Dark soliton control in inhomogeneous optical fibers

Analytic two-dark soliton solutions for a variable–coefficient nonlinear Schrödinger equation are obtained via modified Hirota method. Parallel solitons are observed and soliton control such as the soliton compression is realized with different group velocity dispersion profiles. Besides, soliton interactions are investigated with the interaction distance being adjusted. In addition, soliton repulsive structures as well as attractive ones are obtained with exponential dispersion profile. Results in our research may be useful for the soliton control in inhomogeneous optical fibers, which will be a benefit to the realistic optical communication systems.     

Soliton solutions and Bäcklund transformation for the complex Ginzburg-Landau equation

Symbolically investigated in this paper is the complex Ginzburg-Landau (CGL) equation. With the Hirota method, both bright and dark soliton solutions for the CGL equation are obtained simultaneously. New Bäcklund transformation in the bilinear form is derived. Relevant properties and features are discussed. Solitons can be compressed (amplified) when the nonlinear (linear) dispersion effect is enhanced. Meanwhile, central frequency of the soliton can be affected by the nonlinear and linear dispersion effects. Besides, directions of the movement for the soliton central frequency can be adjusted. Results of this paper would be of certain value to the studies on the soliton compression and amplification.     

New types of exact solutions for the fourth-order dispersive cubic-quintic nonlinear Schrödinger equation

In this study, we use two direct algebraic methods to solve a fourth-order dispersive cubic-quintic nonlinear Schrödinger equation, which is used to describe the propagation of optical pulse in a medium exhibiting a parabolic nonlinearity law. By using complex envelope ansatz method, we first obtain a new dark soliton and bright soliton, which may approach nonzero when the time variable approaches infinity. Then a series of analytical exact solutions are constructed by means of F-expansion method. These solutions include solitary wave solutions of the bell shape, solitary wave solutions of the kink shape, and periodic wave solutions of Jacobian elliptic function.     

On Lie symmetry analysis,conservation laws and solitary waves to a longitudinal wave motion equation

Under investigation in this work is a longitudinal wave motion equation, which describes the solitary waves propagation with dispersion caused by transverse Poisson’s effect in a magneto-electro-elastic circular rod. The Lie symmetry method is employed to study its vector fields and optimal systems, respectively. Furthermore, the symmetry reductions and eight families of soliton wave solutions of the equation are obtained on the basis of the optimal systems, including hyperbolic-type and trigonometric-type solutions. Two of reduced equations are Painlevé-like equations. Finally, by virtue of conservation law multiplier, the complete set of local conservation laws of the equation for the arbitrary constant coefficients is well constructed with a detailed derivation.     

Solitary waves in dusty plasmas with variable dust charge and two temperature ions

Propagation of nonlinear waves in dusty plasmas with variable dust charge and two temperature ions is analyzed. The Kadomtsev–Petviashivili (KP) equation is derived by using the reductive perturbation theory. A Sagdeev potential for this system has been proposed. This potential is used to study the stability conditions and existence of solitonic solutions. Also, it is shown that a rarefactive soliton can be propagates in most of the cases. The soliton energy has been calculated and a linear dispersion relation has been obtained using the standard normal-modes analysis. The effects of variable dust charge on the amplitude, width and energy of the soliton and its effects on the angular frequency of linear wave are discussed too. It is shown that the amplitude of solitary waves of KP equation diverges at critical values of plasma parameters. Solitonic solutions of modified KP equation with finite amplitude in this situation are derived.     

General propagation lattice Boltzmann model for variable-coefficient non-isospectral KdV equation

In this paper, a general propagation lattice Boltzmann model for variable-coefficient non-isospectral Korteweg–de Vries (vc-nKdV) equation, which can describe the interfacial waves in a two layer liquid and Alfvén waves in a collisionless plasma, is proposed by selecting appropriate equilibrium distribution function and adding the compensate function. The Chapman–Enskog analysis shows that the vc-nKdV equation can be recovered correctly from the present model. Numerical simulation for the non-propagating one soliton of this equation in different situations is conducted as validation. It is found that the numerical results match well with the analytical solutions, which demonstrates that the current general propagation lattice Boltzmann model is a satisfactory and efficient method, and could be more stable and accurate than the standard lattice Bhatnagar–Gross–Krook model.     

