Solving Non-Isospectral mKdV Equation and Sine-Gordon Equation Hierarchies with Self-Consistent Sources via Inverse Scattering Transform

N-soliton solutions of the hierarchy of non-isospectral mKdV equation with self-consistent sources and the hierarchy of non-isospectral sine-Gordon equation with self-consistent sources are obtained via the inverse scattering transform.     

KdV Equation with Self-consistent Sources in Non-uniform Media

Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.     

Non-isospectral integrable couplings of Ablowitz-Kaup-Newell-Segur （AKNS） hierarchy with self-consistent sources

A hierarchy of non-isospectral Ablowitz-Kaup-Newell-Segur （AKNS） equations with self-consistent sources is derived. As a general reduction case, a hierarchy of non-isospectral nonlinear SchrSdinger equations （NLSE） with selfconsistent sources is obtained. Moreover, a new non-isospectral integrable coupling of the AKNS soliton hierarchy with self-consistent sources is constructed by using the Kronecker product.     

Exact Solutions of Non-isospectral sine-Gordon Equation with Self-consistent Sources

The non-isospectral sine-Gordon equation with self-consistent sources is derived. Its solutions are obtained by means of Hirota method and Wronskian technique, respectively. Non-isospectral dynamics including one-soliton characteristics, two-soliton scattering, and ghost solitons, are investigated.     

The Qiao--Liu Equation with Self-Consistent Sources and Its Solutions

The Qiao--Liu equation with self-consistent sources (QLESCS) and its Lax representation are derived. A reciprocal transformation for the QLESCS is given. By making use of the reciprocal transformation and the solutions of the mKdV equation with self-consistent sources (mKdVSCS), the solutions of the QLESCS are presented.     

Two New Multi-component BKP Hierarchies

We firstly propose two kinds of new multi-component BKP （mcBKP） hierarchy based on the eigenfunction symmetry reduction and nonstandard reduction, respectively. The first one contains two types of BKP equation with self-consistent sources whose Lax representations are presented. The two mcBKP hierarchies both admit reductions to the k-constrained BKP hierarchy and to integrable （1＋1）-dimensional hierarchy with self-consistent sources, which include two types of SK equation with self-consistent sources and of hi-directional SK equations with self-consistent     

Exact Solutions of Ablowitz-Ladik Hierarchy with Self-Consistent Sources Revisited and Reductions

In the paper, Ablowitz-Ladik hierarchy with new self-consistent sources is investigated. First the source in the hierarchy is described as φ_nφ_n+1, where φ_n is related to the Ablowitz-Ladik spectral problem, instead of the corresponding adjoint spectral problem. Then by means of the inverse scattering transform, the multi-soliton solutions for the hierarchy are obtained. Two reductions to the discrete mKdV and nonlinear Schrödinger hierarchies with self-consistent sources are considered by using the uniqueness of the Jost functions, as well as their N-soliton solutions.     

The multisoliton solutions for the nonisospectral mKPI equation with self-consistent sources

The nonisospectral mKPI equation with self-consistent sources is derived through the linear problem of the nonisospectral mKPI system. The bilinear form of the nonisospectral mKPI equation with self-consistent sources is given and the N-soliton solutions are obtained through Hirota method and Wronskian technique respectively.     

Rosochatius Deformed Soliton Hierarchy with Self-Consistent Sources

Integrable Rosochatius deformations of finite-dimensional integrable systems are generalized to the soliton hierarchy with self-consistent sources. The integrable Rosochatius deformations of the Kaup-Newell hierarchy with self-consistent sources, of the TD hierarchy with self-consistent sources, and of the Jaulent-Miodek hierarchy with self-consistent sources, together with their Lax representations are presented.     

On Camassa-Holm Equation with Self-Consistent Sources and Its Solutions

Regarded as the integrable generalization of Camassa-Holm （CH） equation, the CH equation with self-consistent sources （CHESCS） is derived. The Lax representation of the CHESCS is presented. The conservation laws for CHESCS are constructed. The peakon solution, N-soliton, N-cuspon, N-positon, and N-negaton solutions of CHESCS are obtained by using Darboux transformation and the method of variation of constants.     

