Eckhaus Equation Research Articles

Regarding Deeper Properties of the Fractional Order Kundu-Eckhaus Equation and Massive Thirring Model

In this paper, the fractional natural decomposition method (FNDM) is employed to find the solution for the KunduEckhaus equation and coupled fractional differential equations describing the massive Thirring model. The massive Thirring model consists of a system of two nonlinear complex differential equations, and it plays a dynamic role in quantum field theory. The fractional derivative is considered in the Caputo sense, and the projected algorithm is a graceful mixture of Adomian decomposition scheme with natural transform technique. In order to illustrate and validate the efficiency of the future technique, we analyzed projected phenomena in terms of fractional order. Moreover, the behaviour of the obtained solution has been captured for diverse fractional order. The obtained results elucidate that the projected technique is easy to implement and very effective to analyze the behaviour of complex nonlinear differential equations of fractional order arising in the connected areas of science and engineering.

Soliton solutions for the nonlocal reverse space Kundu–Eckhaus equation via symbolic calculation

The Kundu–Eckhaus (KE) equation describes the propagation of ultrashort femtosecond pulses in optical fibers. In this paper, the Hirota bilinear method is used to deal with nonlocal KE equation which is also called nonlocal integrable nonlinear Schrödinger equation with cubic and quintic nonlinearities. The N-soliton solution for the nonlocal KE equation is derived by virtue of symbolic calculation. Based on that, the exact solution expressions of two-soliton and three-soliton can be obtained. Several propagation situations are demonstrated and discussed for the combinatorial solutions of nonlocal KE equation under different parameters, including one periodic solitary wave evolution, two parallel, perpendicular and periodic solitary waves collision.

A generalized Kundu--Eckhaus equation with an extra-dispersion: pulses configuration

Raman effect is due to self-phase modulation (SPM), which is embedded in Kundu--Eckhaus equation KEE. Here, the objective of this work is to present a generalized KEE by accounting for an extra dispersion which may be induced by Raman scattering. Also, attention is focused to study the effects of the extra dispersion.Which are investigated via obtaining the exact solutions of the new model equation. These solutions are found by the unified method and by introducing a new transformation that ispects soliton- periodic wave collision. We aim to show that a variety of shapes of optical pulses OPs propagation in optical fibers occurs. Waves of multiple geometric structures are observed. Among these waves, hybrid lumps, soliton, cascade, complex chirped, hybrid w-shaped, rhombus (diamond) waves and soliton self phase modulation.The characteristics of the pulses; intensity, frequency, wavelength, polarization, and spectral content are identified. The results found here are of great interest in experimenting the effects of the induced dispersion on pulses configurations. Further, the colliding dynamics are inspected and as it is observed that no rogue or sharp waves formation hold, so the collision is elastic.

Investigation of modulational instability in a coupled Kundu-Ekchaus equation in the presence of modified form of saturable nonlinearity

We study the impact of modified saturable nonlinearity on modulational instability (MI) criteria in a coupled birefringent fiber using pulse propagation technique. Based on the standard linear stability method, we enquire nonlinear self/cross phase modulation on MI in a coupled Kundu–Eckhaus (KE) equation in the absence of nonlinear four-wave mixing expression. Using the modified KE equation, we study the role of modified form of saturable nonlinearity on MI in a birefringent KE model with the adjustable self/cross phase modulation coefficients. In the case of KE without nonlinear saturation and four-wave mixing, we observed that the cross phase modulation (XPM) has less induced the instability region on MI in coupled birefringent fiber system as compared to self phase modulation (SPM) case as shown in Ref. Parasuraman (2021). But, in saturable KE model showed different characteristics of MI and growth rate of profile in a coupled birefringent fiber system in the presence of particular emphasis values of the nonlinear terms as depicted in Ref. Mohanraj (2021). This study shows the novel results of MI criterion in a coupled KE birefringent media with the help of modified type of saturable nonlinearity and both SPM/XPM values. From the above-detailed discussion, the modified form of saturable nonlinearity plays an important role for MI generation in a KE birefringent media. However, the modified nonlinear saturation KE model more helps to generate ultrashort pulses and soliton or solitary wave in a coupled nonlinear birefringent fiber with the help of 3D images.

