Pulse Amplification in Dispersion-Decreasing Fibers with Symbolic Computation

The pulse amplification in the dispersion-decreasing fiber (DDF) isinvestigated via symbolic computation to solve the variable-coefficient higher-order nonlinear Schrödinger equation with the effects of third-order dispersion, self-steepening, and stimulated Raman scattering. The analytic one-soliton solution of this model is obtained with a set of parametricconditions. Based on this solution, the fundamental soliton is shown to be amplified in the DDF. The comparison of the amplitude of pulses for different dispersion profiles of the DDF is also performed through the graphical analysis. The results of this paper would be of certain value to the study of signal amplification and pulse compression.     

Darboux Transformation and Soliton Solutions for Inhomogeneous Coupled Nonlinear Schrodinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrodinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems （i.e. Lax pair）, we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features of picosecond solitons in inhomogeneous optical fibers.     

A Bilinear Baecklund Transformation and N-Soliton-Like Solution of Three Coupled Higher-Order Nonlinear Schroedinger Equations with Symbolic Computation

A bilinear Baecklund transformation is presented for the three coupled higher-order nonlinear Schroedinger equations with the inclusion of the group velocity dispersion, third-order dispersion and Kerr-law nonlinearity, which can describe the dynamics of alpha helical proteins in living systems as well as the propagation of ultrashort pulses in wavelength-division multiplexed system. Starting from the Baecklund transformation, the analytical soliton solution is obtained from a trivial solution. Simultaneously, the N-soliton-like solution in double Wronskian form is constructed, and the corresponding proof is also given via the Wronskian technique. The results obtained from this paper might be valuable in studying the transfer of energy in biophysics and the transmission of light pulses in optical communication systems.     

Sub-ODE＇s New Solutions and Their Applications to Two Nonlinear Partial Differential Equations with Higher-Order Nonlinear Terms

In the present paper, with the aid of symbolic computation, families of new nontrivial solutions of the first-order sub-ODE F12 = AF2 ＋ BF2＋p ＋ CF2＋2p （where F1= dF/dε, p 〉 0） are obtained. To our best knowledge, these nontrivial solutions have not been found in [X.Z. Li and M.L. Wang, Phys. Lett. A 361 （2007） 115] and IS. Zhang, W. Wang, and J.L. Tong, Phys. Lett. A 372 （2008） 3808] and other existent papers until now. Using these nontrivial solutions, the sub-ODE method is described to construct several kinds of exact travelling wave solutions for the generalized KdV-mKdV equation with higher-order nonlinear terms and the generalized ZK equation with higher-order nonlinear terms. By means of this method, many other physically important nonlinear partial differential equations with nonlinear terms of any order can be investigated and new nontrivial solutions can be explicitly obtained with the help of symbolic computation system Maple or Mathematics.     

非线性色散渐减光纤中高峰值脉冲的激发和传输

本文基于变系数的非线性薛定谔方程,数值地讨论高峰值脉冲在色散渐减光纤中的激发和传输。首先,基于变系数非线性薛定谔方程的Peregrine孤子解,解析和数值地讨论精确的Peregrine孤子在色散渐减光纤中的传输特性。其次,通过输入不同的平面波背景上的局域脉冲,研究高峰值脉冲在非线性色散渐减光纤中的激发和传输。结果显示Peregrine孤子在色散渐减光纤中传输时,会产生一个空间和时间都局域化的高峰值单脉冲,并且当啁啾为负时,脉冲的幅值增加,脉宽被压缩。若光纤系统存在增益,脉冲的幅值也会增加。由于非线性光纤中的调制不稳定性过程,不同平面波背景上的小局部扰动都可激发出高峰值脉冲,除了峰值和宽度略有不同外,激发脉冲的形状几乎相同。     

Painleve Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with Symbolic Computation

The new soliton solutions for the variable-coefficient Boussinesq system, whose applications are seen in fluid dynamics, are studied in this paper with symbolic computation. First, the Painleve analysis is used to investigate its integrability properties. For the identified case we give, the Lax pair of the system is found, and then the Darboux transformation is constructed. At last, some new soliton solutions are presented via the Darboux method. Those solutions might be of some value in fluid dynamics.     

