Yunbo Zeng

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.

Two binary Darboux transformations for the KdV hierarchy with self-consistent sources

Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non-auto-Backlund transformation between two nth KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources.

New factorization of the Kaup-Newell hierarchy

Abstract The new factorizations of each equation in the Kaup-Newell (KN) hierarchy into two commuting x - and t n -finite-dimensional integrable Hamiltonian systems (FDIHS) via the higher-order potential-eigenfunction constraints are presented. The integrability, commutativity and Lax representation for these FDIHSs are deduced from the adjoint representations of the auxiliary linear problems for the KN hierarchy. This approach provides a natural way to study soliton hierarchies with source. The zero-curvature representation and Darboux transformation for the KN hierarchy with source related to eigenfunctions are found.

An approach to the deduction of the finite-dimensional integrability from the infinite-dimensional integrability

Abstract A systematic method for deducing the Liouville integrability of a finite-dimensional Hamiltonian system reduced from an infinite-dimensional Hamiltonian system is proposed within the framework of the zero-curvature representation theory. Also a systematic way to treat the higher-order constraints and to obtain the associated infinitely many hierarchies of finite-dimensional integrable Hamiltonian systems is presented.

Generalized Darboux transformations for the KP equation with self-consistent sources

The KP equation with self-consistent sources (KPESCS) is treated in the framework of the constrained KP equation. This offers a natural way to obtain the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation with arbitrary functions in time t for the KPESCS which, in contrast to the binary Darboux transformation of the KP equation, provides a non-auto-Backlund transformation between two KPESCSs with different degrees. The formula for N-times repeated generalized binary Darboux transformation is proposed and enables us to find the N-soliton solution and lump solution as well as some other solutions of the KPESCS.

The constraints of potentials and the finite-dimensional integrable systems

Restricting potential to the space spanned by the eigenvectors of the recursion operator leads to a natural constraint of potential and a finite‐dimensional integrable Hamiltonian system. The general method for proving the consistency of the two systems stemming from the Lax pair and obtaining the constants of the motion for the Hamiltonian system is illustrated by the classical Boussinesq and AKNS hierarchies. By using gauge transformation, similar results for the Jaulent–Miodek and Kaup–Newell hierarchies are presented.

The deduction of the Lax representation for constrained flows from the adjoint representation

For x- and tn-finite-dimensional Hamiltonian systems obtained from the decompositions of zero-curvature equations, it is shown that their Lax representations can be deduced directly from the adjoint representations of the auxiliary linear problems. As a consequence, the zero-curvature representation for soliton hierarchy with source is presented.

Negaton and positon solutions of the soliton equation with self-consistent sources

The Korteweg–de Vries (KdV) equation with self-consistent sources (KdVES) is used as a model to illustrate this method. We present a generalized binary Darboux transformation (GBDT) with an arbitrary time-dependent function for the KdVES as well as the formula for N-times repeated GBDT. This GBDT provides non-auto-Backlund transformation between two KdV equations with different degrees of sources and enables us to construct more general solutions with N arbitrary t-dependent functions. By taking the special t-function, we obtain multisoliton, multipositon, multinegaton, multisoliton–positon, multinegaton–positon and multisoliton–negaton solutions of the KdVES.

Separation of variables for soliton equations via their binary constrained flows

Binary constrained flows of soliton equations admitting 2×2 Lax matrices have 2N degrees of freedom, which is twice as many degrees of freedom than in the case of monoconstrained flows. By using the normal method, their Lax matrices directly give rise to first N pairs of canonical separated variables for their separation of variables. We propose a new method to introduce the other N pairs of canonical separated variables and additional separated equations. The Jacobi inversion problems for binary constrained flows are established. Finally, the factorization of soliton equations by two commuting binary constrained flows and the separability of binary constrained flows enable us to construct the Jacobi inversion problems for some soliton hierarchies.