利用Darboux变换对KdV方程孤子解普遍性质的讨论

该文指出：利用Darboux变换不但可以非常简洁地得到文献［1］关于KdV方程单孤子解和双孤子解，而且便于讨论KdV方程的任意孤子解的性质．通过对KdV方程三孤子解的重点讨论，以及对KdV方程多孤子解的解析分析，得到了关于KdV方程任意阶孤子解的一些非常有意义的普遍结果．这些结果对于人们深入了解孤子相互作用规律具有重要的现实意义．     

On soliton solutions for the Fitzhugh–Nagumo equation with time-dependent coefficients

We consider a generalized Fitzhugh–Nagumo equation exhibiting time-varying coefficients and linear dispersion term. By means of specific solitary wave ansatz and the tanh method, a new variety of soliton solutions are derived. The physical parameters in the soliton solutions are obtained as function of the time-dependent model coefficients. The conditions of existence and uniqueness of solitons are presented. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous media that is described by the variable coefficients Fitzhugh–Nagumo equation. Clearly, adaptive methods are straightforward and concise and their applications for the Fitzhugh–Nagumo equation with t-dependent coefficients enable one to construct soliton-like solutions.     

Nonlinear longitudinal waves in inhomogeneous rods

The work contains two basic results. The first consists in the derivation of an equation for the longitudinal vibrations of a rod which accounts for both nonlinear and dispersion effects as well as effects caused by the inhomogeneity of the material of the rod. This equation is found to be a perturbed Korteweg-de Vries equation. The second result consists in the development of a perturbation method for solving the Cauchy problem for this equation. The solution found describes the deformation of a soliton under the influence of the inhomogeneity of the rod.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 99, pp. 64–73, 1980.     

Optical soliton with damping and frequency chirping in fibre media

Nonlinear Schrödinger (NLS) equation which governs the propagation of single field in the fibre medium with pulse chirping and gain (loss) is considered. The integrability condition of NLS equation is arrived from the linear eigenvalue problem and is studied by using the Painlevé singularity structure analysis. From the Lax pair, soliton solution is constructed by using the Darboux–Bäcklund transformation technique. From the results, we show that the soliton is alive, i.e., pulse area is conserved with the inclusion of damping and pulse chirping effects.     

Analytical studies of the steady solution for optical soliton equation with higher-order dispersion and nonlinear effects

The exact analytical solution of the optical soliton equation with higher-order dispersion and nonlinear effects has been obtained by the method of separating variables. The new type of optical solitary wave solution, which is quite different from the bright and dark soliton solutions, has been found under two special cases. The stability of the solitary wave solutions for the optical soliton equation is discussed. Some new conclusion of the stability are obtained, for the solitary wave solutions of the nonlinear wave equations, by using the Liapunov direct method.     

Traveling-Wave Solutions of the Schwarz–Korteweg–de Vries Equation in 2+1 Dimensions and the Ablowitz–Kaup–Newell–Segur Equation Through Symmetry Reductions

One of the more interesting solutions of the (2+1)-dimensional integrable Schwarz–Korteweg–de Vries (SKdV) equation is the soliton solutions. We previously derived a complete group classification for the SKdV equation in 2+1 dimensions. Using classical Lie symmetries, we now consider traveling-wave reductions with a variable velocity depending on the form of an arbitrary function. The corresponding solutions of the (2+1)-dimensional equation involve up to three arbitrary smooth functions. Consequently, the solutions exhibit a rich variety of qualitative behaviors. In particular, we show the interaction of a Wadati soliton with a line soliton. Moreover, via a Miura transformation, the SKdV is closely related to the Ablowitz–Kaup–Newell–Segur (AKNS) equation in 2+1 dimensions. Using classical Lie symmetries, we consider traveling-wave reductions for the AKNS equation in 2+1 dimensions. It is interesting that neither of the (2+1)-dimensional integrable systems considered admit Virasoro-type subalgebras.     

Fractional derivative method for describing solitons on the surface of deep water

Theoretical and Mathematical Physics - The fractional derivative method is used to take wave dispersion into account in the wave equation when describing the propagation of gravitational soliton...     

