Higher-order Nonlinear Schrodinger Research Articles

The extended F-expansion method and its application for a class of nonlinear evolution equations

Abstract By means of a simple transformation technique, we have shown that the higher-order nonlinear Schrodinger equation in nonlinear optical fibers, a new Hamiltonian amplitude equation, generalized Hirota–Satsuma coupled system and generalized ZK-BBM equation can be reduced to the elliptic-like equation. Then, the extended F-expansion method is used to obtain a series of solutions including the single and the combined nondegenerative Jacobi elliptic function solutions and their degenerative solutions to the above mentioned class of nonlinear evolution equations.

Soliton-like excitations in the one-dimensional electrical transmission line

The dynamics of modulated waves are studied in the one-dimensional discrete nonlinear electrical transmission line. The contribution of the linear dispersive capacitance is taken into account, and it is shown via the reductive perturbation method that the evolution of such waves in this system is governed by the higher-order nonlinear Schrodinger equation. Passing through the Stokes analysis, we establish a generalized criterion for the Benjamin-Feir instability in the network and determine the exact solutions of the obtained wave equation by using the Pathria-Morris approach.

EXACT SOLITON SOLUTIONS FOR THE HIGHER-ORDER NONLINEAR SCHRÖDINGER EQUATION

Based on the complex envelope ansatz method, the projective Riccati equation method and q-deformed hyperbolic functions, a method is developed for constructing a series of exact analytical solutions for higher-order nonlinear Schrödinger (HNLS) equation, which describes propagation of femtosecond light pulse in optical fiber under certain parametric conditions. With the help of symbolic computation, six families of new solitary wave solutions are obtained. The solitary wave solutions obtained by Li et al.18 are special cases of our solutions. The novel soliton solutions can describe W-shaped, bright and dark soliton properties in the same expression and their amplitude may approach nonzero when the time variable approaches infinity. Furthermore, the soliton propagation and solitons interaction scenario are discussed and simulated by computer.

On the Painlevé integrability, periodic wave solutions and soliton solutions of generalized coupled higher-order nonlinear Schrödinger equations

Abstract It is proven that generalized coupled higher-order nonlinear Schrodinger equations possess the Painleve property for two particular choices of parameters, using the Weiss–Tabor–Carnevale method and Kruskal’s simplification. Abundant families of periodic wave solutions are obtained by using the Jacobi elliptic function expansion method with the assistance of symbolic manipulation system, Maple. It is also shown that these solutions exactly degenerate to bright soliton, dark soliton and mixed dark and bright soliton solutions with physical interests.

Combined solitary wave solutions for the inhomogeneous higher-order nonlinear Schrödinger equation

We consider the inhomogeneous higher-order nonlinear Schrodinger equation and explicitly present exact combined solitary wave solutions that can describe the simultaneous propagation of bright and dark solitary waves in a combined form in inhomogeneous fiber media or in optical communication links with distributed parameters. Furthermore, we analyze the features of the solutions, and numerically discuss the stabilities of these solitary waves under slight violations of the parameter conditions and finite initial perturbations. The results show that there exist combined solitary wave solutions in an inhomogeneous fiber system, and the combined solitary wave solutions are stable under slight violations of the parameter conditions and finite initial perturbations. Finally, the interaction between two neighboring combined solitary waves is numerically discussed.

Optical Solitary Waves in the Generalized Higher Order Nonlinear Schrödinger Equation

We study new types solitary wave solutions of the higher order nonlinear Schrodinger equation (HONLSE) with variable coefficients, that is, the generalized higher order nonlinear Schrodinger equati...

A direct truncation method for finding abundant exact solutions and application to the one-dimensional higher-order Schrödinger equation

We suggest a direct truncation technique for finding exact solutions of nonlinear differential equation, this method is based on the WTC test. As an application, abundant new exact stationary solutions of the one-dimensional higher-order nonlinear Schrodinger equation are obtained. These solutions include bright, dark, kink or anti-kink solitary wave solutions, which are dependent of the model and free parameters in the solutions. Algebraic solitary-like solution and new periodic solutions are also obtained. An interesting fact is that some solitary solutions can convert into the periodic solutions and vice versa when the free parameters are changed. (C) 2004 Elsevier Ltd. All rights reserved.

