Kaup Newell Equation Research Articles

Effects of white noise on straddle and soliton dynamics in birefringent fibers using the novel Kaup-Newell equation approach

Effects of white noise on straddle and soliton dynamics in birefringent fibers using the novel Kaup-Newell equation approach

The revised Riemann–Hilbert approach to the Kaup–Newell equation with a non-vanishing boundary condition: Simple poles and higher-order poles

With a non-vanishing boundary condition, we study the Kaup–Newell (KN) equation (or the derivative nonlinear Schrödinger equation) using the Riemann–Hilbert approach. Our study yields four types of Nth order solutions of the KN equation that corresponding to simple poles on or not on the ρ circle (ρ related to the non-vanishing boundary condition), and higher-order poles on or not on the ρ circle of the Riemann–Hilbert problem (RHP). We make revisions to the usual RHP by introducing an integral factor that ensures the RHP satisfies the normalization condition. This is important because the Jost solutions go to an integral factor rather than the unit matrix when the spectral parameter goes to infinity. To consider the cases of higher-order poles, we study the parallelization conditions between the Jost solutions without assuming that the potential has compact support, and present the generalizations of residue conditions of the RHP, which play crucial roles in solving the RHP with higher-order poles. We provide explicit closed-form formulae for four types of Nth order solutions, display the explicit first-order and double-pole solitons as examples and study their properties in more detail, including amplitude, width, and exciting collisions.

Propagation of dark solitons of DNLS equations along a large-scale background

We study dynamics of dark solitons in the theory of the derivative nonlinear Schrödinger equations by the method based on imposing the condition that this dynamics must be Hamiltonian. Combining this condition with Stokes’ remark that relationships for harmonic linear waves and small-amplitude soliton tails satisfy the same linearized equations, so the corresponding solutions can be converted one into the other by replacement of the packet’s wave number k by iκ, κ being the soliton’s inverse half-width, we find the Hamiltonian and the canonical momentum of the soliton’s motion. The Hamilton equations are reduced to the Newton equation whose solutions for some typical situations are compared with exact numerical solutions of the Kaup-Newell DNLS equation.

New exact optical soliton solutions of the derivative nonlinear Schrödinger equation family

In this study, we use a systematic approach named the generalized unified method (GUM) to construct the general exact solutions of the derivative nonlinear Schrödinger (DNLS) family that also includes perturbed terms, which are the Kaup–Newell equation, the Chen–Lee–Liu equation, and the Gerdjikov–Ivanov equation. The GUM provides more general exact solutions with free parameters for nonlinear partial differential equations such that some solutions obtained by different exact solution methods, including the hyperbolic function solutions, the trigonometric function solutions, and the exponential solutions, are derived from these solutions by giving special values to these free parameters. Additionally, the used method reduces a large number of calculations compared to other exact solution methods, enabling computations to be made in a short, effortless, and elegant way. We investigate the DNLS family in this work because of its extensive applications in nonlinear optics. Particularly, the obtained optical soliton solutions of the DNLS family are useful for describing waves in optics and facilitating the interpretation of the propagation of solitons through optical fibers. Furthermore, this work not only contributes significantly to the advancement of soliton dynamics and their applications in photonic systems but also be productively used for more equations that occur in mathematical physics and engineering problems. Finally, 2D and 3D graphs of some derived solutions are plotted to illustrate behaviors of optical soliton.

Optical soliton solutions to the coupled Kaup-Newell equation in birefringent fibers

One of the fastest growing fields of study in nonlinear optics is optical solitons. This article investigates the optical soliton solutions of the coupled Kaup-Newell equation in birefringent fibers without four wave mixing. This model helps explain how modified structures, particularly Alfven waves, propagate in optical fiber and plasma physics. The results presented in this research clearly show how well the new mapping method handle nonlinear Kaup-Newell equation in coupled vector form, in birefringent fiber without four-wave mixing, to extract the analytical soliton solutions. Four-wave mixing expresses a nonlinear optical effect in which four waves interact each other as a result of the third order nonlinearity. As a valuable outcome of this article, a variety of solitons are discovered including bright, dark, periodic, singular bell-shaped, gap, flat-kink, dark-singular, and w-shaped. The existence criteria for these solitons are also obtained that are presented as constraint conditions. The constraint relations guarantee the existence of such solitons. Furthermore, graphs in 2D, 3D and contour are created to show the physical characteristics of the solutions that are produced. The new mapping method is clear and effective, and the solutions provide a wealth of avenues for additional research. The technique can also be functional to other sorts of nonlinear evolution equations in contemporary areas of research. The outcomes presented in this work are new versions which have never been observed before in existing literature.

