Eckhaus Equation Research Articles

Analysis of Fractional Kundu-Eckhaus and Massive Thirring Equations Using a Hybridization Scheme

This paper deals with the study of fractional Kundu-Eckhaus equation (FKEE) and fractional massive Thirring problem (FMTP) that appear in the quantum field theory, weakly nonlinear dispersive water waves, and nonlinear optics. Since the variational iteration method involves integration, the Laplace transform involves convolution theorem in recurrence relation to derive the series solution. To avoid some assumptions and hypothesis, we apply a two-scale approach for such a nonlinear complex model. The fractional differential equation may be transformed into its partner equation using He’s fractional complex transform, and then, the nonlinear elements can be readily handled using the homotopy perturbation method (HPM). Numerical results are derived in a rapid converge series form to improve the accuracy of the scheme greatly. Graphical representations and error distribution show that the two-scale approach is a very convenient tool.

Asymptotic solitons of the focusing Kundu-Eckhaus equation with time-periodic boundary condition

Asymptotic solitons of the focusing Kundu-Eckhaus equation with time-periodic boundary condition

Role of nonlinear saturation on modulational instability in Kundu–Eckhaus equation with the presence of inter modal, XPM and SPM

Role of nonlinear saturation on modulational instability in Kundu–Eckhaus equation with the presence of inter modal, XPM and SPM

Soliton solutions of generalized Kundu-Eckhaus equation with an extra-dispersion via improved modified extended tanh-function technique

Soliton solutions of generalized Kundu-Eckhaus equation with an extra-dispersion via improved modified extended tanh-function technique

Darboux transformation and exact solution to the nonlocal Kundu–Eckhaus equation

In this paper, the n-component nonlocal Kundu–Eckhaus equation is presented. The Darboux transformation to the nonlocal Kundu–Eckhaus equation is constructed to obtain the N-soliton solution to the equation. The difference between the solution obtained here and that of the local Kundu–Eckhaus equation by Darboux transformation is that the solution of the former has symmetric constraints. Besides, the one-exact solution to the nonlocal Kundu–Eckhaus equation is obtained. For example, as for the three-component nonlocal KE equation, the images of the corresponding rogue wave solutions are drawn by taking specific parameters.

Optical soliton solutions to the time-fractional Kundu–Eckhaus equation through the $$(G^{\prime}/G,1/G)$$-expansion technique

Optical soliton solutions to the time-fractional Kundu–Eckhaus equation through the $$(G^{\prime}/G,1/G)$$-expansion technique

Traveling Wave Solutions for Complex Space-Time Fractional Kundu-Eckhaus Equation

In this work, the class of nonlinear complex fractional Kundu-Eckhaus equation is presented with a novel truncated M-fractional derivative. This model is significant and notable in quantum mechanics with good-natured physical characteristics. The motivation for this paper is to construct new solitary and kink wave solutions for the governing equation using the ansatz method. A complex-fractional transformation is applied to convert the fractional Kundu-Eckhaus equation into an ordinary differential equations system. The equilibria of the corresponding dynamical system will be presented to show the existence of traveling wave solutions for the governing model. A novel kink and solitary wave solutions of the governing model are realized by means of the proposed method. In order to gain insight into the underlying dynamics of the obtained solutions, some graphical representations are drawn. For more illustration, several numerical applications are given and analyzed graphically to demonstrate the ability and reliability of the method in dealing with various fractional engineering and physical problems.

Vector form of Kundu - Eckhaus equation and its simplest solutions

Vector form of Kundu - Eckhaus equation and its simplest solutions

Optical modelling of the space-time fractional Eckhaus equation

In this paper, the space-time fractional Eckhaus equation is considered and solved using the a direct method (Khater method) to obtain exact solutions. This meth?od produces more solutions when compared to other known methods. The real solutions of this equation are classified as travelling wave, kink, periodic and sol?itary wave solutions. These solutions are searched with the help of the fractional conformable derivative sense. Some graphs and tables are drawn to interpret the solutions and method. With the interpretation of the results, it is explained that the method used is a reliable, effective, powerful and easily applicable technique for obtaining the solutions of fractional differential equations classes in many fields.