An improved algebra method and its applications in nonlinear wave equations

In this paper, other types of exact solution of a first-order nonlinear ordinary differential equation, which is including the algebraic method, is further investigated. By using the solutions of this equation, we give some types of solutions of the coupled KdV equation, nonlinear dispersion mK( m, n, k) equation, the variant shallow water wave equation, nonlinear dissipative equation and the higher-order nonlinear Schrodinger equation. These solutions includes compacton solutions, solitary pattern solutions, solitary wave solutions, Weierstrass elliptic function solutions, which may be useful to explain some phenomena.

Theoretical analysis of nonlinear pulse propagation in ferrite-dielectric-metal structures based on the nonlinear Schrödinger equation with higher order terms

An analysis of nonlinear magnetostatic surface wave pulse propagation in planar ferrite-dielectric-metal (FDM) structures has been performed. The analysis was based on numerical solutions to the higher-order nonlinear Schrödinger (HONLS) equation, with third-order (D3) and nonlinear (Q) dispersion terms taken into account. The analysis focuses on (1) the crossover dispersion region for FDM structures and the point in wave-number k where the second-order dispersion parameter D2 is a positive maximum, the Lighthill criterion for envelope soliton propagation is satisfied, and D3 is close to zero, and (2) the end points of this crossover region where D2 is zero. All operational HONLS equation parameters were evaluated from analytical dispersion expressions for the FDM structure and for magnetic field and structure parameters which match experiments. For (1), the pulse results indicate nondispersive propagation consistent with envelope solitons. The only effect of the Q term is to decrease slightly the propagation speed. For (2), both end points give nondispersive propagation as well. At the low-k crossover point, this result is consistent with condition D3Q<0 and the existence of an analytic soliton solution. For the high-k crossover point, the nondispersive propagation is likely due simply to the relatively small value of the third-order dispersion D3 parameter. In both cases, one finds a rapidly oscillating wave-packet structure which propagates ahead of the main pulse. For the low-k crossover point, this structure travels at approximately twice the main pulse velocity. For the high-k crossover point, the structure travels with the main pulse. The Q term has a small effect on the pulse amplitude for the low-k crossover point, and serves to reduce the pulse amplitude by 20%–30% at the high-k crossover point.

The Classical Problem of Water Waves: a Reservoir of Integrable and Nearly-Integrable Equations

In this contribution, we describe the simplest, classical problem in water waves, and use this as a vehicle to outline the techniques that we adopt to analyse this particular approach to the derivation of soliton-type equations. The surprise, perhaps, is that such an apparently transparent set of equations (the Euler equation for an incompressible fluid, the equation of mass conservation and the simplest bottom and surface conditions) contains so may different – and important – equations. We will briefly show how such equations are generated, by carefully describing the asymptotic procedures that we adopt. We will then present a number of examples, some of which will be the familiar integrable equations (Korteweg-de Vries (KdV), Nonlinear Schrödinger (NLS)) and also variants of these which are, in a sense, nearly-integrable. These will include, for example, a variable coefficient KdV equation, a nearly concentric KdV equation and higher-order NLS equation. Some of these results have significant consequences for water-wave propagation; for example, the existence of a KdV equation for arbitrary velocity profiles below the surface helps to explain why solitary waves are observed in nature. We conclude with a discussion of some of the more recent, exciting work on the rôle of the Camassa–Holm (CH) equation in water waves. We will also outline how, for this type of problem, a number of interesting and relevant variants of the CH equation can arise.

Generalized method and its application in the higher-order nonlinear Schrodinger equation in nonlinear optical fibres

A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.

Soliton interaction under the influence of higher-order effects

In this paper, we present exact N-soliton solution by employing simple, straightforward Darboux transformation based on the Lax pair for Hirota equation, a higher-order nonlinear Schrödinger (HNLS) equation. As examples, one- and two-soliton solutions in explicit forms are given and their properties are also analyzed. A bound solution without interaction will be theoretically predicted if one can adjust frequency shift for each soliton appropriately. Further, we obtain the approximate eigenvalues by employing two-soliton solution and discuss analytically the interaction between neighboring solitons under the influence of the higher-order effects. It is shown that the combined effects of the higher-order effects can restrain the interaction between neighboring solitons to some extent. The results are proved by directly solving HNLS equation numerically.