Novel solitary wave and periodic solutions for the nonlinear Kaup–Newell equation in optical fibers

Novel solitary wave and periodic solutions for the nonlinear Kaup–Newell equation in optical fibers

New visions of optical soliton to a class of generalized nonlinear Schrödinger equation with triple refractive index and non-local nonlinearity

Because it is used in communication and information technology, the propagation of pulses in optical fibers has become a significant research subject in recent years. Mathematical models have been created to describe the behaviors of optical solitons in optical fibres, such as the Kaup-Newell equation, the nonlinear Schrödinger equation with various types of nonlinearity, and others. A model among these is the newly generalized nonlinear Schrödinger equation with triple refractive index and non-local nonlinearity. To secure optical solitons for a particular case of this novel model, we used the improved modified extended tanh function approach. Different solutions have been acquired such as bright solitons, dark solitons, singular solitons, singular periodic solutions, trigonometric solutions, and hyperbolic solutions. Some of the discovered solutions are shown in 2D and 3D graphs for better comprehension and demonstration.

Integrable reductions of the multi-component Kaup–Newell equations

We investigate the local and nonlocal integrable reductions of the multi-component Kaup–Newell equations and derive a range of novel integrable derivative nonlinear Schrödinger (NLS) equations. Firstly, we demonstrate that the multi-component Kaup–Newell equations can undergo one local reduction and three nonlocal reductions, specifically space-reversal, time-reversal, and space–time-reversal reductions. These reductions result in four multi-component derivative NLS equations. Secondly, focusing on the four-component Kaup–Newell equations, we introduce six joint reductions, which are combinations of the aforementioned local and nonlocal reductions. Through these joint reductions between the local reduction and nonlocal reductions, we discover three scalar nonlocal derivative NLS equations, including a T-reversal derivative NLS equation. Furthermore, through the joint reductions among the nonlocal reductions, we identify three scalar nonlocal derivative NLS equations with four variables, namely (x,t), (-x,t), (x,−t), and (−x,−t). All these reduced systems are integrable systems, meaning they possess infinitely many conserved integrals.

Analytical solutions of the generalized Kaup–Newell equation

The generalized Kaup–Newell equation for describing the pulse propagation in a nonlinear optical medium is considered. The traveling wave reduction of this equation is taken into account. The method of direct transformations is used to construct a general solution of a nonlinear ordinary differential equation. Analytical solutions are found, expressed in terms of elliptic integrals in canonical form. The implicit function method is applied to find some degenerate solutions under constraints on the model parameters. Among the solutions obtained there are doubly periodic solutions on the complex plane, as well as optical solitons.

Dispersive optical soliton solutions in birefringent fibers with stochastic Kaup–Newell equation having multiplicative white noise

In this article, we study for the first time the stochastic Kaup–Newell equation in birefringent fibers. Three integration algorithms are applied, namely, the unified Riccati equation expansion method, the addendum to Kudryashov's method, and the addendum to modified sub‐ODE method. Many types of optical soliton solutions such as Jacobi‐elliptic function solutions, Weierstrass elliptic function solutions, bright solitons, dark solitons, singular solitons, and periodic function solutions are obtained. Numerical schemes give a visual perspective to the solutions derived analytically.

Dynamics and optical soliton solutions in birefringent fibers for the coupled Kaup–Newell equation via planar dynamical system analysis method

In this paper, the dynamical behavior and optical soliton of the coupled Kaup–Newell equation are studied. Firstly, the phase portraits of the coupled Kaup–Newell equation are drawn by using Maple software. Secondly, the optical soliton solutions of the coupled Kaup–Newell equation are construct by the bifurcation theory of planar dynamical system together with the “three-step method” of Professor Li’s method. Finally, in order to better explain the wave propagation in birefringent fiber, the three-dimensional graphic, the two-dimensional graphic, contour plot and polar plot of some optical soliton solutions are drawn.

Novel analytical solutions and optical soliton structures of fractional-order perturbed Kaup–Newell model and its application

The Kaup–Newell equation is used to model sub-picoseconds pulses that travel throughout optical fibers. The fractional-order perturbed Kaup–Newell model, which represents extensive waves parallel to the field of magnetic, is examined. In this paper, two analytical techniques named, improved F-expansion and generalized exp[Formula: see text]-expansion techniques, are employed and new analytical solutions in generalized forms like bright solitons, dark solitons, multi-peak solitons, peakon solitons, periodic solitons and further wave results are assembled. These soliton solutions and other waves findings have important applications in applied sciences. The configurations of some solutions are shown in the form of graphs through assigning precise values to parameters, and their dynamics are described. The illustrated novel structures of some solutions also assist engineers and scientists in better grasping the physical phenomena of this fractional model. A comparison analysis has been given to explain the originality of the current findings compared to the previously achieved results. The results of computer simulations show that the procedures described are effective, simple, and efficient.

