Multisoliton solutions of the (2+1)-dimensional Harry Dym equation

The exact N-soliton solutions of the (2+1)-dimensional Harry Dym equation are constructed analytically. Different types of two-soliton interactions are singled out in the general N-soliton solution. The existence of inelastic soliton interaction and two-soliton resonances are shown.     

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.     

Multi-hump bright solitons in a Schrödinger–mKdV system

We consider the problem of energy transport in a Davydov model along an anharmonic crystal medium obeying quartic longitudinal interactions corresponding to rigid interacting particles. The Zabusky and Kruskal unidirectional continuum limit of the original discrete equations reduces, in the long wave approximation, to a coupled system between the linear Schrödinger (LS) equation and the modified Korteweg–de Vries (mKdV) equation. Single- and two-hump bright soliton solutions for this LS–mKdV system are predicted to exist by variational means and numerically confirmed. The one-hump bright solitons are found to be the anharmonic supersonic analogue of the Davydov's solitons while the two-hump (in both components) bright solitons are found to be a novel type of soliton consisting of a two-soliton solution of mKdV trapped by the wave function associated to the LS equation. This two-hump soliton solution, as a two component solution, represents a new class of polaron solution to be contrasted with the two-soliton interaction phenomena from soliton theory, as revealed by a variational approach and direct numerical results for the two-soliton solution.     

一类非线性耦合系统的复合型双孤子新解

首先给出一种函数变换,把一类非线性耦合系统化为两个第一种椭圆方程组.然后利用第一种椭圆方程的新解与B?cklund变换,构造了一类非线性耦合系统的无穷序列复合型双孤子新解.     

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.     

Comment on Revision of Kaup-Newell＇s Works on IST for DNLS Equation

In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.     

Quantitative analysis of soliton interactions based on the exact solutions of the nonlinear Schrödinger equation

We make a quantitative study on the soliton interactions in the nonlinear Schrödinger equation (NLSE) and its variable-coefficient (vc) counterpart. For the regular two-soliton and double-pole solutions of the NLSE, we employ the asymptotic analysis method to obtain the expressions of asymptotic solitons, and analyze the interaction properties based on the soliton physical quantities (especially the soliton accelerations and interaction forces); whereas for the bounded two-soliton solution, we numerically calculate the soliton center positions and accelerations, and discuss the soliton interaction scenarios in three typical bounded cases. Via some variable transformations, we also obtain the inhomogeneous regular two-soliton and double-pole solutions for the vcNLSE with an integrable condition. Based on the expressions of asymptotic solitons, we quantitatively study the two-soliton interactions with some inhomogeneous dispersion profiles, particularly discuss the influence of the variable dispersion function f(t) on the soliton interaction dynamics.     

The Two-Soliton Bound-State Solution in an Order-Parameter-Preserving Ant iferromagnet

Employing the Dyson-Maleev transformation and the coherent state ansatz, the two-soliton bound-state solution in an order-parameter-preserving antiferromagnet is investigated by using the method of multiple scales and long-wavelength approximation.     

A certain critical two-soliton solution of the nonlinear Schroedinger equation

An exact two-soliton solution of the nonlinear Schroedinger equation is derived by using the Hirota direct approach. This solution describes such a critical process that two still solitons separated infinitely approach and then pass through each other and keep straight on infinitely.     

A Special Two-Soliton Solution of sine-Gordon Equation

A special two-soliton solution of sine-Gordon equation is obtained by using the Hirota direct method. It is shown in a mass-centre system how two kinks move and interact with each other.     

Soliton dynamics in the Wess-Zumino-Novikov-Witten model

Exact solutions for the classical Wess-Zumino-Novikov-Witten model are constructed by the matrix Darboux transformation method. The behavior of one- and two-soliton solutions is analyzed. The formula for an N-soliton solution is given. The corresponding linearized problem is considered.     

Pfaffian Solutions and Resonant Interaction Properties of a Coupled BKP Lattice

In this paper, we give a coupled lattice equation with the help of Hirota operators, which comes from a special BKP lattice. Two-soliton and three-soliton solutions to the coupled system are constructed. Furthermore,resonant interaction of the two-soliton solution is analyzed in detail. Under some special resonant condition, it is shown that low soliton can propagate faster than high one. Finally, the N-soliton solution is presented in the Pfaffian form.     

