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 hell-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.

Interaction Between Line Soliton and Algebraic Soliton for Asymmetric Nizhnik-Novikov-Veselov Equation

Starting from the variable separation approach, the algebraic soliton solution and the solution describing the interaction between line soliton and algebraic soliton are obtained by selecting appropriate seed solution for （2＋1）-dimensional ANNV equation. The behaviors of interactions are discussed in detail both analytically and graphically. It is shown that there are two kinds of singular interactions between line soliton and algebraic soliton： 1） the resonant interaction where the algebraic soliton propagates together with the line soliton and persists infinitely; 2） the extremely repulsive interaction where the algebraic soliton affects the motion of the line soliton infinitely apart.

Green＇s Function Method for Perturbed sine-Gordon Equation

The usual Green's function method is introduced to show the completeness of the squared Jost solutions of the multi-soliton case in this work. Since sine-Gordon equation contains second derivative in time, the known procedure with generalized Marchenko equation to show completeness relation of the eigenfunctions of linearized equation is unavailable. And the explicit expressions of Jost solutions are not necessary here. Thus a general method of direct perturbation method for the perturbed sine-Gordon equation is developed.

Oscillating Solitons for （2＋1）-Dimensional Nonlinear Models

Using extended homogeneous balance method and variable separation hypothesis, we found new variable separation solutions with three arbitrary functions of the （2＋1）-dimensional dispersive long-wave equations, Based on derived solutions, we revealed abundant oscillating solitons such as dromion, multi-dromion, solitoff, solitary waves, and so on, by selecting appropriate functions.

Soliton Solutions for a Negative Order AKNS Equation

In this short paper,bilinear form of a negative order AKNS equation is given.The N-soliton solutions areobtained through Hiorta's direct method.

Some Special Solutions of Three Nonlinear Lattices

Some soliton solutions and periodic solutions of hybrid lattice, discretized mKdV lattice, and modified Volterra lattice have been obtained by introducing a new method. This approach allows us to directly construct some explicit exact solutions for polynomial nonlinear differential-difference equations.

Restudy of Structures and Interactions of Solitons in （2＋1）-Dimensional Nizhnik-Novikov-Veselov Equations

Some new structures and interactions of solitons for the （2＋1）-dimensional Nizhnik-Novikov-Veselov equation are revealed with the help of the idea of the bilinear method and variable separation approach. The solutions to describe the interactions between two dromions, between a line soliton and a y-periodic soliton, and between two y-periodic solitons are included in our results. Detailed behaviors of interaction are illustrated both analytically and in graphically. Our analysis shows that the interaction properties between two solitons are related to the form of interaction constant. The form of interaction constant and the dispersion relationship are related to the form of the seed solution （u0, v0, w0 ） in Backlund transformation.

A Bilinear Bcklund Transformation and N-Soliton-Like Solution of Three Coupled Higher-Order Nonlinear Schrdinger 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.

Collisions of Two Spatial Solitons in Inhomogeneous Nonlinear Media

Collisions of spatial solitons occurring in the nonlinear Schroeinger equation with harmonic potential are studied, using conservation laws and the split-step Fourier method. We find an analytical solution for the separation distance between the spatial solitons in an inhomogeneous nonlinear medium when the light beam is self-trapped in the transverse dimension. In the self-focusing nonlinear media the spatial solitons can be transmitted stably, and the interaction between spatial solitons is enhanced due to the linear focusing effect （and also diminished for the linear defocusing effect）. In the self-defocusing nonlinear media, in the absence of self-trapping or in the presence of linear self-defocusing, no transmission of stable spatial solitons is possible. However, in such media the linear focusing effect can be exactly compensated, and the spatial solitons can propagate through.

An Elastic Interaction Model of Vortices

Soliton theory plays an important role in nonlinear physics.The elastic interaction among solitons is oneof the most important properties for integrable systems.In this Letter, an elastic vortex interaction model is proposed.It is found that the momenta, vortex momenta and the energies of every one vortex and the interaction energies of everytwo vortices are conserved.

A Generalized Method and Exact Solutions in Bose-Einstein Condensates in an Expulsive Parabolic Potential

In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the onedimensional nonlinear Schrfdinger equation which describes the dynamics of solitons in Bose-Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function＇s soliton solutions, but also Jacobi elliptic function solutions and Weierstra.ss elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient.