Variable-coefficient higher-order nonlinear Schrödinger model in optical fibers: New transformation with burstons,brightons and symbolic computation

In a realistic fiber of weakly dispersive and nonlinear dielectrics with distributed parameters, a variable-coefficient higher-order nonlinear Schrödinger (vcHNLS) model can be used to describe the femtosecond pulse propagation, applicable to, e.g., the design of ultrafast signal-routing and dispersion-managed fiber-transmission systems. In this Letter, new transformation is proposed, by virtue of symbolic computation, from a vcHNLS model to its known constant-coefficient counterpart without amplification/absorption. Features of the transformation are analyzed, and constraints on the variable coefficients are presented. Such physically/optically interesting examples as the variable-coefficient burstons and brightons are constructed in explicit forms with their properties discussed. Burstons and brightons are potentially observable with future optical-fiber experiments.     

Multi-soliton solutions and a Bäcklund transformation for a generalized variable-coefficient higher-order nonlinear Schrödinger equation with symbolic computation

In this paper, by virtue of symbolic computation, the investigation is made on a generalized variable-coefficient higher-order nonlinear Schrödinger equation with varying higher-order effects and gain or loss, which can describe the femtosecond optical pulse propagation in a monomode dielectric waveguide. A modified dependent variable transformation is introduced into the bilinear method to transform such an equation into a variable-coefficient bilinear form. Based on the formal parameter expansion technique, the multi-soliton solutions of this equation are obtained through the bilinear form under sets of parametric constraints. A Bäcklund transformation in bilinear form is also obtained for the first time in this paper. Finally, discussions on the analytic soliton solutions are given and various propagation situations are illustrated.     

Symbolic computation on soliton solutions for variable-coefficient nonlinear Schr?dinger equation in nonlinear optics

Soliton interaction and control using the dispersion-decreasing fibers with potential applications to the design of high-speed optical devices and ultralarge capacity transmission systems are investigated based on solving the variable-coefficient nonlinear Schr?dinger equation with symbolic computation. Via the Hirota method, analytic two- and three-soliton solutions for that model are obtained, with their relevant properties and features illustrated. Dispersion-decreasing fibers with different profiles are found to be able to control the soliton velocity. Additionally, through the asymptotic analysis for the two-soliton solutions, we point out that the interaction between two solitons is elastic. Finally, a new approach to controll the soliton interaction using the dispersion-decreasing fiber with the Gaussian profile is suggested.     

Various Methods for Constructing Auto-Baicklund Transformations for a Generalized Variable-Coefficient Korteweg-de Vries Model from Plasmas and Fluid Dynamics

In this paper, under the Painleve-integrable condition, the auto-Biicklund transformations in different forms for a variable-coefficient Korteweg-de Vries model with physical interests are obtained through various methods including the Hirota method, truncated Painleve expansion method, extendedvariable-coefficient balancing-act method, and Lax pair. Additionally, the compatibility for the truncated Painleve expansion method and extended variable-coetfficient balancing-act method is testified.     

Soliton Solutions and Bilinear B/icklund Transformation for Generalized Nonlinear Schr6dinger Equation with Radial Symmetry

Investigated in this paper is the generalized nonlinear Schrodinger equation with radial symmetry. With the help of symbolic computation, the one-, two-, and N-soliton solutions are obtained through the bilinear method. B~cklund transformation in the bilinear form is presented, through which a new solution is constructed. Graphically, we have found that the solitons are symmetric about x = O, while the soliton pulse width and amplitude will change along with the distance and time during the propagation.     

Exact Solutions to a （3＋l）-Dimensional Variable-Coefficient Kadomtsev-Petviashvilli Equation via the Bilinear Method and Wronskian Technique

By truncating the Painleve expansion at the constant level term, the Hirota bilinear form is obtained for a （3＋1）-dimensional variable-coefficient Kadomtsev Petviashvili equation. Based on its bilinear form, solitary-wave solutions are constructed via the ε-expansion method and the corresponding graphical analysis is given. Furthermore, the exact solution in the Wronskian form is presented and proved by direct substitution into the bilinear equation.     

A Maple Package on Symbolic Computation of Hirota Bilinear Form for Nonlinear Equations

An improved algorithm for symbolic computation of Hirota bilinear forms of KdV-type equations with logarithmic transformations is presented. In the algorithm, the general assumption of Hirota bilinear form is successfully reduced based on the property of uniformity in rank. Furthermore, we discard the integral operation in the traditional algorithm. The software package HBFTrans is written in Maple and its running effectiveness is tested by a variety soliton equations.     

