New Multi-soliton Solutions for the （2＋1）-Dimensional Kadomtsev-Petviashvili Equation

In this paper, an explicit N-fold Darboux transformation with multi-parameters for both a （1＋1）- dimensional Broer-Kaup （BK） equation and a （1＋1）-dimensional high-order Broer-Kaup equation is constructed with the help of a gauge transformation of their spectral problems. By using the Darboux transformation and new basic solutions of the spectral problems, 2N-soliton solutions of the BK equation, the high-order BK equation, and the Kadomtsev-Petviashvili （KP） equation are obtained.     

Painleve Analysis and Darboux Transformation for a Variable-Coefficient Boussinesq System in Fluid Dynamics with Symbolic Computation

The new soliton solutions for the variable-coefficient Boussinesq system, whose applications are seen in fluid dynamics, are studied in this paper with symbolic computation. First, the Painleve analysis is used to investigate its integrability properties. For the identified case we give, the Lax pair of the system is found, and then the Darboux transformation is constructed. At last, some new soliton solutions are presented via the Darboux method. Those solutions might be of some value in fluid dynamics.     

Darboux Transformation and Soliton Solutions for a Variable-Coefficient Modified Kortweg-de Vries Model from Fluid Mechanics,Ocean Dynamics,and Plasma Mechanics

This paper is to investigate a variable-coefficient modified Kortweg-de Vries （vc-mKdV） model, which describes some situations from fluid mechanics, ocean dynamics, and plasma mechanics. By the AblowRz-Kaup-NewellSegur procedure and symbolic computation, the Lax pair of the vc-MKdV model is derived. Then, based on the aforementioned Lax pair, the Darboux transformation is constructed and a new one-soliton-like solution is obtained as weft Features of the one-soliton-like solution are analyzed and graphically discussed to illustrate the influence of the variable coefficients in the solitonlike propagation.     

Broer-Kaup系统的达布变换和新的精确解

利用Broer-Kaup系统Lax对的两个基本解，构造了该系统一个新的达布变换，并利用该变换求出了Broer-Kaup系统一组新的双向孤子解. 关键词： Lax对 达布变换 孤子解     

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrfdinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrodinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrfdinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrfdinger equation is given.``     

A New Negative Discrete Hierarchy and Its N-Fold Darboux Transformation

Starting from a matrix discrete spectral problem, we derive a negative discrete hierarchy. It is shown that the hierarchy is integrable in the Liouville sense and possesses a bi-Hamiltonian structure. Furthermore, its N-fold Darboux transformation is established with the help of gauge transformation of Lax pair. As an application of the Darboux transformation, some new exact solutions for a discrete equation in the negative hierarchy are obtained.     

A Unified and Explicit Construction of N-Soliton Solutions for the Nonlinear Schrödinger Equation

An explicit N-fold Darboux transformation with multiparameters for nonlinear Schrödinger equation is constructed with the help of its Lax pairs and a reduction technique. According to this Darboux transformation, the solutions of the nonlinear Schrödinger equation are reduced to solving a linear algebraic system, from which a unified and explicit formulation of N-soliton solutions with multiparameters for the nonlinear Schrödinger equation is given.     

Degenerate Solutions of the Nonlinear Self-Dual Network Equation

The N-fold Darboux transformation(DT) T_n~([N]) of the nonlinear self-dual network equation is given in terms of the determinant representation. The elements in determinants are composed of the eigenvalues λ_j(j = 1, 2..., N)and the corresponding eigenfunctions of the associated Lax equation. Using this representation, the N-soliton solutions of the nonlinear self-dual network equation are given from the zero "seed" solution by the N-fold DT. A general form of the N-degenerate soliton is constructed from the determinants of N-soliton by a special limit λ_j →λ_1 and by using the higher-order Taylor expansion. For 2-degenerate and 3-degenerate solitons, approximate orbits are given analytically,which provide excellent fit of exact trajectories. These orbits have a time-dependent "phase shift", namely ln(t~2).     

Lump and lump-soliton interaction solutions for an integrable variable coefficient Kadomtsev-Petviashvili equation

For a variable coefficient Kadomtsev-Petviashvili (KP) equation the Lax pair as well as conjugate Lax pair are derived from the Painleve analysis.The N-fold binary Darboux transformation is presented in a compact form.As an application,the multi-lump,higher-order lump and general lump-soliton interaction solutions for the variable coefficient KP equation are obtained.Typical lump structures with amplitudes exponentially decaying to zero as the time tends to infinity and interactions between one lump and one soliton are shown.     

Hamiltonian Systems and Darboux Transformation Associated with a 3 × 3 Matrix Spectral Problem

Starting from a 3 × 3 matrix spectral problem, we derive a hierarchy of nonlinear equations. It is shown that the hierarchy possesses bi-Hamiltonian structure. Under the symmetry constraints between the potentials and the eigenfunctions, Lax pair and adjoint Lax pairs including partial part and temporal part are nonlinearied into two finitedimensional Hamiltonian systems (FDHS) in Liouville sense. Moreover, an explicit N-fold Darboux transformation for CDNS equation is constructed with the help of a gauge transformation of the spectral problem.     

