Kaup Newell Equation Research Articles

Solitons to the derivative nonlinear Schrödinger equation: Double Wronskians and reductions

In this paper, we derive solutions to the derivative nonlinear Schrödinger equation, which are associated to real and complex discrete eigenvalues of the Kaup–Newell spectral problem. These solutions are obtained by investigating double Wronskian solutions of the coupled Kaup–Newell equations and their reductions by means of bilinear method and a reduction technique. The reduced equations include the derivative nonlinear Schrödinger equation and its nonlocal version. Some obtained solutions allow not only periodic behavior, but also solitons on periodic background. Dynamics are illustrated.

Chirped envelope patterns of sup-pico-second pulse propagation through an optical fiber

We study the optical chirped envelope patterns of sub-pico-second pulse propagation through an optical fiber, which is modeled by the Kaup–Newell equation with the form of derivative nonlinear Schrodinger equation. The complete discrimination system for polynomial method is applied to give all chirped envelope solutions. A family of propagation patterns including the bright, dark, singular and elliptical type solutions are obtained.

Optical solitons and other solutions to Kaup–Newell equation with Jacobi elliptic function expansion method

In this paper, we studied Kaup–Newell (KN) equation in coupled vector form without four-wave mixing terms in birefringent fibers. We employed Jacobi elliptic function expansion method in order to demonstrate sub-pico-second optical soliton solutions. Beside bright and dark solitons, Jacobi elliptic function solutions and hyperbolic solutions are obtained. Moreover, the graphs for some solution are presented.

Analysis of optical solitons solutions of two nonlinear models using analytical technique

<abstract><p>Looking for the exact solutions in the form of optical solitons of nonlinear partial differential equations has become very famous to analyze the core structures of physical phenomena. In this paper, we have constructed some various type of optical solitons solutions for the Kaup-Newell equation (KNE) and Biswas-Arshad equation (BAE) via the generalized Kudryashov method (GKM). The conquered solutions help to understand the dynamic behavior of different physical phenomena. These solutions are specific, novel, correct and may be beneficial for edifying precise nonlinear physical phenomena in nonlinear dynamical schemes. Graphical recreations for some of the acquired solutions are offered.</p></abstract>

On the Riemann–Hilbert problem of a generalized derivative nonlinear Schrödinger equation

In this work, we present a unified transformation method directly by using the inverse scattering method for a generalized derivative nonlinear Schrödinger (DNLS) equation. By establishing a matrix Riemann–Hilbert problem and reconstructing potential function q(x, t) from eigenfunctions in the inverse problem, the initial-boundary value problems for the generalized DNLS equation on the half-line are discussed. Moreover, we also obtain that the spectral functions f(η), s(η), F(η), S(η) are not independent of each other, but meet an important global relation. As applications, the generalized DNLS equation can be reduced to the Kaup–Newell equation and Chen–Lee–Liu equation on the half-line.

A Riemann-Hilbert Approach to the Multicomponent Kaup-Newell Equation

A Riemann-Hilbert approach is developed to the multicomponent Kaup-Newell equation. The formula is presented of N -soliton solutions through an identity jump matrix related to the inverse scattering problems with reflectionless potential.

Optical solitons in birefringent fibers of Kaup-Newell's equation with extended simplest equation method

The area of telecommunications has experienced a considerable growth in the last few decades due to remarkable development in the field of optical fibers. Optical solitons form the basic fabric in the area of telecommunication industry. The captivating technology of sub-pico second pulses that propagate through optical fibers is modeled with Kaup-Newell’s (KN) equation. In this paper, we introduce the KN equation without four-wave mixing (FWM) terms in birefringent fibers, we study this equation, and we obtain different types of solutions. Beside bright, dark and singular solitons, other types of solutions are obtained. Moreover, for the physical illustration of the obtained solutions, the graphs are presented.