Phase engineering of chirped rogue waves in Bose–Einstein condensates with a variable scattering length in an expulsive potential

We consider a cubic Gross-Pitaevskii (GP) equation governing the dynamics of Bose-Einstein condensates (BECs) with time-dependent coefficients in front of the cubic term and inverted parabolic potential. Under a special condition imposed on the coefficients, a combination of phase-imprint and modified lens-type transformations converts the GP equation into the integrable Kundu-Eckhaus (KE) one with constant coefficients, which contains the quintic nonlinearity and the Raman-like term producing the self-frequency shift. The condition for the baseband modulational instability of CW states is derived, providing the possibility of generation of chirped rogue waves (RWs) in the underlying matter-wave (BEC) model. Using known RW solutions of the KE equation, we present explicit first- and second-order chirped RW states. The chirp of the first- and second-order RWs is independent of the phase imprint. Detailed solutions are presented for the following configurations: (i) the nonlinearity exponentially varying in time; (ii) time-periodic modulation of the nonlinearity; (iii) a stepwise time modulation of the strength of the expulsive potential. Singularities of the local chirp coincide with valleys of the corresponding RWs. The results demonstrate that the temporal modulation of the s-wave scattering length and strength of the inverted parabolic potential can be used to manipulate the evolution of rogue matter waves in BEC.

Saturation effects on modulational instability in birefringent media with the help of Kundu–Ekchaus equation

The present work shows the role of nonlinear saturation effects on modulational instability (MI) in a birefringent media with the help of nonlinear wave propagation technique. Using the Kundu–Eckhaus (KE) equation, we investigate both self phase and cross phase modulation effect on MI in a birefringent media without four wave mixing term. We consider only KE equation model without the existence of four wave mixing expression and the standard linear stability theory is presented to the appropriate model. In the case of modified KE equation, we discuss the role of MI in a birefringent fiber system with the help of conventional type of saturation effect. In the case of KE without four wave mixing, we observed that the SPM plays a vital role on MI in coupled birefringent fiber compared to XPM as shown in Ref. Parasuraman (2021). But, In saturable KE has different MI characteristic as well as growth rate of profile in birefringent model with the help of particular emphasis values of the nonlinear parameters. However, the nonlinear saturation KE model more helps to generate soliton or solitary wave in coupled nonlinear birefringent fiber in the presence of both SPM and XPM.

The [formula omitted]-fold degenerate Darboux transformation of the three component Kundu–Eckhaus equations: Novel interactions between soliton and positon solution

In this paper, an explicit representation of the N-fold degenerate Darboux transformation (DT) for the three component Kundu–Eckhaus equations, which can be viewed as a model to describe the effect of quintic nonlinearity on the ultra-short optical pulse propagation in non-Kerr media, has been derived. The DT is expressed by the initial eigenfunctions, spectral parameters and ‘seed’ solution. As applications of DT, the abundant and novel interactions between soliton and positon are discussed in detail, such as elastic interactions, inelastic interactions and bound states.

Non-classical Lie symmetry and conservation laws of the nonlinear time-fractional Kundu–Eckhaus (KE) equation

Non-classical Lie symmetry and conservation laws of the nonlinear time-fractional Kundu–Eckhaus (KE) equation

Modulational instability criterion for optical wave propagation in birefringent fiber of Kundu–Eckhaus equation

We study the modulational instability of nonlinear wave propagation in birefringent fiber of the Kundu–Eckhaus equation without four wave mixing term and analysis the effect of self phase modulation and cross phase modulation. The model of Kundu–Eckhaus equation in the absence of four wave mixing term are consider and apply the linear stability analysis to the Kundu–Eckhaus equation. Modulational instability criteria are found from linearized equations for small perturbations. Using modulational instability criteria, the self phase modulation and cross phase modulation effects on optical wave propagation in birefringent fiber are discussed.