New Explicit Solutions of （1 ＋ 1）-Dimensional Variable-Coefficient Broer-Kaup System

By using the compatibility method, many explicit solutions of the （1＋1）-dimensional variable-coefficient Broer Kaup system are constructed, which include new solutions expressed by error function, Bessel function, exponential function, and Airy function. Some figures of the solutions are given by the symbolic computation system Maple.     

N-fold Darboux Transformation for Integrable Couplings of AKNS Equations

For the integrable couplings of Ablowitz-Kaup-Newell-Segur(ICAKNS) equations, N-fold Darboux transformation(DT) TN, which is a 4 × 4 matrix, is constructed in this paper. Each element of this matrix is expressed by a ratio of the(4N + 1)-order determinant and 4N-order determinant of eigenfunctions. By making use of these formulae,the determinant expressions of N-transformed new solutions p~([N ]), q~([N ]), r~([N ])and s~([N ])are generated by this N-fold DT.Furthermore, when the reduced conditions q =-p*and s =-r*are chosen, we obtain determinant representations of N-fold DT and N-transformed solutions for the integrable couplings of nonlinear Schr?dinger(ICNLS) equations.Starting from the zero seed solutions, one-soliton solutions are explicitly given as an example.     

Broer-Kaup Systems from Darboux Transformation Related Symmetry Constraints of Kadomtsev-Petviashvili Equation

Using the Darboux transformation related symmetry constraints of the Kadomtsev-Petviashvili (KP) equation, we get some integrable (1+1)-dimensional and (2+1)-dimensional multi-component Broer-Kaup (BK) systems.     

Darboux Transformation and Grammian Solutions for Nonisospectral Modified Kadomtsev-Petviashvili Equation with Symbolic Computation

In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painleve branches through the Painleve analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.     

Explicit Solutions for a New （2＋1）-Dimensional Coupled mKdV Equation

An integrable （2＋1）-dimensional coupled mKdV equation is decomposed into two （1 ＋1）-dimensional soliton systems, which is produced from the compatible condition of three spectral problems. With the help of decomposition and the Darboux transformation of two （1＋1）-dimensional soliton systems, some interesting explicit solutions of these soliton equations are obtained.     

Soliton interactions and asymptotic state analysis in a discrete nonlocal nonlinear self-dual network equation of reverse-space type

We propose a reverse-space nonlocal nonlinear self-dual network equation under special symmetry reduction,which may have potential applications in electric circuits.Nonlocal infinitely many conservation laws are constructed based on its Lax pair.Nonlocal discrete generalized(m,N?m)-fold Darboux transformation is extended and applied to solve this system.As an application of the method,we obtain multi-soliton solutions in zero seed background via the nonlocal discrete N-fold Darboux transformation and rational solutions from nonzero-seed background via the nonlocal discrete generalized(1,N?1)-fold Darboux transformation,respectively.By using the asymptotic and graphic analysis,structures of one-,two-,three-and four-soliton solutions are shown and discussed graphically.We find that single component field in this nonlocal system displays unstable soliton structure whereas the combined potential terms exhibit stable soliton structures.It is shown that the soliton structures are quite different between discrete local and nonlocal systems.Results given in this paper may be helpful for understanding the electrical signals propagation.     

Darboux Transformation and Soliton Solutions for Inhomogeneous Coupled Nonlinear Schrodinger Equations with Symbolic Computation

With the aid of computation, we consider the variable-coefficient coupled nonlinear Schrodinger equations with the effects of group-velocity dispersion, self-phase modulation and cross-phase modulation, which have potential applications in the long-distance communication of two-pulse propagation in inhomogeneous optical fibers. Based on the obtained nonisospectral linear eigenvalue problems （i.e. Lax pair）, we construct the Darboux transformation for such a model to derive the optical soliton solutions. In addition, through the one- and two-soliton-like solutions, we graphically discuss the features of picosecond solitons in inhomogeneous optical fibers.     

Darboux Transformations and Soliton Solutions for Classical Boussinesq-Burgers Equation

Two basic Darboux transformations of a spectral problem associated with a classical Boussinesq-Burgers equation are presented in this letter. They are used to generate new solutions of the classical Boussinesq-Burgers equation.     

LAX PAIR,BINARY DARBOUX TRANSFORMATION AND NEW GRAMMIAN SOLUTIONS OF NONISOSPECTRAL KADOMTSEV–PETVIASHVILI EQUATION WITH THE TWO-SINGULAR-MANIFOLD METHOD

In this letter, the two-singular-manifold method is applied to the (2+1)-dimensional nonisospectral Kadomtsev–Petviashvili equation with two Painlevé expansion branches to determine auto-Bäcklund transformation, Lax pairs and Darboux transformation. Based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the Nth iterated transformation formula in the form of Grammian is also presented. By using these Darboux transformations, we obtain some new grammian solutions.     

Darboux Transformation and New Soliton-like Solutions for (1+1)-Dimensional Higher-Order Broer-Kaup System

In this letter, we construct a kind of new Darboux transformation for the (1+1)-dimensional higher-order Broer-Kaup (HBK) system with the help of a gauge transformation of a spectral problem. By applying this new Darboux transformation, some new soliton-like solutions of the (1+1)-dimensional HBK system are obtained.