Solitons for a generalized variable-coefficient nonlinear SchrSdinger equation

In this paper, we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear SchrSdinger equation （NLS） with an arbitrary time-dependent linear potential which describes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensations. Under some reasonable assumptions, one-soliton and two-soliton solutions are constructed analytically by the Hirota method. From our results, some previous one- and two- soliton solutions for some NLS-type equations can be recovered by some appropriate selection of the various parameters. Some figures are given to demonstrate some properties of the one- and the two-soliton and the discussion about the integrability property and the Hirota method is given finally.     

Solitons in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length in a Harmonic Trap

We obtain the integrable relation for the one-dimensional nonlinear Schrodinger equations which describes the dynamics of a Bos-Einstein Condensates with time-dependent scattering length in a harmonic potential. The exact one- and two-soliton solutions are constructed analytically by using the Hirota method. Then we further discuss the dynamics of the one soliton and the interactions between two solitons in currently experimental conditions.     

Breathers and solitons for the coupled nonlinear Schr?dinger system in three-spine α-helical protein

We mainly investigate the variable-coefficient 3-coupled nonlinear Schr?dinger(NLS) system, which describes soliton dynamics in the three-spine α-helical protein with inhomogeneous effect. The variable-coefficient NLS equation is transformed into the constant coefficient NLS equation by similarity transformation firstly. The Hirota method is used to solve the constant coefficient NLS equation, and then we get the one-and two-breather solutions of the variable-coefficient NLS equation. The results show that, in the background of plane waves and periodic waves, the breather can be transformed into some forms of combined soliton solutions. The influence of different parameters on the soliton solution and the collision between two solitons are discussed by some graphs in detail. Our results are helpful to study the soliton dynamics inα-helical protein.     

Interaction properties of solitons for a couple of nonlinear evolution equations

We study one-and two-soliton solutions for the Cahn–Allen(CA) equation and the Brethorton equation. The CA equation has broad spectrum of applications especially in anti-phase boundary motion and it is used in phase-field models.While the Brethorton equation is a model for dispersive wave systems, it is used to find the resonant nonlinear interaction among three linear modes. We use the Hirota bilinear method to obtain one-and two-soliton solutions to the CA equation and the Brethorton equation.     

Bilinear Backlund transformation and explicit solutions for a nonlinear evolution equation

The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The Backlund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.     

Rational and Periodic Wave Solutions of Two-Dimensional Boussinesq Equation

Two new exact, rational and periodic wave solutions are derived for the two-dimensional Boussinesq equation. For the first solution it is obtained by performing an appropriate limiting procedure on the soliton solutions obtained by Hirota bilinear method. The second one in terms of Riemann theta function is explicitly presented by virtue of Hirota bilinear method and its asymptotic property is also analyzed in detail. Moreover, it is of interest to note that classical soliton solutions can be reduced from the periodic wave solutions.     

Direct Approach to Construct the Periodic Wave Solutions for Two Nonlinear Evolution Equations

With symbolic computation, the Hirota method and Riemann theta function are employed to directly construct the periodic wave solutions for the Hirota-Satsuma equation for shallow water waves and Boiti-Leon-Manna- Pempinelli equation. Then, the corresponding figures of the periodic wave solutions are given. Fhrthermore, it is shown that the known soliton solutions can be reduced from the periodic wave solutions.     

Bilinear Bcklund transformation and explicit solutions for a nonlinear evolution equation

The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The B\"{a}cklund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.     

Pfaffian-Type Soliton Solution to a Multi-Component Coupled Ito Equation

Multi-soliton solution to a multi-component coupled Ito system is investigated based on the Hirota bilinear method. By virtue of the perturbation method, we firstly derive one- and two-soliton solutions for the coupled Ito system possessing four components. Then the multi-soliton solution for the multi-component coupled Ito system is summarized into a general form expressed by pfaffians. Finally, this general pfaffian-type soliton solution is proved by pfaffian techniques.     

Bright and dark soliton solutions in growing Bose-Einstein condensates

This paper develops the Hirota method carefully for applying into the growing model of quasi-one-dimensional Bose—Einstein condensations with attractive and repulsive interaction, respectively. After a tedious calculation it obtains the exact bright and dark soliton solutions analytically. It shows that the growing model has the important effect on the soliton amplitude and the time-dependent potential only contributes to the phase and phase velocity. A detailed analysis for the asymptotic behaviour of two-soliton solutions shows that the collision of two soliton is elastic.     

Solitons in Bose-Einstein Condensates with Time-Dependent Atomic Scattering Length in a Harmonic Trap

We obtain the integrable relation for the one-dimensional nonlinear Schrödinger equations which describes the dynamics of a Bose-Einstein Condensates with time-dependent scattering length in a harmonic potential. The exact one- and two-soliton solutions are constructed analytically by using the Hirota method. Then we further discuss the dynamics of the one soliton and the interactions between two solitons in currently experimental conditions.     

