Higher-order Nonlinear Schrodinger Equation Research Articles

Femtosecond Pulse Propagation in Optical Fibers Under Higher Order Effects: A Collective Variable Approach

Employing collective variable approach, femtosecond pulse propagation has been investigated in optical fibers using the higher order nonlinear Schrodinger equation. In order to view the pulse dynamics along the propagation distance, variation of different pulse parameters, called collective variables, such as pulse amplitude, width, chirp, pulse center and frequency has been investigated by numerically solving the set of ordinary equations obtained from collective variable approach.

A generalized auxiliary equation method and its applications

In this paper, a generalized auxiliary equation method with the aid of the computer symbolic computation system Maple is proposed to construct more exact solutions of nonlinear evolution equations, namely, the higher-order nonlinear Schrodinger equation, the Whitham–Broer–Kaup system, and the generalized Zakharov equations. As a result, some new types of exact travelling wave solutions are obtained, including soliton-like solutions, trigonometric function solutions, exponential solutions, and rational solutions. The method is straightforward and concise, and its applications are promising.

Solitary wave interaction for a higher-order nonlinear Schrödinger equation

Solitary wave interaction for a higher-order version of the nonlinear Schrodinger (NLS) equation is examined. An asymptotic transformation is used to transform a higher-order NLS equation to a higher-order member of the NLS integrable hierarchy, if an algebraic relationship between the higher-order coefficients is satisfied. The transformation is used to derive the higher-order one- and two-soliton solutions; in general, the N -soliton solution can be derived. It is shown that the higher-order collision is asymptotically elastic and analytical expressions are found for the higher-order phase and coordinate shifts. Numerical simulations of the interaction of two higher-order solitary waves are also performed. Two examples are considered, one satisfies the algebraic relationship derived from asymptotic theory, and the other does not. For the example which satisfies the algebraic relationship, the numerical results confirm that the collision is elastic. The numerical and theoretical predictions for the higher-order phase and coordinate shifts are also in strong agreement. For the example which does not satisfy the algebraic relationship, the numerical results show that the collision is inelastic and radiation is shed by the solitary wave collision. As the bed of radiation shed by the waves decays very slowly (like t-1/2), it is computationally infeasible to calculate the final phase and coordinate shifts for the inelastic example. An asymptotic conservation law is derived and used to test the finite-difference scheme for the numerical solutions. © 2007 Oxford University Press.

Generation and propagation of subpicosecond pulse train

Higher-order nonlinear Schrodinger equation with the Hirota constraint conditions is considered, and an analytic solution, which can describe the modulational instability process, is presented. Based on the solution, a new pulse train without continuous wave (CW) background is generated in quadratures and the propagation of the pulse train is discussed in detail by simulating numerically. The results show that, unlike the propagation of the picosecond pulse train, under the effects of the higher-order terms, the pulse train cannot propagate along the fibre when the energy is very high; however, for some medium energy the pulse train can stably propagate. We also investigate the stability of the pulse train against violation of the Hirota conditions, and the results show that the pulse train can still propagate stably when the Hirota conditions are broken.

General Solution to a Bilinear Reduction of the Higher Order Nonlinear Schrödinger Equation

The general solution is found to a bilinear reduction of the higher order nonlinear Schrodinger equation. Except for the previously known special cases having multisoliton solutions, the solution has the form of a single envelope solitary wave or wave train for all other values of the parameters. We conjecture that the generic solution is limited to such a narrow class of functions because the bilinear system is overdetermined.

Sub-ODE method and solitary wave solutions for higher order nonlinear Schrödinger equation

With the aid of an ordinary differential equation (ODE) involving an arbitrary positive power of dependent variable and its various positive solutions, three types of solitary wave solution of the higher order nonlinear Schrodinger equation (NLSE) with non-Kerr terms have been found out, which are the bell type solitary waves, the kink type solitary waves and the algebraic solitary waves, provided that the coefficients of the higher order NLSE satisfy certain constraint conditions.

Exact bright soliton solution for a family of coupled higher-order nonlinear Schrödinger equation in inhomogeneous optical fiber media

In this paper, we present a family of coupled higher-order nonlinear Schrodinger equation describing the optical soliton pulse propagating in inhomogeneous optical fiber media. The exact N-soliton solution and its characteristics of stabilities and novel elastic collision properties are studied in detail. As an example, we give the relative numerical evolutions by a soliton control system to discuss the pulses propagation characteristics.

Exact solutions for the high-order dispersive cubic-quintic nonlinear Schrödinger equation by the extended hyperbolic auxiliary equation method

Abstract By using the extended hyperbolic auxiliary equation method, we present explicit exact solutions of the high-order nonlinear Schrodinger equation with the third-order and fourth-order dispersion and the cubic-quintic nonlinear terms, describing the propagation of extremely short pulses. These solutions include trigonometric function type and exact solitary wave solutions of hyperbolic function type. Among these solutions, some are found for the first time.

The subsidiary elliptic-like equation and the exact solutions of the higher-order nonlinear Schrödinger equation

Abstract By using the homogeneous balance principle, the exact solutions of the subsidiary elliptic-like equation are derived with the aid of the coupled projective Riccati equations. The higher-order nonlinear Schrodinger equation describing the propagation of ultrashort pulses in nonlinear optical fibers is researched and its various exact solutions are derived by means of the simple transformation and the subsidiary elliptic-like equation mentioned above.

New solitons for the Hirota equation and generalized higher-order nonlinear Schrödinger equation with variable coefficients

In this paper, multisoliton solutions of the Hirota equation with variable coefficients are obtained by the Darboux transformation based on the Ablowitz–Kaup–Newell–Segur technology. As an example, we discuss the evolutional behaviour of a two-soliton solution in a soliton control fibre system. The results reveal that one may control the interaction between the pulses by choosing the third-order dispersion parameters d4 and h appropriately. Meanwhile, more generalized forms of bright soliton and dark soliton solutions of generalized higher order nonlinear Schrodinger equations (GHONLSE) with variable coefficients are obtained by the extended tanh-function method. Moreover, new bright and dark combined solitary wave, kink solitary wave and M-shaped solitary wave to GHONLSE with variable coefficients are firstly reported in this paper. Especially, the term proportional to α1 resulting from the group velocity decides the group velocity and the phase shift of these new solitary waves.

Exact solutions to two higher order nonlinear Schrödinger equations

Abstract Using the homogeneous balance principle and F-expansion method, the exact solutions to two higher order nonlinear Schrodinger equations which describe the propagation of femtosecond pulses in nonlinear fibres are obtained with the aid of a set of subsidiary higher order ordinary differential equations (sub-equations for short).

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.

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.

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.

Cross-phase modulational instability in an elliptical birefringent fiber with higher order nonlinearity and dispersion

We obtain conditions for the occurrence of cross-phase modulational instability in the normal dispersion regime for the coupled higher order nonlinear Schrodinger equation with higher order dispersion and nonlinear terms.