Analytical and numerical treatments for the Kaup–Newell dynamical equation

The present paper examines both analytically and numerically the Kaup–Newell equation being an important class of nonlinear Schrödinger equations with lots of applications in optical fibers. The Riccati equation method is employed for the analytical study and revealed quite a number of interesting soliton solutions including bright-singular, dark-singular, and singular soliton solutions to mention a few; while the Adomian’s improved decomposition method is adopted on the hand for the numerical investigation. More, since bright, dark and singular solitons are very important types of solitons that arising in laminar jet and nonlinear dispersive media; we considered certain hyperbolic ansatz to construct exact soliton solutions for the numerical comparative study. The presented numerical scheme thus validated the exact analytical solutions and turned out to possess a high-level of accuracy. We finally provided some comparison tables and depictions to support the reported results after inviting the classical Adomian’s method to assess the devised numerical technique.

New solutions for chirped optical solitons related to Kaup-Newell equation: Application to plasma physics

In this work, some new periodic and localized analytical solutions in the framework of the Kaup-Newell equation (KNE) which is considered one of the most popular forms of the derivative nonlinear Schrödinger equation (DNLSE) are constructed. The evolution equation is reduced to a cubic-quintic Duffing equation (CQDE) using traveling wave transform for describing the dynamics of nonlinear structures that can exist and propagate in nonlinear and dispersive media (e.g. optical fiber and plasma physics). The CQDE is solved in terms of the Jacobian elliptic functions; cn, sn, sc, F, and Π and a general solution in the form of cnoidal waves is obtained. Here, the solution of CQDE is solved based on the Jacobian elliptic functions theory. The obtained solutions are in the form of elliptic periodic, chirped solitons, and trigonometric. These solutions are generalized as compared to other published solutions. Also, the chirped soliton solution can be derived directly from the cnoidal solution as a special case. The model is applied to the description of the propagation of modulated structures in optical fiber and in plasma physics especially Alfvén waves.

Rogue Waves in the Generalized Derivative Nonlinear Schrödinger Equations

General rogue waves are derived for the generalized derivative nonlinear Schrodinger (GDNLS) equations by a bilinear Kadomtsev-Petviashvili (KP) reduction method. These GDNLS equations contain the Kaup-Newell equation, the Chen-Lee-Liu equation and the Gerdjikov-Ivanov equation as special cases. In this bilinear framework, it is shown that rogue waves to all members of these equations are expressed by the same bilinear solution. Compared to previous bilinear KP reduction methods for rogue waves in other integrable equations, an important improvement in our current KP reduction procedure is a new parameterization of internal parameters in rogue waves. Under this new parameterization, the rogue wave expressions through elementary Schur polynomials are much simpler. In addition, the rogue wave with the highest peak amplitude at each order can be obtained by setting all those internal parameters to zero, and this maximum peak amplitude at order $N$ turns out to be $2N+1$ times the background amplitude, independent of the individual GDNLS equation and the background wavenumber. It is also reported that these GDNLS equations can be decomposed into two different bilinear systems which require different KP reductions, but the resulting rogue waves remain the same. Dynamics of rogue waves in the GDNLS equations is also analyzed. It is shown that the wavenumber of the constant background strongly affects the orientation and duration of the rogue wave. In addition, some new rogue patterns are presented.

On the Riemann-Hilbert problem of the Kundu equation

The Kundu equation as a special case of the complex Ginzburg–Landau equation can be used to describe a slice of phenomena in physics and mechanics. In this paper, we analyzed the Kundu equation on the half-line by the Fokas method and proved that the potential function u(z, t) of the Kundu equation can be uniquely expressed by the solution of Riemann-Hilbert (RH) problem. It also includes the RH problem of the derivative nonlinear Schrödinger equation (also known as Kaup-Newell equation) (if ε=0), Chen–Lee–Liu equation (if ε=14) and Gerjikov–Ivanov equation (if ε=12) on the half-line.

N-Soliton Solution of The Kundu-Type Equation Via Riemann-Hilbert Approach

In this article, we focus on investigating the Kundu-type equation with zero boundary condition at infinity. Based on the analytical and symmetric properties of eigenfunctions and spectral matrix of its Lax pair, a Riemann-Hilbert problem for the initial value problem of the Kundu-type equation is constructed. Further through solving the regular and nonregular Riemann-Hilbert problem, a kind of general N-soliton solution of the Kundu-type equation are presented. As special cases of this result, the N-soliton solution of the Kaup-Newell equation, Chen-Lee-Liu equation, and Gerjikov-Ivanov equation can be obtained respectively by choosing different parameters.

