A hybrid type of soliton equations with self-consistent sources: KP and Toda cases

Source generation procedure is applied to construct a hybrid type of soliton equations with self-consistent sources (SESCSs). The examples include the KP equation with self-consistent sources (KPESCS) and two-dimensional TodaESCS. One typical feature for this hybrid type of SESCSs is that soliton solutions of these new systems contain arbitrary functions of a linear combination of two independent variables, which is different from the normal SESCSs where soliton solutions only contain arbitrary functions of one independent variable. What's more, the obtained two hybrid SESCSs can be reduced to two different simpler SESCSs respectively.     

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.     

Some types of solutions and generalized binary Darboux transformation for the mKP equation with self-consistent sources

In this paper, an mKP equation with self-consistent sources (mKPESCSs) is structured in the framework of the constrained mKP equation. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation and the N-times repeated Darboux transformation with arbitrary functions at time t for the mKPESCSs which offers a non-auto-Bäcklund transformation between two mKPESCSs with different degrees of sources. With the help of these transformations, some new solutions for the mKPESCSs such as soliton solutions, rational solutions, breather type solutions and exponential solutions are found by taking the special initial solution for auxiliary linear problems and the special functions of t-time.     

The non-isospectral modified Kadomtsev-Petviashvili equation with self-consistent sources and its coupled system

A new algebraic method called ‘source generation procedure’ is applied to construct non-isospectral soliton equations with self-consistent sources. As results, the non-isospectral modified Kadomtsev-Petviashvili equation with self-consistent sources (mKPESCS) and its Gramm-type determinant solutions are obtained by using the source generation procedure. Furthermore, a new coupled system of the non-isospectral mKPESCS and its Pfaffian solutions are constructed.     

Positon Solutions of the KdV Equation with Self-Consistent Sources

We construct a binary Darboux transformation with an arbitrary time function for the KdV equation with self-consistent sources. With this transformation, we obtain positon solutions of the KdV equation with self-consistent sources. We also discuss the properties of these solutions.     

Lie symmetries of soliton equations with self-consistent sources via source generation procedure

In this paper, the ‘source generation’ procedure (SGP) proposed by Hu and Wang [X.B. Hu, H.Y. Wang, Construction of dKP and BKP equation with self-consistent sources, Inverse Problems 22 (2006) 1903-1920] is utilized to derive Lie symmetries of bilinear soliton equations with self-consistent sources (SESCS) such as KPESCS, BKPESCS, and differential-difference KPESCS. Furthermore, it is shown that these Lie symmetries constitute generators of the corresponding Lie symmetry algebras.     

The exact solutions to a Ragnisco-Tu hierarchy with self-consistent sources

The Ragnisco-Tu hierarchy with self-consistent sources is derived. The exact solutions of the hierarchy are obtained via the inverse scattering transform (IST). An explicit form for a solution of the Ragnisco-Tu equation is presented.     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.     

Integrability of the semi-discrete Toda equation with self-consistent sources

A new procedure to construct and solve soliton equations with self-consistent sources (SESCSs) is applied to the semi-discrete Toda equation, based on its bilinear from. Bilinear Bäcklund transformation (BT) for the semi-discrete Toda ESCS is presented. Starting from the BT, a Lax pair is derived for the semi-discrete Toda ESCS.     

On the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources

A new procedure called ‘source generation’ is applied to construct non-isospectral soliton equations with self-consistent sources. As results, the non-isospectral Kadomtsev–Petviashvili equation with self-consistent sources (KPESCS) and its Gramm-type determinant solutions are obtained. Furthermore, the non-isospectral Pfaffianized-KP equation with self-consistent sources is constructed. This coupled system can not only be reduced to the non-isospectral Pfaffianized-KP equation, but also reduced to the non-isospectral KPESCS.     