Evolution of dark optical soliton in birefringent fiber of Kundu–Eckhaus equation with four wave mixing and inter-modal dispersion

The dark soliton propagation in birefringent fiber of Kundu–Eckhaus equation with four wave mixing and inter-modal dispersion is investigated. We consider the Kundu–Eckhaus (KE) equation with the inclusion of four wave mixing term and inter-modal dispersion term. The developed Jacobi elliptic function expansion method is used to construct the solution of KE equation. New results are presented which includes dark soliton solution and using it to analysis the inter-model dispersion in birefringent fiber of the KE equation with four wave mixing.

A [formula omitted]-dressing approach to the Kundu-Eckhaus equation

Starting from a local 2×2 matrix ∂¯-equation with non-normalization boundary conditions, the spatial and time spectral problems associated with the Kundu-Eckhaus (KE) equation are derived via two linear constraint equations. The gauge equivalence among the KE equation, NLS equation and Heisenberg chain equation are given. A KE hierarchy with source is proposed by using recursive operator. The N-solitions of the KE equation are constructed based the ∂¯-equation by choosing a special spectral transformation matrix. Further the explicit one- and two-soliton solutions are obtained.

Optical Solutions of the Kundu-Eckhaus Equation Via Two Different Methods

This work is devoted to obtaining new optical solutions to the Kundu-Eckhaus (KE) equation which is believed to play a crucial part in the area of nonlinear optics. Two different methods, the exp(−𝜑 (ε)) method with the exponential rational function approach have been utilized. Both methods are efficient in finding the analytical solutions of many nonlinear partial differential equations and fractional differential equations. Results obtained in this research are dissimilar to the ones in the literature and the solutions are controlled by relocating them back to the primary equation. Finally, it can be stated that optical solutions have a promising future.

Inverse scattering method for the Kundu-Eckhaus equation with zero/nonzero boundary conditions

Abstract In this paper, the inverse scattering approach is applied to the Kundu-Eckhaus equation with two cases of zero boundary condition (ZBC) and nonzero boundary conditions (NZBCs) at infinity. Firstly, we obtain the exact formulae of soliton solutions of three cases of N simple poles, one higher-order pole and multiple higher-order poles via the associated Riemann-Hilbert problem (RHP). Moreover, given the initial data that allow for the presence of discrete spectrum, the higher-order rogue waves of the equation are presented. For the case of NZBCs, we can construct the infinite order rogue waves through developing a suitable RHP. Finally, by choosing different parameters, we aim to show some prominent characteristics of this solution and express them graphically in detail. Our results should be helpful to further explore and enrich the related nonlinear wave phenomena.

Rogue wave and multi-pole solutions for the focusing Kundu–Eckhaus Equation with nonzero background via Riemann–Hilbert problem method

In this work, we use the inverse scattering transform method to consider the focusing Kundu–Eckhaus (KE) equation with nonzero background (NZBG) at infinity. Based on the analytical, symmetric, asymptotic properties of eigenfunctions, the inverse problem is solved via a matrix Riemann–Hilbert problem (RHP). The general multi-pole solutions are given in terms of the solution of the associated RHP. And the formula of the N simple-pole soliton solutions are obtained, too. We show that the Peregrine’s rational solution can be viewed as some appropriate limit of the simple-pole soliton solutions at branch point. Furthermore, by taking some other proper limits, the two and three simple-pole soliton solutions can yield to the double- and triple-pole solutions for focusing KE equation with NZBG. The effect of the parameter $$\beta $$ , characterizing the strength of the non-Kerr (quintic) nonlinear and the self-frequency shift effect, and some typical collisions between solutions of focusing KE equation are graphically displayed.