广义耦合KdV孤子方程的达布变换及其偶孤子解

根据广义耦合KdV孤子方程的Lax对, 借助谱问题的规范变换, 一个包含多参数的达布变换被构造出来. 利用达布变换来产生广义耦合KdV孤子方程的偶孤子解, 并且用行列式的形式来表达广义耦合KdV孤子方程的偶孤子解. 作为应用, 广义耦合KdV孤子方程的偶孤子解的前两个例子被给出.     

A study on the (2 + 1)‐dimensional KdV4 equation derived by using the KdV recursion operator

We derive a new ( 2 + 1)‐dimensional Korteweg–de Vries 4 (KdV4) equation by using the recursion operator of the KdV equation. This study shows that the new KdV4 equation possess multiple soliton solutions the same as the multiple soliton solutions of the KdV hierarchy, but differ only in the dispersion relations. We also derive other traveling wave solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

Observation of two soliton propagation in an erbium doped inhomogeneous lossy fiber with phase modulation

We consider optical pulse propagation in an Erbium doped inhomogeneous lossy optical fiber with time dependent phase modulation, which is governed by a system of Generalized Inhomogeneous Nonlinear Schrödinger Maxwell–Bloch (GINLS–MB) equation. Multi-soliton propagation is studied analytically by means of deriving associated Lax pair and the soliton solutions are obtained using Darboux transformation. By suitably adjusting the group velocity dispersion and nonlinearity parameter, we discuss various soliton dynamics such as periodic distributed amplification, pulse compression etc. In each case, we demonstrate the influence of inhomogeneous parameter. Finally we investigate the pulse compression through nonlinear tunneling.     

Solitary wave solutions for high dispersive cubic-quintic nonlinear Schrödinger equation

We consider the higher-order dispersive nonlinear Schrödinger equation including fourth-order dispersion effects and a quintic nonlinearity. This equation describes the propagation of femtosecond light pulses in a medium that exhibits a parabolic nonlinearity law. By adopting the ansatz solution of Li et al. [Zhonghao Li, Lu Li, Huiping Tian, Guosheng Zhou. New types of solitary wave solutions for the higher-order nonlinear Schrödinger equation. Phys Rev Lett 2000;84:4096], we find two different solitary wave solutions under certain parametric conditions. These solutions are in the form of bright and dark soliton solutions.     

Method of Immersion and Its Parametric Realization for the Korteweg–De Vries Equation

By using the method of immersion (imbedding) proposed in the author's previous works, we describe the space S of initial conditions of the Cauchy problem for the general differential Korteweg–de Vries equation. The space S is called a stationary soliton Korteweg–de Vries manifold because "stationary projections" of solitons fall into the space S. In addition, we introduce the notion of a space of Sturm–Liouville operators over a soliton Korteweg–de Vries manifold. For real functions and parameters, we formulate the spectral theorem for a commutative Lax pair over a real stationary soliton Korteweg–de Vries manifold.     

Dynamics of a coupled modified (1 + 1)‐dimensional Toda equation of BKP type

In this paper, we present a new coupled modified (1 + 1)‐dimensional Toda equation of BKP type (Kadomtsev‐Petviashvilli equation of B‐type), which is a reduction of the (2 + 1)‐dimensional Toda equation. Two‐soliton and three‐soliton solutions to the coupled system are derived. Furthermore, the N‐soliton solution is presented in the form of Pfaffian. The asymptotic analysis of two‐soliton solutions is studied to explain their collision properties. It is shown that the coupled system exhibit richer interaction phenomena including soliton fission, fusion, and mixed collision. Copyright © 2015 John Wiley & Sons, Ltd.     

Solitons in the system of coupled Hirota–Maxwell–Bloch equations

We propose the system of coupled Hirota–Maxwell–Bloch equations which governs the propagation of optical pulses in an erbium doped nonlinear fibre with higher order dispersion, self-steepening and self induced transparency (SIT) effects. The Lax pair is explicitly constructed and the soliton solution is obtained using the Darboux–Bäcklund transformations. Hence, the system is found to admit soliton type lossless wave propagation.