context, aiding in the comprehension of the significance of our findings.

Nonlocal Kundu–Eckhaus equation: integrability, Riemann–Hilbert approach and Cauchy problem with step-like initial data

Nonlocal Kundu–Eckhaus equation: integrability, Riemann–Hilbert approach and Cauchy problem with step-like initial data

A new approach for solving fractional Kundu-Eckhaus equation and fractional massive Thirring model using controlled Picard’s technique with [formula omitted]-Laplace transform

This paper introduces a novel approach called the Controlled Picard technique along with ρ-Laplace transform for simulating the solutions of the fractional Kundu–Eckhaus equation (FKEE) and time-fractional massive Thirring model (FMTM). Significant applications of these equations can be found in nonlinear optics, weakly dynamic dispersive waves of water, and the study of quantum fields. Our proposed approach merges the Controlled Picard technique with the ρ-Laplace transform, resulting in an elegant framework that effectively solves nonlinear fractional equations without the need for Lagrange multipliers or Adomian polynomials. The inclusion of a small parameter h enhances convergence and is particularly advantageous for nonlinear differential equations. We demonstrate the accuracy and convergence of our method by comparing it with other analytical approaches. In addition, we offer graphical depictions of the numerical results to demonstrate the effectiveness as well as reliability of our suggested methodology. Overall, our approach is highly efficient, accurate, and straightforward to implement.

Exact solutions and modulation instability analysis of a generalized Kundu-Eckhaus equation with extra-dispersion in optical fibers

This study aims to examine the nonlinear partial differential equation known as the (1+1)-dimensional generalized Kundu-Eckhaus equation with extra-dispersion, which is used to model the transmission of ultra-short femtosecond pulses in an optical fiber. Two versatile techniques, namely the extended G′G2 -expansion as well as the extended exp(−ϕ(ξ)) -expansion techniques, are utilized to generate numerous precise answers. Diverse novel collections of exact traveling wave solutions, such as bright solitons, dark solitons, singular solitons, W-shape solutions, M-shape solutions, and rational solutions, are identified as a result. Several of the acquired solutions are interpreted physically through the use of figures. In addition, the modulation instability analysis of the considered equation is performed and presented via 3D and 2D graphs. In the field of nonlinear sciences, the proposed methods have great value and can be applied to other nonlinear evolutionary equations that are used to represent nonlinear physical models.

Multi-parallelized PINNs for the inverse problem study of NLS typed equations in optical fiber communications: Discovery on diverse high-order terms and variable coefficients

In this paper, we focus on the inverse problem study of nonlinear Schrödinger (NLS) typed equations in optical fiber communicaitons. As an extension of the physics-informed neural network (PINN), multi-parallelized PINNs are constructed and trained for the discovery of diverse high-order terms and variable coefficients. We firstly study various constant-coefficient combinations of a generalized high-order NLS typed equation, where the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation and the Kundu-Eckhaus equation are included. With small amount of exact solutions available to us, we predict the value of multiple coefficients under different cases to deduce the undetermined terms of the generalized equation based on the multi-parallelized PINN. Different categories of NLS typed equations are then inferred. In the meantime, high accuracy numerical solutions on localized regions can be accordingly obtained.The parameter discovery of NLS typed equations with variable coefficients has also been carried out based on the extended network, including analysis on interaction behaviors and the periodic phenomenon of solutions. According to outputs of multi-parallelized PINNs, we compare the numerical solutions and the predicted variable coefficients with exact results. Error analysis are then performed to check the accuracy of prediction, where both the absolute error and the mean squared error are given. Compared with the traditional PINN, our model exerts its state-of-art power in the inverse problem study of nonlinear systems, where different high-order terms and variable-coefficient terms can be clearly predicted while deducing diverse types of localized numerical solutions with lower fitting error and less data consumption. As the self-steepening pulses without self-phase modulation and the Raman effect are closely related to the inferred high-order terms of NLS typed equations, our research will serve as experimental basis in the field of optical fiber communications.

Analysis of solitary wave solutions in the fractional-order Kundu–Eckhaus system

The area of fractional partial differential equations has recently become prominent for its ability to accurately simulate complex physical events. The search for traveling wave solutions for fractional partial differential equations is a difficult task, which has led to the creation of numerous mathematical approaches to tackle this problem. The primary objective of this research work is to provide optical soliton solutions for the Frictional Kundu–Eckhaus equation (FKEe) by utilizing generalized coefficients. This strategy utilizes the Riccati–Bernoulli sub-ODE technique to effectively discover the most favorable traveling wave solutions for fractional partial differential equations. As a result, it facilitates the extraction of optical solitons and intricate wave solutions. The Backlund transformation is used to methodically construct a sequence of solutions for the specified equations. The study additionally showcases 3D and Density graphics that visually depict chosen solutions for certain parameter selections, hence improving the understanding of the outcomes.

Optical solitons of time fractional Kundu–Eckhaus equation and massive Thirring system arises in quantum field theory

Optical solitons of time fractional Kundu–Eckhaus equation and massive Thirring system arises in quantum field theory

Integrable turbulence and statistical characteristics of chaotic wave field in the Kundu-Eckhaus equation.

Integrable turbulence, as an irregular behavior in dynamic systems, has attracted a lot of attention in integrable and Hamiltonian systems. This article focuses on the studies of integrable turbulence phenomena of the Kundu-Eckhaus (KE) equationas well as the generation of rogue waves from the numerical and statistical viewpoints. First, via the Fourier collocation method, we obtain the spectral portraits of different analytical solutions. Second, we perform the numerical simulation on the KE equationunder the initial condition of a plane wave with random noise to simulate the chaotic wave fields. Then, we analyze the influences of standard deviation and correlation length on the integrable turbulence and amplitude of wave field. It's found that both of the two parameters have positive effects on the generation probability of rogue wave caused by the interactions. But only the variation of standard deviation can lead to the transition from the breather turbulence to soliton turbulence. Furthermore, by analyzing the effects of additional higher-order nonlinear terms on the chaotic wave field, we find that those two higher-order nonlinear effects in the KE equationcan lead to a larger amplitude of the chaotic wave field and a higher probability of generating rouge waves compared with the NLS equation.