Hirota method for the nonlinear Schrödinger equation with an arbitrary linear time-dependent potential

In this paper, a Hirota method is developed for applying to the nonlinear Schrödinger equation with an arbitrary time-dependent linear potential which denotes the dynamics of soliton solutions in quasi-one-dimensional Bose-Einstein condensation. The nonlinear Schrödinger equation is decoupled to two equations carefully. With a reasonable assumption the one- and two-soliton solutions are constructed analytically in the presence of an arbitrary time-dependent linear potential.     

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.     

Solitons for the cubic-quintic nonlinear Schrödinger equation with Raman effect in nonlinear optics

Considering the ultrashort optical soliton propagation in the non-Kerr media, the cubic-quintic nonlinear Schrödinger equation with Raman effect is studied through the dependent variable transformation and Hirota method. Based on symbolic computation, the bilinear form, the explicit one- and two-soliton solutions for the equation are presented. The constraint parametric condition for the existence of soliton solutions is also derived. Propagation characteristics and interaction behaviors of the solitons are graphically shown and discussed: (1) Overtaking elastic interactions of the two solitons; (2) periodic attraction and repulsion of the bounded states of two solitons; (3) propagation in parallel of the two solitons.     

Bright and dark solitons in the normal dispersion regime of inhomogeneous optical fibers: Soliton interaction and soliton control

Symbolically investigated in this paper is a nonlinear Schrödinger equation with the varying dispersion and nonlinearity for the propagation of optical pulses in the normal dispersion regime of inhomogeneous optical fibers. With the aid of the Hirota method, analytic one- and two-soliton solutions are obtained. Relevant properties of physical and optical interest are illustrated. Different from the previous results, both the bright and dark solitons are hereby derived in the normal dispersion regime of the inhomogeneous optical fibers. Moreover, different dispersion profiles of the dispersion-decreasing fibers can be used to realize the soliton control. Finally, soliton interaction is discussed with the soliton control confirmed to have no influence on the interaction. The results might be of certain value for the study of the signal generator and soliton control.     

Higher-Dimensional KdV Equations and Their Soliton Solutions

A (2+1)-dimensional KdV equation is obtained by use of Hirota method, which possesses N-soliton solution, specially its exact two-soliton solution is presented. By employing a proper algebraic transformation and the Riccati equation, a type of bell-shape soliton solutions are produced via regarding the variable in the Riccati equation as the independent variable. Finally, we extend the above (2+1)-dimensional KdV equation into (3+1)-dimensional equation, the two-soliton solutions are given.     

Excitations of nonlinear local waves described by the sinh-Gordon equation with a variable coefficient

We explore novel excitations in the form of nonlinear local waves, which are described by the sinh-Gordon (SHG) equation with a variable coefficient. With the aid of the self-similarity transformation, we establish the relationship between solutions of the SHG equation with a variable coefficient and those of the standard SHG equation. Then, using the Hirota bilinear method, we obtain a more general bilinear form for the standard SHG equation and find new one- and two-soliton waves whose forms involve two arbitrary self-similarity functions. By an appropriate choice of the smooth self-similarity functions, we determine and display novel localized waves, and discuss their properties. The method used here can be extended to the three- and higher order soliton solutions.     

Some types of dark soliton interactions in inhomogeneous optical fibers

Dark solitons are the subject of intense theoretical and experimental studies in nonlinear optics due to their unique characteristics compared with bright solitons. In this paper, the variable coefficient high-order nonlinear Schrödinger equation in the inhomogeneous optical fiber is investigated. Via the Hirota bilinear method and symbolic computation, the analytic dark two-soliton solutions are obtained. With the suitable choices of functions and coefficients for the obtained dark two-soliton solutions, some new phenomena are presented for the first time. The influences on phases and amplitudes of soliton interactions are detailed analyzed. Moreover, sets of double-triangle structures and methods of changing the propagation direction of dark solitons are introduced. Finally, by choosing suitable functions of the fourth-order dispersion parameter, the arch-structure and M-structure interactions are revealed. Results may be potentially useful in designing all-optical switches and optical fibers.     

Hirota method for oblique solitons in two-dimensional supersonic nonlinear Schrödinger flow

In a previous work El et al. (2006) [1] exact stable oblique soliton solutions were revealed in two-dimensional nonlinear Schrödinger flow. In this work we show that single soliton solution can be expressed within the Hirota bilinear formalism. An attempt to build two-soliton solutions shows that the system is “close” to integrability provided that the angle between the solitons is small and/or we are in the hypersonic limit.