Construction of Soliton-Cnoidal Wave Interaction Solution for the (2+1)-Dimensional Breaking Soliton Equation

In this paper, the truncated Painlev′e analysis and the consistent tanh expansion(CTE) method are developed for the(2+1)-dimensional breaking soliton equation. As a result, the soliton-cnoidal wave interaction solution of the equation is explicitly given, which is difficult to be found by other traditional methods. When the value of the Jacobi elliptic function modulus m = 1, the soliton-cnoidal wave interaction solution reduces back to the two-soliton solution. The method can also be extended to other types of nonlinear evolution equations in mathematical physics.     

Pfaffian Solutions and Resonant Interaction Properties of a Coupled BKP Lattice

In this paper, we give a coupled lattice equation with the help of Hirota operators, which comes from a special BKP lattice. Two-soliton and three-soliton solutions to the coupled system are constructed. Furthermore, resonant interaction of the two-soliton solution is analyzed in detail. Under some special resonant condition, it is shown that low soliton can propagate faster than high one. Finally, the N-soliton solution is presented in the Pfaffian form.     

Solitons for a generalized variable-coefficient nonlinear Schrdinger equation

In this paper,we investigate some exact soliton solutions for a generalized variable-coefficients nonlinear Schrdinger 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.     

Nonparaxial effects on the propagation and scattering of a polarized optical pulse

Propagation characteristics of a polarized optical solitary pulse are analyzed by taking into account the effect of nonparaxiality and mutual interaction. To start with, a pair of generalized nonlinear Schrodinger equations is deduced through an operator approach. Stationary solutions of such a system are then analyzed numerically through a boundary value problem in two stages, with and without the nonparaxial effect. In the second stage, the propagating form of the corresponding spatial soliton is studied by an extended split step algorithm ETDRK. The initial profile is considered to be both a one- and two-soliton solution, to visualize the event of scattering and fusion. From this data, we have computed the intensity, root mean square spectral width, and chirp of a single soliton as it propagates. In the case of the two-soliton solution, we observe that for source parameter values, the fusion is more favored than scattering. It is observed that nonparaxiality and the interaction between A(x) and A(y) tends to destroy the periodic behaviors of these parameters. Lastly, we have investigated the modulational instability of the system as function of frequency detuning and nonparaxiality. The form of the gain is discussed as a function of nonparaxiality.     

Exact soliton solution of spin chain with an external magnetic field in linear wave background

Employing a simple, straightforward Darboux transformation we construct exact N-soliton solution for anisotropic spin chain driven by an external magnetic field in linear wave background. As a special case the explicit one- and two-soliton solution dressed by the linear wave corresponding to magnon in quantum theory is obtained analytically and its property is discussed in detail. The dispersion law, effective soliton mass, and the energy of each soliton are investigated as well. Our result show that the stability criterion of soliton is related with anisotropic parameter and the amplitude of the linear wave.     

Soliton dynamics of symmetry-endowed two-soliton solutions of the nonlinear Schrodinger equation

A comprehensive analysis is presented of the propagation of symmetry-endowed two-soliton solutions under the influence of various perturbations important in nonlinear optics. Thus, we begin by introducing the analytical expressions of these two-soliton solutions. Then, by considering perturbations which preserve the initial symmetry of the two-soliton solutions, the dependence of the soliton parameters on the propagation distance is determined by using an adiabatic perturbation method. As perturbations of this kind, important for soliton-based communication systems, we consider the bandwidth-limited amplification, nonlinear amplification, and amplitude and phase modulation. Moreover, the results obtained by the adiabatic perturbation method are compared with those obtained by direct numerical simulations of the corresponding governing differential equations. (c) 2000 American Institute of Physics.     

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.     

Traveling-wave-type gravitational soliton solutions

With the traveling-wave condition of Yan and Ge, the traveling-wave-type gravitational two-soliton solutions are generated from a flat metric by using the inverse scattering method (ISM) of Belinsky and Zakharov (BZ). It is shown that when the traveling-wave condition is added to the condition required by the BZ technique the exact general solutions of the vacuum gravitational field equations can be found by straightforward integration; in the general solutions there are three arbitrary functions. A special solution with solitonic character in the ordinary sense is also given.