Runliang Lin

Integration of the soliton hierarchy with self-consistent sources

In contrast with the soliton equations, the evolution of the eigenfunctions in the Lax representation of soliton equation with self-consistent sources (SESCS) possesses singularity. We present a general method to treat the singularity to determine the evolution of scattering data. The AKNS hierarchy with self-consistent sources, the MKdV hierarchy with self-consistent sources, the nonlinear Schrodinger equation hierarchy with self-consistent sources, the Kaup–Newell hierarchy with self-consistent sources and the derivative nonlinear Schrodinger equation hierarchy with self-consistent sources are integrated directly by using the inverse scattering method. The N soliton solutions for some SESCS are presented. It is shown that the insertion of a source may cause the variation of the velocity of soliton. This approach can be applied to all other (1+1)-dimensional soliton hierarchies.

Solving the KdV hierarchy with self-consistent sources by inverse scattering method

The evolution of the eigenfunctions in the Lax representation of the KdV hierarchy with self-consistent sources possesses singularity. By proposing a method to treat the singularity to determine the evolution of scattering data, the KdV hierarchy with self-consistent sources is integrated by the inverse scattering method. The soliton solutions of these equations are obtained. It is shown that the insertion of a source may cause the variation of the speed of soliton. This approach can be applied to other (1+1)-dimensional soliton hierarchies.

A new extended q-deformed KP hierarchy

Abstract A method is proposed in this paper to construct a new extended q-deformed KP (q-KP) hiearchy and its Lax representation. This new extended q-KP hierarchy contains two types of q-deformed KP equation with self-consistent sources, and its two kinds of reductions give the q-deformed Gelfand-Dickey hierarchy with self-consistent sources and the constrained q-deformed KP hierarchy, which include two types of q-deformed KdV equation with sources and two types of q-deformed Boussinesq equation with sources. All of these results reduce to the classical ones when q goes to 1. This provides a general way to construct (2+1)- and (1+1)-dimensional q-deformed soliton equations with sources and their Lax representations.

The q-deformed mKP hierarchy with self-consistent sources, Wronskian solutions and solitons

Based on the eigenfunction symmetry constraint of the q-deformed modified KP hierarchy, a q-deformed mKP hierarchy with self-consistent sources (q-mKPHSCSs) is constructed. The q-mKPHSCSs contain two types of q-deformed mKP equation with self-consistent sources. By the combination of the dressing method and the method of variation of constants, a generalized dressing approach is proposed to solve the q-deformed KP hierarchy with self-consistent sources (q-KPHSCSs). Using the gauge transformation between the q-KPHSCSs and the q-mKPHSCSs, the q-deformed Wronskian solutions for the q-KPHSCSs and the q-mKPHSCSs are obtained. The one-soliton solutions for the q-deformed KP (mKP) equation with a source are given explicitly.

A Generalized Dressing Approach for Solving the Extended KP and the Extended mKP Hierarchy

A combination of dressing method and variation of constants as well as a formula for constructing the eigenfunction is used to solve the extended KP hierarchy, which is a hierarchy with one more series of time flow and based on the symmetry constraint of KP hierarchy. Similarly, extended mKP hierarchy is formulated and its zero-curvature form, Lax representation, and reductions are presented. Via gauge transformation, it is easy to transform dressing solutions of extended KP hierarchy to the solutions of extended mKP hierarchy. Wronskian solutions of extended KP and extended mKP hierarchies are constructed explicitly.

Restricted flows and the soliton equation with self-consistent sources.

The KdV equation is used as an example to illustrate the relation between the restricted flows and the soliton equation with self-consistent sources. Inspired by the results on the Backlund transformation for the restricted flows (by V.B. Kuznetsov et al.), we constructed two types of Darboux transformations for the KdV equation with self-consistent sources (KdVES). These Darboux transformations are used to get some explicit solutions of the KdVES, which include soliton, rational, positon, and negaton solutions.

An extended two-dimensional Toda lattice hierarchy and two-dimensional Toda lattice with self-consistent sources

We extend the two-dimensional Toda lattice hierarchy (2DTLH) by its squared eigenfunction symmetries. This extended 2DTLH (ex2DTLH) includes the two-dimensional Toda lattice equation with self-consistent sources (2DTLSCS) as its first nontrivial equation. Lax representation of ex2DTLH is also presented. With the help of the Lax representation, we construct a nonauto-Backlund Darboux transformation (DT) for 2DTLSCS by applying the variation of constants to 2DTLSCS auto-Backlund DT. This DT enables us to find many solutions to 2DTLSCS, including solitons, rational solutions, positons, negatons, and complexitons.

Discrete KP equation with self-consistent sources

Abstract We show that the discrete Kadomtsev–Petviashvili (KP) equation with sources obtained recently by the “source generalization” method can be incorporated into the squared eigenfunction symmetry extension procedure. Moreover, using the known correspondence between Darboux-type transformations and additional independent variables, we demonstrate that the equation with sources can be derived from Hirotas discrete KP equations but in a space of higher dimension. In this way we uncover the origin of the source terms as coming from multidimensional consistency of the Hirota system itself.

Families of dynamical r-matrices and Jacobi inversion problem for nonlinear evolution equations

Three families of dynamical r-matrices are explicitly constructed by means of constrained flows of nonlinear evolution equations (NLEE) associated with polynomial second-order spectral problems. The dynamical classical Yang–Baxter equations for the three linear r-matrix algebras are presented with explicit expression for the extra term. The separation variables and the Jacobi inversion problems for the x- and tn-constrained flow of the NLEEs are found. The factorization of NLEE allows us to obtain the Jacobi inversion problem for the NLEE by combining the above two Jacobi inversion problems. This provides a method of separation of variables to solve the NLEEs.

Bilinear Identities and Hirota's Bilinear Forms for an Extended Kadomtsev-Petviashvili Hierarchy

Abstract In this paper, we construct the bilinear identities for the wave functions of an extended Kadomtsev-Petviashvili (KP) hierarchy, which is the KP hierarchy with particular extended flows. By introducing an auxiliary parameter, whose flow corresponds to the so-called squared eigenfunction symmetry of KP hierarchy, we find the tau-function for this extended KP hierarchy. It is shown that the bilinear identities will generate all the Hirotas bilinear equations for the zero-curvature forms of the extended KP hierarchy, which includes two types of KP equation with self-consistent sources (KPSCS). The Hirotas bilinear equations obtained in this paper for the KPSCS are in different forms by comparing with the existing results.