Complexiton solutions of the Korteweg–de Vries equation with self-consistent sources

Complexiton solutions to the Korteweg–de Vires equation with self-consistent sources are presented. The basic technique adopted is the Darboux transformation. The resulting solutions provide evidence that soliton equations with self-consistent sources can have complexiton solutions, in addition to soliton, positon and negaton solutions. This also implies that soliton equations with self-consistent sources possess some kind of analytical characteristics that linear differential equations possess and brings ideas toward classification of exact explicit solutions of nonlinear integrable differential equations.     

Discrete three-dimensional three wave interaction equation with self-consistent sources

A discrete three-dimensional three wave interaction equation with self-consistent sources is constructed using the source generation procedure. The algebraic structure of the resulting fully discrete system is clarified by presenting its discrete Gram-type determinant solution. It is shown that the discrete three-dimensional three wave interaction equation with self-consistent sources has a continuum limit into the three-dimensional three wave interaction equation with self-consistent sources.     

超经典Boussinesq系统的守恒律和自相容源

本文研究了超经典Boussinesq系统.利用已有的超经典Boussinesq方程族及其超哈密顿结构,构造了带自相容源的超经典Boussinesq方程族,并通过引入变量F和G,获得了超经典Boussinesq方程族的守恒律.     

A simple but efficient approach for studying on nonlinear differential-difference equations

In this paper, we present a new approach for constructing exact solutions to nonlinear differential-difference equations (NLDDEs). By applying the new method, we have studied the saturable discrete nonlinear Schrodinger equation (SDNLSE) and obtained a number of new exact localized solutions, including discrete bright soliton solution, dark soliton solution, bright and dark soliton solution, alternating phase bright soliton solution, alternating phase dark soliton solution and alternating phase bright and dark soliton solution, provided that a special relation is bound on the coefficients of the equation among the solutions obtained.     

The negative extended KdV equation with self-consistent sources: soliton, positon and negaton

The negative extended KdV equation with self-consistent sources (eKdV⁻ESCSs) is firstly presented and the associated linear auxiliary equations are derived. The generalized binary Darboux transformation (DT) is applied to construct some new solutions of the eKdV⁻ESCSs such as singular N-soliton solution, N-soliton solution with finite amplitude, N-positon solution and N-negaton solution. The properties of these solutions are analyzed. Moreover, the interactions of two solitons, positon and negaton, positon and soliton, and two positons are discussed.     

Some new soliton equations with self-consistent sources

By using the Hirota bilinear method and some direct variable separation assumptions, a new method is proposed to derive soliton equations with self-consistent sources and their nonlinear variable separation solutions.     

Broer-Kaup-Kupershmidt族的非线性双可积耦合及其自相容源

基于新的非半单矩阵李代数,介绍了构造孤子族非线性双可积耦合的方法,由相应的变分恒等式给出了孤子族非线性双可积耦合的Hamilton结构.作为应用,给出了Broer-Kaup-Kupershmidt族的非线性双可积耦合及其Hamilton结构.最后指出了文献中的一些错误,利用源生成理论建立了新的公式,并导出了带自相容源Broer-Kaup-Kupershmidt族的非线性双可积耦合方程.     

New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations

In this paper, we consider the Benjamin Bona Mahony equation (BBM), and we obtain new exact solutions for it by using a generalization of the well-known tanh-coth method. New periodic and soliton solutions for the Generalized BBM and Burgers-BBM equations are formally derived.     

A new extended matrix KP hierarchy and its solutions

Starting from the matrix KP hierarchy and adding a new τ_B flow, we obtain a new extended matrix KP hierarchy and its Lax representation with the symmetry constraint on squared eigenfunctions taken into account. The new hierarchy contains two sets of times t_A and τ_B and also eigenfunctions and adjoint eigenfunctions as components. We propose a generalized dressing method for solving the extended matrix KP hierarchy and present some solutions. We study the soliton solutions of two types of (2+1)-dimensional AKNS equations with self-consistent sources and two types of Davey-Stewartson equations with selfconsistent sources.