Effect of modified saturable nonlinearity on the instability spectra in Kundu–Eckhaus equation with the inclusion of SPM, inter modal, and XPM

We examine the impact of intermodal dispersion on the modulation instability (MI) of an optical soliton in the Kundu-Eckhaus (KE) equation in the presence of cross phase modulation (XPM), modified saturable nonlinearity (MSNL), and self phase modulation (SPM). We consider the two component KE model with intermodal dispersion and MSNL, but without the four wave mixing term. Because the rays connected to various modes traverse the optical fiber at various effective lengths, intermodal dispersion arises in multimode fibers. As a result, light in the various modes spreads out temporally along the fiber. The MI criterion for the KE equation is established using the traditional linear stability analysis. When XPM and SPM are present, it is possible to determine, how inter modal dispersion and MSNL will impact the MI of an optical soliton in a birefringent fiber using the MI criterion. According to the findings, changed saturable nonlinearity significantly influences the MI dynamical behavior. In a typical dispersion scenario, an increase in the nonlinear saturation parameter can lead to an increase in the MI soliton. However, the anomalous dispersion regime exhibits the contrary behavior. In a two core birefringent KE model, the MSN and intermodal dispersion are capable of producing ultra-short pulses and a soliton. Our research will contribute to increasing the fiber optic telecommunication network's capacity and quality.

The dressing method and dynamics of soliton solutions for the Kundu–Eckhaus equation

The boundary value problem for the focusing Kundu–Eckhaus equation with nonzero boundary conditions is studied by the Dbar dressing method in this work. A Dbar problem with non-canonical normalization condition at infinity is introduced to investigate the soliton solution. The eigenfunction of Dbar problem is meromorphic outside annulus with center 0, which is used to construct the Lax pair of the Kundu–Eckhaus equation with nonzero boundary conditions, which is a crucial step to further search for the soliton solution. Furthermore, the original nonlinear evolution equation and conservation law are obtained by means of choosing a special distribution matrix. Moreover, the N-soliton solutions of the focusing Kundu–Eckhaus equation with nonzero boundary conditions are discussed based on the symmetries and distribution. As concrete examples, the dynamic behaviors of the one-breather solution and the two-breather solution are analyzed graphically by considering different parameters.

Soliton Solutions and Other Solutions for Kundu–Eckhaus Equation with Quintic Nonlinearity and Raman Effect Using the Improved Modified Extended Tanh-Function Method

Our paper studies the optical solitons for the Kundu–Eckhaus (KE) equation with quintic nonlinearity and Raman effect. By applying the improved modified extended tanh-function method, many soliton solutions can be obtained such as bright soliton solutions, dark soliton solutions, and the singular soliton solution. In addition, we can obtain various types of solutions, namely, singular periodic solutions, exponential solutions, rational solutions, Jacobi elliptic solutions and Weierstrass elliptic doubly periodic solutions. Moreover, some selected solutions are illustrated graphically to show the physical nature and the characteristics of the obtained solutions.

New solitary wave and computational solitons for Kundu–Eckhaus equation

The goal of this research is to find novel optical solutions to the Kundu–Eckhaus equation, which possess crucial roles in the field of nonlinear optics. A collective variable (CV) strategy is adopted to solve governing equation including the Raman effect and quintic nonlinearity. This method is a suitable to deal with both conservative and non-conservative systems by exposing a set of equations of motion regardless of nonlinearities or dissipative components. The parameters employed in this approach are chirp, temporal position, phase, amplitude, frequency and width, namely, collective variables. The fourth order Runge–Kutta technique is a well-known numerical scheme that aims towards the solution of the resulting system of ordinary differential equations representing the variables involved in the pulse ansatz. This technique presents the evolution of pulse parameters with regard to propagation variables. The graphical profiles at suitable values of pulse parameters are also provided. The unified technique is also applied to find soliton solutions. The obtained solution is a periodic solitary wave, showed graphically. The results developed in this article are found to be new in the literature and the approach utilized, can be applied to solve a variety of nonlinear problems in the mathematical sciences.