Darboux transformation of the generalized mixed nonlinear Schrödinger equation revisited.

In this paper, we present an explicit Darboux transformation of the generalized mixed nonlinear Schrödinger (GMNLS) equation. The compact determinant representation of the n-fold Darboux transformation of the GMNLS equation is constructed and the nth-order solution is built. We further prove that only the even-fold Darboux transformation and the even-order solution of the GMNLS equation can, respectively, be reduced to the Darboux transformation and solution of the Kundu-Eckhaus equation. Furthermore, two different kinds of explicit one-soliton solutions of the GMNLS equation are constructed and discussed.

Kinky breathers, multi-peak and multi-wave soliton solutions for the nonlinear propagation of Kundu–Eckhaus dynamical model

The multi-wave solutions for nonlinear Kundu–Eckhaus (KE) equation are obtained using logarithmic transformation and symbolic computation using the function method. Three-wave method, double exponential and homoclinic breather approach are used to get these solutions. We study the conflict between our results and considerably-known results and state that the solutions reached here are new. By specifying the suitable values for the parameter, the drawings of the solutions obtained are shown in this paper.

On traveling wave solutions of the Kundu–Eckhaus equation

The Kundu–Eckhaus equation is considered taking into account the traveling wave reduction. Nonlinear ordinary differential equation is shown to pass the weak Painlev e test in case when the Cauchy problem for the Kundu–Eckhaus equation is solved by the inverse scattering transform. However the general solution of nonlinear ordinary differential equation exist for more general case. This solution is given in this paper.

Soliton solutions of Kundu-Eckhaus equation in birefringent optical fiber with inter-modal dispersion

We study the dynamics of optical soliton in birefringent optical fiber with the effect of inter-modal dispersion. We consider the model of the Kundu-Eckhaus (KE) equation for birefringent optical fiber with the inclusion of inter-modal dispersion. Multisoliton solutions of KE equation with the presence of inter-modal dispersion are found analytically by using mapping method and a symbolic computation system. Initially, the solutions are expressed in terms of the Jacobi elliptic function which is also recovered by the use of limiting case of modulus of ellipticity. With the help of graphical illustrations of our solutions, we understand the effect of inter-modal dispersion on the soliton evolution in optical birefringent optical fiber.

Bright and dark soliton solutions for the complex Kundu-Eckhaus equation

The main target of this article is to realize bright and dark soliton solutions to the complex Kundu-Eckhaus equation which represents a general form of integrable system that is gauge-equivalent to the mixed nonlinear Schrödinger equation. This soliton solution which represents the propagation of the ultra-short femtosecond pulses and the rogue wave in an optical fiber achieved for the first time in the framework of the solitary wave ansatz method. Furthermore, in this connection at the same time and parallel the extended simple equation method has been successfully applied to achieve new impressive solitary wave solutions to this model. A comparison between the obtained results and that satisfied in previous work is established.

The Riemann–Hilbert approach to focusing Kundu–Eckhaus equation with non-zero boundary conditions

In this paper, we investigate the focusing Kundu–Eckhaus equation with non-zero boundary conditions. An appropriate two-sheeted Riemann surface is introduced to map the spectral parameter [Formula: see text] into a single-valued parameter [Formula: see text]. Starting from the Lax pair of Kundu–Eckhaus equation, two kinds of Jost solutions are constructed. Further, their asymptotic, analyticity, symmetries as well as spectral matrix are analyzed in detail. It is shown that the solution of the Kundu–Eckhaus equation with non-zero boundary conditions can be characterized with a matrix Riemann–Hilbert problem. Then a formula of [Formula: see text]-soliton solutions is derived by solving the Riemann–Hilbert problem. As applications of the [Formula: see text]-soliton formula, the first-order explicit soliton solutions with different dynamical features are obtained and analyzed.