Bilinear forms and dark soliton behaviors for a higher-order variable-coefficient nonlinear Schrödinger equation in an inhomogeneous alpha helical protein

In this paper, a higher-order variable-coefficient nonlinear Schrödinger equation is studied, which describes an inhomogeneous alpha helical protein with higher-order excitation and interaction under the continuum approximation. With the aid of auxiliary function, we obtain the variable-coefficient Hirota’s bilinear equations under a set of integrable constraints. Using the Hirota’s method and symbolic computation, we derive the dark one-, two- and N-soliton solutions. Influences of the variable coefficients on the soliton velocity, amplitude, and shape are analyzed. For instance, when the variable coefficients are the linear and quadratic functions of time, since the pharmacological efficacy in specific sites of the alpha helical protein diffuses linearly and quadratically as time goes on, we obtain a parabolic and cubic soliton. Interactions between/among the two, three, and four solitons with different values of variable coefficients are also discussed with the results including the parabolic, cubic, periodical, and stationary solitons.     

Nonlinear polarization rotation-induced pulse shaping in a stretched-pulse ytterbium-doped fiber laser

We report on controllable pulse shaping in a Yb-doped stretched-pulse fiber laser followed by a high-power chirped pulse amplifier. We demonstrate that the pulses after an extra-cavity grating pair change their intensity profile from Lorentz to Gaussian and then to sech2 shapes by adjusting the intra-cavity polarization through a quarter-wave plate inside the fiber laser cavity. The laser pulses with different pulse shapes exhibit pulse-to-pulse amplitude fluctuation of -- 1.02%, while the sech2-shaped pulse train is provided with a more stable free-running repetition rate as a result of the stronger self-phase modulation in the fiber laser cavity than Lorentz- and Gaussian-shaped pulse trains.     

Integrability study on a generalized (2+1)-dimensional variable-coefficient Gardner model with symbolic computation

Gardner model describes certain nonlinear elastic structures, ion-acoustic waves in plasmas, and shear flows in ocean and atmosphere. In this paper, by virtue of the computerized symbolic computation, the integrability of a generalized (2+1)-dimensional variable-coefficient Gardner model is investigated. Painleve? integrability conditions are derived among the coefficient functions, which reduce all the coefficient functions to be proportional only to γ(t), the coefficient of the cubic nonlinear term u(2)u(x). Then, an independent transformation of the variable t transforms the reduced γ(t)-dependent equation into a constant-coefficient integrable one. Painleve? test shows that this is the only case when our original generalized (2+1)-dimensional variable-coefficient Gardner model is integrable.     

Wronskian and Grammian Determinant Solutions for a Variable-Coefficient Kadomtsev-Petviashvili Equation

In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.     

Wronskian Form of N-Solitonic Solution for a Variable-Coefficient Korteweg-de Vries Equation with Nonuniformities

By the symbolic computation and Hirota method, the bilinear form and an auto-Backlund transformation for a variable-coemcient Korteweg-de Vries equation with nonuniformities are given. Then, the N-solitonic solution in terms of Wronskian form is obtained and verified. In addition, it is shown that the （N - 1）- and N-solitonic solutions satisfy the auto-Backlund transformation through the Wronskian technique.     

色散渐减光纤中自相似脉冲传输特性研究

研究了在高阶色散影响下,自相似脉冲在具有正常色散的色散渐减光纤中的演化情况.结果表明:当考虑高阶色散的影响时,脉冲的啁啾仍然具有很强的线性特性,只是中心变得不对称,产生中心漂移.这种啁啾特性使得自相似脉冲在时域中的抛物线形状产生畸形,导致了脉冲峰向一边延迟,并使脉冲的中心位置漂移,同时伴随着脉冲边沿的振荡.但是通过采用色散补偿技术,自相似脉冲强的线性啁啾仍然可以得到高质量的飞秒量级压缩脉冲,与忽略三阶色散影响时得到的压缩脉冲的脉宽近似相等.     

Darboux Transformation and Soliton Solutions for a Variable-Coefficient Modified Kortweg-de Vries Model from Fluid Mechanics,Ocean Dynamics,and Plasma Mechanics

This paper is to investigate a variable-coefficient modified Kortweg-de Vries （vc-mKdV） model, which describes some situations from fluid mechanics, ocean dynamics, and plasma mechanics. By the AblowRz-Kaup-NewellSegur procedure and symbolic computation, the Lax pair of the vc-MKdV model is derived. Then, based on the aforementioned Lax pair, the Darboux transformation is constructed and a new one-soliton-like solution is obtained as weft Features of the one-soliton-like solution are analyzed and graphically discussed to illustrate the influence of the variable coefficients in the solitonlike propagation.