Effect of inter modal dispersion on modulational instability of optical soliton in Kundu–Eckhaus equation with the presence of SPM and XPM

We investigate analytically the role of inter modal dispersion on modulation instability of optical soliton in Kundu–Eckhaus (KE) equation with the presence of self phase modulation (SPM) and cross phase modulation (XPM). We consider the model of two component Kundu–Eckhaus with the inclusion of inter modal dispersion and in absence of four wave mixing term. The modulational instability (MI) criterion for KE equation is obtained based on the standard linear stability analysis. The effect of inter modal dispersion on MI of optical soliton in birefringent fiber with the presence of SPM and XPM is analyzed by using MI criterion. Through graphical illustration of MI gain, the role of inter modal dispersion on MI in optical fiber for the case of absence and presence of SPM and XPM are discussed.

Riemann–Hilbert approach and $$N$$-soliton solutions of the two-component Kundu–Eckhaus equation

Riemann–Hilbert approach and $$N$$-soliton solutions of the two-component Kundu–Eckhaus equation

Chirped self-similar localized pulses on a continuous wave background in presence of cubic–quintic nonlinearity and self-frequency shift

We investigate the nonlinear pulse propagation through an inhomogeneous single-mode optical fiber where the evolution dynamics is governed by the generalized higher-order nonlinear Schrödinger equation with varying group velocity dispersion, cubic and quintic nonlinearities, self-frequency shift arising from stimulated Raman scattering, and linear gain or loss. A new class of soliton pulse solutions for a wave field is constructed based on the similarity transformation connecting with the constant-coefficient Kundu–Eckhaus equation. This class describes soliton structures that propagate self-similarity on a continuous-wave (cw) background, and acquires a nonlinear chirp as it propagates in the inhomogeneous fiber medium. It is found that the nonlinear chirp associated with these self-similar pulses originates essentially from the self-frequency shift effect. The conditions on fiber parameters for the existence of those nonlinearly chirped self-similar localized pulses are also given. As a practical example, we studied their evolution dynamics in a specified soliton control system. The solution stability against perturbation is checked by direct numerical simulation.

Omnipresent instability criterion of birefringent Kundu-Eckhaus model with nonic nonlinearity

This paper intends to present an investigation of the ultra shortpulse propagation under the nonlinear birefringent Kundu-Eckhaus (KE) model. The dynamics of modulational instability(MI) is revealed with the linear stability analysis of the KE equation with higher-order, the cubic, and nonic nonlinearity effects. The role of cubic-nonic nonlinearity on MI gain spectra in KE fiber system with self and cross phase modulation is studied for the first time. The impact of nonic nonlinearity is shown to be much more prominent over its cubic-quintic counterpart. The presence of cubic-nonic combo nonlinearity lets the MI gain profile to be adjusted with the rate of self-phase modulation (SPM). This new MI gain growth dynamics controlled by the SPM parameter will be useful on soliton generation in birefringent KE fiber media.

Investigation of optical solitons to the nonlinear complex Kundu-Eckhaus and Zakharov-Kuznetsov-Benjamin-Bona-Mahony equations in conformable.

This research manuscript focuses on the extraction of dark-bright solitons and periodic wave distributions of two models, namely, the Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation and complex Kundu–Eckhaus equation with conformable derivative. To reach these important properties, the generalized exponential rational function method is considered. To observe wave distributions in periodic and singular sense, dynamical behaviour modulus of solutions are also archived. Strain conditions for validity of results obtained in this paper are also reported.

Cubic-quartic solitons in birefringent fibers with Kundu–Eckhaus equation using two integration schemes

In this paper, we deduce the soliton solutions of the Kundu–Eckhaus equation in birefringent fibers. The addendum to Kudryashov’s method and the modified Kudryashov’s method have been used. The combo bright-singular soliton, dark soliton, bright soliton and the singular soliton solutions are existed under constraint conditions.

On the existence of chirped algebraic solitary waves in optical fibers governed by Kundu–Eckhaus equation

This paper reports the first analytical demonstration of propagation of algebraic solitary waves accompanied with a nonlinear chirp in optical fibers governed by the Kundu–Eckhaus equation. Both chirped dark and bright algebraic solitary waves can exist in the system with higher-order effects such as Raman effect and quintic nonlinearity. The chirp associated with each of these algebraic solitary waves is shown to be dependent on the intensity of the wave. It is found that the Raman process has a profound impact in the chirp of these localized pulses.