On （2＋1）-Dimensional Non-isospectral Toda Lattice Hierarchy and Integrable Coupling System

By considering （2＋1）-dimensional non-isospectral discrete zero curvature equation, the （2＋1）-dimensional non-isospectral Toda lattice hierarchy is constructed in this article. It follows that some reductions of the （2＋1）- dimensional Toda lattice hierarchy are given. Finally, the （2＋1）-dimensional integrable coupling system of the Toda lattice hierarchy is obtained through enlarging spectral problem.     

Integral-Type Darboux Transformations for the mKdV Hierarchy with Self-Consistent Sources

We present integral-type Darboux transformation for the mKdV hierarchy and for the mKdV hierarchy withself-consistent sources. In contrast with the normal Darboux transformation, the integral-type Darboux transformationscan offer non-auto-Backlund transformation between two (2n 1)-th mKdV equations with self-consistent sources withdifferent degrees. This kind of Darboux transformation enables us to construct the N-soliton solution for the mKdVhierarchy with self-consistent sources. We also propose the formulas for the m times repeated integral-type Darbouxtransformations for both mKdV hierarchy and mKdV hierarchy with self-consistent sources.     

THE TRANSFORMATION BETWEEN THE AKNS HIERARCHY AND THE KN HIERARCHY WITH SELF-CONSISTENT SOURCES

It is shown that the AKNS hierarchy with self-consistent sources can transform to KN hierarchy with self-consistent sources through a transformation operator and gauge transformation. Besides, there exists transformation in their conservation laws and Hamiltonian structures.     

New Rogue Wave Solutions of (1+2)-Dimensional Non-Isospectral KP-II Equation

The generalized binary Darboux transformation for the (1+2)-dimensional non-isospectral KP-II equation is presented. Moreover, as a direct application, the new rogue wave solutions for the (1+2)-dimensional non-isospectral KP-II equation are constructed by the generalized binary Darboux transformation.     

A New Integrable Couplings of Classical-Boussinesq Hierarchy withSelf-Consistent Sources

A kind of integrable couplings of soliton equations hierarchy with self-consistent sources associated with \tilde{sl}(4) is presented by Yu. Based on this method, we construct a new integrable couplings of the classical-Boussinesq hierarchy with self-consistent sources by using of loop algebra \tilde{sl}(4). In this paper, we also point out that there exist some errors in Yu's paper and have corrected these errors and set up new formula. The method can be generalized other soliton hierarchy with self-consistent sources.     

Scattering of Solitons of Modified KdV Equation with Self-consistent Sources

This paper investigates in detail the dynamics of the modified KdV equation with self-consistent sources, including characteristics of one-soliton, scattering conditions and phase shifts of two solitons, degenerate case of two solitons and ＂ghost＂ solitons, etc. Co-moving coordinate frames are employed in asymptotic analysis.     

New extension of dispersionless Harry Dym hierarchy

The symmetry constraint for dispersionless Harry Dym (dHD) hierarchy is derived for the first time by taking dispersionless limit of that for 2+1 dimensional Harry Dym hierarchy. Then, the dHD is extended by means of the symmetry constraint which we derived. From the zero-curvature equation of the new extended dHD hierarchy, two types of dHD equations with self-consistent sources (dHDESCS) together with their associated conservation equations are obtained. Moreover, the hodograph solutions to the first type of dHDESCS are given. Finally, Bäcklund transformation between the extended dispersionless mKP hierarchy and extended dHD Hierarchy are also constructed.     

Rosochatius Deformed Soliton Hierarchy with Self-Consistent Sources

Integrable Rosochatius deformations of finite-dimensional integrable systems are generalized to the soliton hierarchy with self-consistent sources. The integrable Rosochatius deformations of the Kaup-Newell hierarchy with self-consistent sources, of the TD hierarchy with self-consistent sources, and of the Jaulent-Miodek hierarchy with self- consistent sources, together with their Lax representations are presented.     

A Limit Symmetry of Modified KdV Equation and Its Applications

In this letter we consider a limit symmetry of the modified KdV equation and its application. The similarity reduction leads to limit solutions of the modified KdV equation. Besides, a modified KdV equation with new self-consistent sources is obtained and its solutions are derived.     

A Direct Method of Hamiltonian Structure

A direct method of constructing the Hamiltonian structure of the soliton hierarchy with self-consistent sources is proposed through computing the functional derivative under some constraints. The Hamiltonian functional is related with the conservation densities of the corresponding hierarchy. Three examples and their two reductions are given.