Soliton resolution and asymptotic stability of N-solutions for the defocusing Kundu–Eckhaus equation with nonzero boundary conditions

In this paper, we address interesting soliton resolution, asymptotic stability of N-soliton solutions and the Painlevé asymptotics for the Kundu-Eckhaus (KE) equation with nonzero boundary conditions iqt+qxx−2(∣q∣2−1)q+4β2(∣q∣4−1)q+4iβ∣q∣2xq=0,q(x,0)=q0(x)∼±1,x→±∞. The key to proving these results is to establish the formulation of a Riemann-Hilbert (RH) problem associated with the above Cauchy problem and find its connection with the RH problem of the defocusing NLS equation. With the ∂¯ -steepest descent method and the results of the defocusing NLS equation, we find complete leading order approximation formulas for the defocusing KE equation on the whole (x,t) half-plane including soliton resolution and asymptotic stability of N-soliton solutions in a solitonic region, Zakharov-Shabat asymptotics in a solitonless region and the Painlevé asymptotics in two transition regions.

Stability of kink, anti kink and dark soliton solution of nonlocal Kundu–Eckhaus equation

We study the modulational instability and soliton solution for nonlocal Kundu–Eckhaus (KE) equation which is distinctly different from KE equation. KE equation can describe the dynamical behavior of optical pulse envelopes in birefringent fiber. Although, the present equation has novel space, which induce new physical effects and inspire novel physical applications. Therefore, through the Jacobi elliptic function method, the elliptic function solution is derived for the nonlocal KE. Deng et al. (2022) found the N-soliton solution for nonlocal KE equation by the use of Hirota bilinear method. Though, the present problem of the solution is new which is Jacobi elliptic function solution. It is found that the solution exhibits a variety of forms of soliton as dark, kink and antikink soliton due to distinct choices of values of focusing and defocusing parameters. Importantly, with the use of the Jacobi elliptic solution, the role of focusing and defocusing parameters of cubic, quintic nonlinearity and self frequency shift on optical soliton in birefringent fiber for nonlocal KE equation is discussed. The linear stability analysis is used to construct the gain of nonlocal Kundu–Eckhaus equation and using it to analysis the stability/instability of various forms of soliton.

Dynamic analysis on optical pulses via modified PINNs: Soliton solutions, rogue waves and parameter discovery of the CQ-NLSE

Under investigation in this paper is the cubic–quintic nonlinear Schrödinger equation, which describes the propagation of optical on resonant-frequency fields in the inhomogeneous fiber. According to abundant previous researches on the model, exact soliton solutions and rogue wave solutions have been derived through Darboux transformation. The modulation instability phenomenon has been analyzed to evaluate the ability of an initially perturbed plane wave to split into localized energy packets when propagating in a dispersive and nonlinear medium.Numerical solutions with high accuracy are needed in fields of production and engineering. Nonetheless, the data acquisition costs of the optical pulse transmission system is high, which will limit the accuracy and the efficiency of typical numerical and data-driven methods. With the physical knowledge embedded into neural networks in the form of loss function, the problem of big data dependence has been solved. For dynamic analysis on optical pulses with small amount of known information, we strive to obtain high accuracy numerical solutions. Considering the case that the cubic–quintic nonlinear Schrödinger equation is converted to the Kundu–Eckhaus equation with simplified coefficient constraints through variable transformation, we construct modified physics-informed neural networks, where conversions on the input and output are attached to deep neural networks. Training networks with the given initial and boundary data, we effectively derive the expected soliton and rogue wave solutions, where the approximated one-soliton, two-soliton, first-order and second-order rogue waves are included. In general, the modified network reaches low prediction errors with small data available. With the coefficients of equations, the weights and the bias of networks combined as parameters to be trained, we deduce the corresponding value of condition settings for different systems. Moreover, we simulate diverse localized waves in the context of nonlinear electrical transmissions with different environment settings and compare the evolution process to reach conclusions on the parameter discovery.

Numerical inverse scattering transform for the focusing and defocusing Kundu–Eckhaus equations

In this paper, we develop the numerical inverse scattering transform (NIST) for the focusing and defocusing Kundu–Eckhaus (KE) equations. The NIST consists of numerical direct scattering and numerical inverse scattering. In numerical direct scattering, tanh mapping is introduced to improve the Chebyshev collocation method, which can help us obtain high-precision scattering data. Benefitting from the Deift–Zhou nonlinear steepest descent method and Olver’s numerical method, we can effectively complete the calculation for numerical inverse scattering. Different from traditional methods, the NIST does not require advance time during the calculation, and it will be more effective in a long time. Thus the NIST is of great significance in studying the long-time evolution for solutions.

Optical solitary wave solutions in generalized determinant form for Kundu–Eckhaus equation

The Kundu–Eckhaus (KE) equation describes the propagation of ultra-short femtosecond pulses in optical fibers. In this paper, on the basis of Hirota bilinear form of KE equation, a complex matrix is introduced into the differential relation satisfied by determinant elements. By constructing relations among the matrix elements, the solution in generalized double Wronskian determinant form for the KE equation is derived. When the complex matrix introduced in the differential relation of the determinant elements take diagonal matrix and Jordan block matrix respectively, the soliton solutions and Jordan block solutions of the KE equation are obtained and propagation situations are discussed via different parameters.