Yishen Li

The constraint of the Kadomtsev-Petviashvili equation and its special solutions

Abstract By constraining the potential of the Kadomtsev-Petviashvili (KP) equation to its co-invariants expressed in terms of the squared eigenfunctions, the KP equation is reduced to a (1 + 1)-dimensional system consisting of the generalized multicomponent nonlinear Schrodinger and modified Korteweg-de Vries equations. So any solution of this system gives rise to a solution of the KP equation, such as new kinds of soliton-like solutions and a solution periodic in the x direction.

Darboux transformations of classical Boussinesq system and its new solutions

Abstract Two basic Darboux transformations of a spectral problem associated with a classical Boussinesq system are considered in this letter. They are used to generate new solutions of the classical Boussinesq system. An interesting solution of double-peak soliton has been found.

Constraints of the 2+1 dimensional integrable soliton systems

The authors show that the linear systems associated with some integrable hierarchies of the soliton equations in 2+1 dimensions can be constrained to integrable hierarchies in 1+1 dimensions such that submanifolds solutions of the given systems in 2+1 can be obtained by solving the resulting integrable systems in 1+1 dimensions. The constraints of the KP hierarchy to the AKNS and Burgers hierarchies respectively are shown in detail and the results of these for the modified KP and 2+1 dimensional analogue of the Caudrey-Dodd-Gibbon-Kotera-Sawata equations to several integrable systems in 1+1 are given.

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.

Darboux transformations of classical Boussinesq system and its multi-soliton solutions

Abstract In this Letter, we present the third kind of Darboux transformation of the classical Boussinesq system, and discuss its relationship with the two basic Darboux transformations. By applying the Darboux transformations, we obtain the solutions of multiple soliton interactions, including overtaking and head-on collisions.

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.

Integrable Hamiltonian systems related to the polynomial eigenvalue problem

The independent integrals of motion in involution for the Hamiltonian system related to the second‐order polynomial eigenvalue problem are constructed by using relevant recursion formula. The hierarchy of Hamiltonian systems obtained from the above problem and the time part of the Lax pair are shown to be completely integrable and they are shown to commute with each other. Furthermore, their solution solves the evolution equation associated with the Lax pair.

Hamiltonian structure of the super evolution equation

The constrained variational calculus proposed in previous papers [Y. Zheng, Y. Li, and D. Chen, Sci. Sinica A 24, 138 (1986); G. Tu, Kexue Tangbao 29, 1227 (1984)] is generalized to the supersymmetric case. Utilizing this method to some super AKNS system, which has soliton solutions and conserved quantities, their equations of motion with the Hamiltonian structure can be written as a 4×4 matrix. This is the supersymmetric symplectic matrix and hn are Hamiltonians of the system. The conserved quantities worked out in recent literature are just the first few terms of our analytic expression.

Two choices of the gauge transformation for the AKNS hierarchy through the constrained KP hierarchy

On the basis of the equivalence between the AKNS hierarchy and the cKP hierarchy with the constraint k=1, we point out that there exist two choices to keep the form of the Lax operator when we perform the gauge transformation for the AKNS hierarchy, which results in two classes of functions to trigger the gauge transformation. For the second choice, two theorems for two types of gauge transformation are established. Several new and more general forms of tau-functions for the AKNS hierarchy are obtained by means of gauge transformations of both types. The union of the two choices leads to new forms of τ-functions. We generate the AKNS hierarchy from the “free” Lax operator L(0)=∂ via a chain of gauge transformations.

THE DETERMINANT REPRESENTATION OF THE GAUGE TRANSFORMATION OPERATORS

The determinant representation of the gauge transformation operators is establised. In this process, the generalized Wronskian determinant is introduced. As a simple application, the authors present a construction of the special τ-function obtained firstly by Chau et al. (Commun. Math. Phys., 149(1992), 263), which involves the generalized Wronskian determinant. Also, some properties of this determinant are given.