Shifted nonlocal Kundu type equations: Soliton solutions

We study the shifted nonlocal reductions of the integrable coupled Kundu type system. We then consider particular cases of this system; namely Chen–Lee–Liu, Gerdjikov–Ivanov, and Kaup–Newell systems. We obtain one- and two-soliton solutions of these systems and their shifted nonlocal reductions by the Hirota bilinear method. We present particular examples for one- and two-soliton solutions of the reduced shifted nonlocal Chen–Lee–Liu, Gerdjikov–Ivanov, and Kaup–Newell equations.

The generalized super Kaup–Newell equation and its super bi-Hamiltonian structure

In this work, from the generalized Kaup–Newell (KN) equation spectral problem, the generalized super KN equation and its bi-Hamiltonian structure are constructed based on a Lie super-algebras and the super-trace identity. Besides, a generalized super KN equation with self-consistent sources is also discussed.

A non-linear study of optical solitons for Kaup-Newell equation without four-wave mixing

Nonlinear science is a fundamental science frontier that include studies and the common properties of nonlinear phenomena. This article is devoted to the study of sub-pico second optical pluses in birefringent fibers for Kaup-Newell equation (KNE) without four-wave mixing. Three prominent integrations techniques are successfully implemented on KNE in coupled vector form. Variety of soliton solutions namely dark, bright, periodic singular, singular and bright-singular combo solitons are constructed for the KNE in birefringent fibers. The obtained solutions are reckoned with their respective existence criterion. In addition, two-dimensional and three-dimensional graphs are drawn to exhibit the physical behavior of the obtained solutions.

Dark, bright and singular optical solutions of the Kaup–Newell model with two analytical integration schemes

We have searched for optical solutions of Kaup–Newell equation (KNE) used in nonlinear optical and plasma physics, which have a very important role in the class of derivative nonlinear Schrödinger equations (dNLSEs). To obtain exact wave solutions of the proposed model, two analytical techniques, recently presented as new Kudryashov’s method and the generalized projective Riccati equations method (GPREM) have been applied for the first time. These two methods are based on traveling wave transformation and homogeneous balance principles . The presented nonlinear partial differential equation (NLPDE) has been converted to ordinary differential equation (ODE) using traveling wave transformation. Thus, new analytical solutions have been obtained by applying the proposed methods. The new solitons and other traveling wave solutions acquired will be a motivation to researchers working in engineering and applied physics. Various constraints have been used to ensure the validity of the obtained solutions and to get the most appropriate solutions. To illustrate the physical phenomena of this model, 2D, 3D and contour plots of some derived solutions have been depicted with explanatory graphics.

Cubic–quartic optical solitons in birefringent fibers with Kaup–Newell equation using different arithmatic algorithms

This paper studies cubic–quartic optical solitons in birefringent fiber with the Kaup–Newell equation for the first time. Bright and singular solitons are recovered by the addendum to Kudryashov’s method and unified Riccati equation expansion method. These solitons depend on certain parameter restrictions which are also listed in this paper.

Optical solitons in birefringent fibers with Kaup–Newell equation using two integration schemes

This paper recovers optical solitons in birefringent fibers with the governing system that is derived from the Kaup–Newell equation. Two integration schemes are employed in this study. They are the direct algebraic and unified Riccati equation approaches. Singular, dark and bright solitons are retrieved. Jacobi elliptic and periodic solutions are also yielded. These solutions depend on certain parameter restrictions.

A new form of general soliton solutions and multiple zeros solutions for a higher-order Kaup–Newell equation

Due to the fact that the higher-order Kaup–Newell (KN) system has more complex and diverse solutions than the classical second-order flow KN system, the research on it has attracted much attention. In this paper, we consider a higher-order KN equation with third-order dispersion and fifth-order nonlinearity. Based on the theory of the inverse scattering, the matrix Riemann–Hilbert problem is established. Through the dressing method, the solution matrix with simple zeros without reflection is constructed. In particular, a new form of solution is given which is more direct and simpler than previous methods. In addition, through the determinant solution matrix, the vivid diagrams and dynamic analysis of single-soliton solution and two-soliton solution are given in detail. Finally, by using the technique of limit, we construct the general solution matrix in the case of multiple zeros, and the examples of solutions for the cases of double zeros, triple zeros, single–double zeros, and double–double zeros are especially shown.

Optical solitons for the Kaup–Newell equation by collective variables method

In this work, we present a collective variable (CV) approach to establish dispersive solitary wave solutions for the Kaup–Newell Equation (KNE). The full CV theory has been utilized to enunciate the soliton molecules through its ground-laying parameters including the power of each pulse, phase and center-of-mass. Additionally, the dynamics of an ultra short pulse has been analyzed by using CV. This work may be utilized for various dynamics of solitons as well as the influence the amplitude, temporal position, frequency, phase and chirp on the solitons’ nonlinear parameters. Moreover, the numerical simulations have been designed by means of appropriate parameter values to explain more on the obtained results.