DARK SOLITONS IN OPTICAL FIBERS WITH HIGHER ORDER EFFECTS

We consider the higher order nonlinear Schrödinger (HNLS) equation, which governs the nonlinear wave propagation in optical fibers with higher order effects. Lax pair associated with the integrable HNLS equation for the pulse propagation in normal dispersion regime of the fiber media is constructed with the help of Ablowitz–Kaup–Newell–Segur method. Using Hirota bilinear method, dark soliton solution is explicitly derived. Similar study is also carried out for simultaneous propagation of N nonlinear pulses in the normal dispersion regime of the fiber system with higher order effects.

A higher-order model for transverse waves in a generalized elastic solid

In the present study, the nonlinear modulation of transverse waves propagating in a generalized elastic solid is studied using a multi-scale expansion of quasi-monochromatic wave solutions. In particular, to include the higher-order nonlinear and dispersive effects in the evolution equations, higher-order perturbation equations are considered, and it is shown that the modulation of two transverse waves is governed by a pair of the coupled higher-order nonlinear Schrödinger (HONLS) equations. In the absence of one of the transverse waves, the coupled HONLS equations reduce to the single HONLS equation that has already been obtained in the context of nonlinear optics. Some special solutions to the coupled HONLS equations are also presented.

Propagation of dark solitons in a system of coupled higher-order nonlinear Schrödinger equations

In this paper, we consider the higher-order linear and nonlinear effects in optics and analyse a coupled system of higher-order nonlinear Schrodinger equations to identify the conditions for dark-soliton propagation through Painleve analysis, to supplement the known bright-soliton conditions. We also construct the explicit Lax pair and the Hirota bilinear form is used to generate one and two dark solitons.

BRIGHT AND DARK SOLITARY WAVES IN KERR MEDIUM WITH HIGHER ORDER LINEAR AND NONLINEAR DISPERSIONS

The existence of both bright and dark solitary wave solutions has been demonstrated for higher order nonlinear Schrödinger (HNLS) equation containing up to the fourth order linear and the second order nonlinear dispersion. We found that the solutions are completely characterized by the material properties and free from arbitrary parameters in contrast to solutions of the pure NLS equation. The two types of solution were further shown to have complementary parameter space and share a singularity line of asymptotic values ±∞ for the solitary envelope velocities.

DEEP REDUCTION OF DARBOUX TRANSFORMATION TO A 3×3 SPECTRAL PROBLEM

The Darboux tranformation of a 3×3 spectral problem which is associated with the higher-order nonlinear Schrodinger equation is given. Some solutions of the higher-order nonlinear Schrodinger equation are provided by taking different "seeds".

A Coupled Higher-Order Nonlinear Schrodinger Equation Including Higher-Order Bright and Dark Solitons

We suggest a generalized Lax pair on a Hermi- tian symmetric space to generate a new coupled higher-order nonlinear Schrodinger equation of a dual type which contains both bright and dark soliton equations depending on parameters in the Lax pair. Through the generalized ways of reduction and the scaling transformation for the coupled higher-order nonlinear Schrodinger equation, two integrable types of higher-order dark soliton equations and their extensions to vector equations are newly derived in addition to the corresponding equations of the known higher-order bright solitons. Analytical discus- sion on a general scalar solution of the higher- order dark soliton equation is then made in de- tail.

Derivation of a higher order nonlinear Schrödinger equation for weakly nonlinear Rossby waves

In the paper, with the help of a perturbation expansion method a new higher order nonlinear Schrödinger (HNLS) equation is derived to describe nonlinear modulated Rossby waves in the geophysical fluid. Using this equation, the modulational wave trains are discussed. It is found that the higher order terms favor the instability growth of modulational disturbances superimposed on uniform Rossby wave trains, but the instability region becomes narrower. In addition, the latitude and uniform background basic flow are found to affect the instability growth rate and instability region of uniform Rossby wave train. However, for a geostrophic flow the background basic flow does not affect the modulational instability of uniform Rossby wave train.

Multisoliton solutions and integrability aspects of coupled higher-order nonlinear Schrodinger equations

Using Painleve singularity structure analysis, we show that coupled higher-order nonlinear Schrodinger (CHNLS) equations admit the Painleve property. Using the results of the Painleve analysis, we succeed in Hirota bilinearizing the CHNLS equations for the integrable cases. Solving the Hirota bilinear equations, one soliton and two soliton solutions are explicitly obtained. Lax pairs are explicitly constructed.