N-Soliton Solution in Wronskian Form for a Generalized Variable-Coefficient Korteweg--de Vries Equation

We concentrate on finding exact solutions for a generalized variable-coefficient Korteweg-de Vries equation of physically significance. The analytic N-soliton solution in Wronskian form for such a model is postulated and verified by direct substituting the solution into the bilinear form by virtue of the Wronskian technique. Additionally, the bilinear auto-Backlund transformation between the （ N - 1）- and N-soliton solutions is verified.     

N-Soliton Solution of a Generalized Hirota-Satsuma Coupled KdV Equation and Its Reduction

Based on the Hirota method and the perturbation technique, the N-soliton solution of a generalized Hirota-Satsuma coupled KdV equation is obtained. Further, the N-soliton solution of a complex coupled KdV equation is given by reducing.     

N-soliton solution of a coupled integrable dispersionless equation

<正>A new coupled integrable dispersionless equation is presented by considering a spectral problem.A Darboux transformation for the resulting coupled integrable dispersionless equation is constructed with the help of spectral problems.As an application,the N-soliton solution of the coupled integrable dispersionless equation is explicitly given.     

Soliton Collision in a Ferromagnetic Spin Chain Driven by a Magnetic Field

By employing an inverse-scattering transformation, the exact solution of an N-soliton train in the spin chain with a time-dependent magnetic field is derived. As a special case the one-soliton solution (N = 1) exhibits explicitly the spin precession around the magnetic field and periodic shape-variation induced by the time varying field as well. In terms of the general solution, inelastic two-soliton collision is analysed. The inelastic collision, by which we mean the soliton shape-change before and after collision, appears generally due to the time varying field.     

N-soliton solutions for the nonlocal two-wave interaction system via the Riemann–Hilbert method

In this paper, a nonlocal two-wave interaction system from the Manakov hierarchy is investigated via the Riemann–Hilbert approach. Based on the spectral analysis of the Lax pair, a Riemann–Hilbert problem for the nonlocal two-wave interaction system is constructed. By discussing the solutions of this Riemann–Hilbert problem in both the regular and nonregular cases, we explicitly present the N-soliton solution formula of the nonlocal two-wave interaction system. Moreover,the dynamical behaviour of the single-soliton solution is shown graphically.     

Bcklund Transformations and Solutions of a Generalized Kadomtsev-Petviashvili Equation

In this paper,the bilinear form of a generalized Kadomtsev-Petviashvili equation is obtained by applying the binary Bell polynomials.The N-soliton solution and one periodic wave solution are presented by use of the Hirota direct method and the Riemann theta function,respectively.And then the asymptotic analysis demonstrates one periodic wave solution can be reduced to one soliton solution.In the end,the bilinear Bcklund transformations are derived.     

Wronskian Form Solutions for a Variable Coefficient Kadomtsev–Petviashvili Equation

Starting from a simple transformation, and with the aid of symbolic computation, we establish the relation- ship between the solution of a generalized variable coefficient Kadomtsev-Petviashvili （vKP） equation and the solution of a system of linear partial differential equations. According to this relation, we obtain Wronskian form solutions of the vKP equation, and further present N-soliton-like solutions for some degenerated forms of the vKP equation. Moreover, we also discuss the influences of arbitrary constants on the soliton and N-soliton solutions of the KPII equation.     

Wronskian Form of N-Soliton Solution for the （2＋1）-Dimensional Breaking Soliton Equation

The Wronskian form of N-soliton solution for the （2-kl）-dimensional breaking soliton equation is obtained by resorting to the Hirota direct method.     

Explicit solutions of Boussinesq--Burgers equation

Darboux transformation with multi-parameters for the Boussinesq--Burgers (B--B) equation is derived. For an application, some important explicit solutions of the B--B equation are obtained, including 2N-soliton solution and periodic solution. Finally, some elegant and interesting figures are plotted.     

Grammian Determinant Solution and Pfaffianization for a （3＋1）-Dimensional Soliton Equation

Based on the Pfaffian derivative formulae, a Grammian determinant solution for a （3＋1）-dimensional soliton equation is obtained. Moreover, the Pfaffianization procedure is applied for the equation to generate a new coupled system. At last, a Gram-type Pfaffian solution to the new coupled system is given.     

QED corrections and chemical properties of Eka-Hg

In this paper, we present a family of coupled higher-order nonlinear Schr?dinger 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.     

N-Soliton Solutions of （2＋1）-Dimensional Non-isospectral AKNS System

The bilinear form of the （2＋1）-dimensional non-isospectral AKNS system is derived. Its N-soliton solutions are obtained by using the Hirota method. As a reduction, a （2＋1）-dimensional non-isospectral Schrodinger equation and its N-soliton solutions are constructed.     

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.     

Controlled Manipulation with a Bose--Einstein Condensates N-Soliton Train under the Influence of Harmonic and Tilted Periodic Potentials

A model of the perturbed complex Toda chain （PCTC） to describe the dynamics of a Bose-Einstein condensate （BEC） N-soliton train trapped in an applied combined external potential consisting of both a weak harmonic and tilted periodic component is first developed. Using the developed theory, the BEC N-soliton train dynamics is shown to be well approximated by 4N coupled nonlinear differential equations, which describe the fundamental interactions in the system arising from the interplay of amplitude, velocity, centre-of-mass position, and phase. The simplified analytic theory allows for an efficient and convenient method for characterizing the BEC N-soliton train behaviour. It further gives the critical values of the strength of the potential for which one or more localized states can be extracted from a soliton train and demonstrates that the BEC N-soliton train can move selectively from one lattice site to another by simply manipulating the strength of the potential.     

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.