A Higher-Dimensional Hirota Condition and Its Judging Method

When a one-dimensional nonlinear evolution equation could be transformed into a bilinear differential form as F（Dt, Dx）f . f = O, Hirota proposed a condition for the above evolution equation to have arbitrary N-soliton solutions, we call it the 1-dimensional Hirota condition. As far as higher-dimensional nonlinear evolution equations go, a similar condition is established in this paper, also we call it a higher-dimensional Hirota condition, a corresponding judging theory is given. As its applications, a few two-dimensional KdV-type equations possessing arbitrary N-soliton solutions are obtained.

Soliton Solutions to Coupled Integrable Dispersionless Equations

In this paper, by introducing a new transformation, the bilinear form of the coupled integrable dispersionless （CID） equations is derived. It will be shown that this bilineax form is easier to perform the standard Hirota process. One-, two-, and three-soliton solutions are presented. Furthermore, the N-soliton solutions axe derived.

New Compacton Solutions of Nonlinearly Dispersive R（m, n） Equations

In this paper, we establish exact solutions for the .R（m,n） equations by using an sn-cn metnou,As a result, abundant new cornpactons, i,e, solitons with the absence of infinite wings, new type of Jacobi elliptic function, solitary wave and periodic wave solutions, of this equation are obtained with minimal calculations. The properties of the R（m, n） equations are shown in figures.

Properties of Controllable Soliton Switching in Optical Lattices with Longitudinal Exponential-Asymptotic Modulation

The properties of controllable soliton switching in Kerr-type optical lattices with different modulation are investigated theoretically and simulated numerically.The results show that the optical lattices can be available for all- optical soliton switching through utilization for length-scale competition effects.And through longitudinal exponential- asymptotic modulation for the linear refractive index,the properties of soliton switching in the optical lattices can be improved.The number of output channels of soliton switching can be controlled by the parameters such as incident angle,asymptotic rate of longitudinal modulation,guiding parameter and form factor.

Incoherently Coupled Anti-dark Soliton Families in Cubic-quintic Nonlinear Media

A theory is presented to predict that a novel type of incoherently coupled spatial soliton families, named anti-dark soliton families, can exist in cubic-quintie nonlinear media. The intensity profiles, associated phase distribution and propagation properties are discussed in detail.

Determinant Solutions to a (3+1)-Dimensional Generalized KP Equation with Variable Coefficients

1 Introduction Although partial differential equations that govern the motion of solitons are nonlinear, many of them can be put into the bilinear form. Hirota, in 1971, developed an ingenious method to obtain exact solutions to nonlinear partial differential equations in the soliton theory, such as the KdV equation, the Boussinesq equation and the KP equation （see [1-2]）.

Vortex and Gaussian Solitons of (3+1)-Dimensional Spatially Modulated Cubic-Quintic-Septimal Nonlinear Schrdinger Equation with External Potential

Based on the variable separation principle and the similarity transformation, vortex soliton solution of a （3＋1）-dimensional cubie-quintic-septimal nonlinear Schrodinger equation with spatially modulated nonlinearity under the external potential are obtained in the spatially modulated cubic-quintic-septimal nonlinear media. If the topological charge m = 0 and m ≠0, Gaussian solitons and vortex solitons can be constructed respectively. The shapes of vortex soliton possess similar structures when the value of l - m is same. Moreover, all phases of vortex solitons exist m-jump with the change of every jump as 2π/m-jumps, and thus totally realize the azimuthal change of 21r around their cores.

Painlev Property, Bcklund Transformations and Rouge Wave Solutions of (3+1)-Dimensional Burgers Equation

Burgers equation is the simplest one in soliton theory, which has been widely applied in almost all the physical branches. In this paper, we discuss the Painleve property of the （3＋1）-dimensional Burgers equation, and then Becklund transformation is derived according to the truncated expansion of the obtained Painleve analysis. Using the Backlund transformation, we find the rouge wave solutions to the equation via the multilinear variable separation approach. And we aiso give an exact solution obtained by general variable separation approach, which is proved to possess abundant structures.

A Study of the Behavioral Controllability for Superposed Kuznetsov–Ma Soliton of the (3+1)-Dimensional Generalized Nonlinear Schr?dinger Equation

Abstract The （3＋1 ）-dimensional variable-coetfficient nonlinear SchrSdinger equation with linear and parabolic traps is studied, and an exact Kuznetsov-Ma soliton solution in certain parameter conditions is derived. These precise expressions indicate that diffraction and chirp factors influence phase, center and widths, while the gain/loss parameter only affects peaks. By adjusting the relation between the maximum accumulated time Tm and the accumulated time To based on maximum amplitude of Kuznetsov Ma soliton, postpone, maintenance and restraint of superposed Kuznetsov-Ma solitons are investigated.