Sub pico-second optical pulses in birefringent fibers for Kaup–Newell equation with cutting-edge integration technologies

This research article studies sub pico-second optical pulses in birefringent fibers. The Kaup–Newell equation in coupled vector form is firstly introduced. Bright, dark and singular solitons are obtained by modified simple and trial function principles.

First integrals and solutions of the traveling wave reduction for the Triki–Biswas equation

The integrability of the traveling wave reduction for the Triki–Biswas equation is considered. Two first integrals for this equation are found. It is shown that the Triki–Biswas equation is transformed to the nonlinear ordinary differential equation of the first order. The general solution of the traveling wave reduction for the Triki–Biswas equation is written using the quadrature. The general solution of the traveling wave reduction for the Kaup–Newell equation is given. Special solutions of the Triki–Biswas equation are presented.

Stochastic perturbation of sub-pico second envelope solitons for Triki–Biswas equation with multi-photon absorption and bandpass filters

This paper studies sub-pico second solitons modeled by Triki–Biswas equation that is a generalized version of Kaup–Newell equation in presence of stochastic perturbation terms, in addition to deterministic perturbations which are due to multi-photon absorption and bandpass filters. The soliton mean free velocity is recovered after analyzing the corresponding Langevin equation.

Bright and singular optical solitons for Kaup–Newell equation with two fundamental integration norms

This paper retrieves soliton solutions to Kaup–Newell's equation in optical fibers. Two fundamental integration schemes, namely the csch-function method and traveling wave hypothesis, obtain singular and bright soliton solutions respectively. The existence criteria for these solitons are also presented.

Solutions and connections of nonlocal derivative nonlinear Schrödinger equations

All possible nonlocal versions of the derivative nonlinear Schrodinger equations are derived by the nonlocal reduction from the Chen–Lee–Liu equation, the Kaup–Newell equation and the Gerdjikov–Ivanov equation which are gauge equivalent to each other. Their solutions are obtained by composing constraint conditions on the double Wronskian solution of the Chen–Lee–Liu equation and the nonlocal analogues of the gauge transformations among them. Through the Jordan decomposition theorem, those solutions of the reduced equations from the Chen–Lee–Liu equation can be written as canonical form within real field.

A τ-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter

A matrix spectral problem is researched with an arbitrary parameter. Through zero curvature equations, two hierarchies are constructed of isospectral and nonisospectral generalized derivative nonlinear schrödinger equations. The resulting hierarchies include the Kaup-Newell equation, the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation, the modified Korteweg-de Vries equation, the Sharma-Tasso-Olever equation and a new equation as special reductions. The integro-differential operator related to the isospectral and nonisospectral hierarchies is shown to be not only a hereditary but also a strong symmetry of the whole isospectral hierarchy. For the isospectral hierarchy, the corresponding τ -symmetries are generated from the nonisospectral hierarchy and form an infinite-dimensional symmetry algebra with the K-symmetries.

Chirped envelope optical solitons for Kaup–Newell equation

We investigate the Kaup-Newell equation that represents one of the forms of derivative nonlinear Schrödinger equation. The model applies to the description of sub-pico-second pulse propagation through an optical fiber. A special complex envelope traveling-wave method is applied to find a nonlinear equation with a fifth-degree nonlinear term describing the dynamics of field amplitude in the nonlinear media. It is shown that the phase associated to the obtained pulses has a nontrivial form and possesses two intensity dependent chirping terms in addition to the simplest linear contribution. A class of soliton solutions of the bright, dark and singular type are derived for the first time. The requirements concerning the optical material parameters for the existence of these chirped structures are also discussed.

Stability Analysis and Conservation Laws via Multiplier Approach for the Perturbed Kaup-Newell Equation

Stability Analysis and Conservation Laws via Multiplier Approach for the Perturbed Kaup-Newell Equation