https://scholarworks.utrgv.edu/cgi/viewcontent.cgi?article=1028&context=mss_fac

A (2+1)-dimensional sine-Gordon and sinh-Gordon equations with symmetries and kink wave solutions

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure. All these nonlinear systems have {\it r}-matrices, and are completely integrable in Liouville's sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary.

Generalized r-matrix structure and algebro-geometric solution for integrable systems

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of the usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-Levi, etc, are derived from the reductions of this structure. All these nonlinear systems have r-matrices, and are completely integrable in Liouville's sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary.

Generalized r-Matrix Structure and Algebro-Geometric Solution for Integrable System

(2+1)-dimensional local and nonlocal reductions of the negative AKNS system: soliton solutions

Soliton dynamics to the multi-component complex coupled integrable dispersionless equation

A new production form for a hierarchy of nonlinear evolution equations (NLEEs) is given in this paper The form contains productions of isospectral and non-isospectral hierarchy Under this form a generalized structure of Lax representations for the hierarchy of NLEEs is this presented As a concrete example, the Levi-hierarchy of evolution equations are discussed at the end of this paper

Generalized structure of lax representations for nonlinear evolution equation

It is proven that on a symplectic submanifold the restricted c-KdV flow is just the interpolating Hamiltonian flow of invariant for the restricted Toda flow, which is an integrable symplectic map of Neumann type. They share the common Lax matrix, dynamical r-matrix and system of involutive conserved integrals. Furthermore, the procedure of separation of variables is considered for the restricted c-KdV flow of Neumann type.

On the Restricted Toda and c-KdV Flows of Neumann Type

A new method is presented to generate finite dimensional integrable systems. Our starting point is a generalized Lax matrix instead of usual Lax pair. Then a unisonant r-matrix structure and a set of generalized Hamiltonian functions are constructed. It can be clearly seen that various constrained integrable flows by nonlinearization method, such as the c-AKNS, c-MKdV, c-Toda, etc., are derived from the reduction of this structure. Furthermore, some new integrable flows are produced.

http://iopscience.iop.org/article/10.1088/0256-307X/17/4/001/pdf

Qiao Zhi-Jun

Papers

Generalized structure of lax representations for nonlinear evolution equation

On the Restricted Toda and c-KdV Flows of Neumann Type

A Unisonant r-Matrix Structure of Integrable Systems and Its Reductions