The generalized Kupershmidt deformation for the fifth-order coupled KdV equations hierarchy

Based on the Kupershmidt deformation, we propose the generalized Kupershmidt deformation (GKD) to construct new systems from integrable bi-Hamiltonian system. As applications, the generalized Kupershmidt deformation of the fifth-order coupled KdV equations hierarchy with self-consistent sources and its Lax representation are presented.     

The generalized Kupershmidt deformation for constructing new discrete integrable systems

It is known that the KdV6 equation can be represented as the Kupershmidt deformation of the KdV equation. We propose a generalized Kupershmidt deformation for constructing new discrete integrable systems starting from the bi-Hamiltonian structure of a discrete integrable system. We consider the Toda, Kac-van Moerbeke, and Ablowitz-Ladik hierarchies and obtain Lax representations for these new deformed systems. The generalized Kupershmidt deformation provides a new way to construct discrete integrable systems.     

A generalized AKNS hierarchy,bi-Hamiltonian structure,and Darboux transformation

A new generalized AKNS hierarchy is presented starting from a 4 × 4 matrix spectral problem with four potentials. Its generalized bi-Hamiltonian structure is also investigated by using the trace identity. Moreover, the special coupled nonlinear equation, the coupled KdV equation, the KdV equation, the coupled mKdV equation and the mKdV equation are produced from the generalized AKNS hierarchy. Most importantly, a Darboux transformation for the generalized AKNS hierarchy is established with the aid of the gauge transformation between the corresponding 4 × 4 matrix spectral problem, by which multiple soliton solutions of the generalized AKNS hierarchy are obtained. As a reduction, a Darboux transformation of the mKdV equation and its new analytical positon, negaton and complexiton solutions are given.     

The Bi-Hamiltonian Theory of the Harry Dym Equation

We describe how the Harry Dym equation fits into the the bi-Hamiltonian formalism for the Korteweg–de Vries equation and other soliton equations. This is achieved using a certain Poisson pencil constructed from two compatible Poisson structures. We obtain an analogue of the Kadomtsev–Petviashivili hierarchy whose reduction leads to the Harry Dym hierarchy. We call such a system the HD–KP hierarchy. We then construct an infinite system of ordinary differential equations (in infinitely many variables) that is equivalent to the HD–KP hierarchy. Its role is analogous to the role of the Central System in the Kadomtsev–Petviashivili hierarchy.     

Mechanical systems related to the Schrödinger spectral problem

We present a variety of integrable mechanical systems which are embedded into the KdV and the Harry Dym hierarchies of soliton equations as their restricted flows. Integrable structures of these systems are systematically derived from the properties of the underlying hierarchies of equations. It is illustrated by the key example of the Garnier system which describes the motion of a particle in a quartic ‘wine bottle’ potential.     

A generalized AKNS hierarchy and its bi-Hamiltonian structures

First we construct a new isospectral problem with 8 potentials in the present paper. And then a new Lax pair is presented. By making use of Tu scheme, a class of new soliton hierarchy of equations is derived, which is integrable in the sense of Liouville and possesses bi-Hamiltonian structures. After making some reductions, the well-known AKNS hierarchy and other hierarchies of evolution equations are obtained. Finally, in order to illustrate that soliton hierarchy obtained in the paper possesses bi-Hamiltonian structures exactly, we prove that the linear combination of two-Hamiltonian operators admitted are also a Hamiltonian operator constantly. We point out that two Hamiltonian operators obtained of the system are directly derived from a recurrence relations, not from a recurrence operator.     

AN UNITED APPROACH TO BOTH THE TM HIERARCHY AND THE COUPLED KdV HIERARCHY

A united model of both the TM hierarchy and the coupled KdV hierarchy is proposed. By using the trace identity, the bi-Hamiltonian structure of the corresponding hierarchy is established. The isospectral problem is nonlinearized as a new completely integrable Hamiltonian system in Liouville sense.     

New exact solutions for a generalization of the Korteweg-de Vries equation (KdV6)

The generalized tanh-coth method is used to construct periodic and soliton solutions for a new integrable system, which has been derived from an integrable sixth-order nonlinear wave equation (KdV6). The system is formed by two equations. One of the equations may be considered as a Korteweg-de Vries equation with a source and the second equation is a third-order linear differential equation.     

MBIUS-TODA HIERARCHY AND ITS INTEGRABILITY

In this paper, we construct a new integrable equation called Mbius-Toda equation which is a generalization of q-Toda equation. Meanwhile its soliton solutions are constructed to show its integrable property. Further the Lax pairs of the Mbius-Toda equation and a whole integrable Mbius-Toda hierarchy are also constructed. To show the integrability, the bi-Hamiltonian structure and tau symmetry of the Mbius-Toda hierarchy are given and this leads to the existence of the tau function.     

Solving soliton equations with self-consistent sources by method of variation of parameters

Starting from the solutions of soliton equations and corresponding eigenfunctions obtained by Darboux transformation, we present a new method to solve soliton equations with self-consistent sources (SESCS) based on method of variation of parameters. The KdV equation with self-consistent sources (KdVSCS) is used as a model to illustrate this new method. In addition, we apply this method to construct some new solutions of the derivative nonlinear Schrödinger equation with self-consistent sources (DNLSSCS) such as phase solution, dark soliton solution, bright soliton solution and breather-type solution.     

超经典Boussinesq系统的守恒律和自相容源

本文研究了超经典Boussinesq系统.利用已有的超经典Boussinesq方程族及其超哈密顿结构,构造了带自相容源的超经典Boussinesq方程族,并通过引入变量F和G,获得了超经典Boussinesq方程族的守恒律.     

On Classification of Integrable Nonevolutionary Equations

We study partial differential equations of second order (in time) that possess a hierarchy of infinitely many higher symmetries. The famous Boussinesq equation is a member of this class after the extension of the differential polynomial ring. We develop the perturbative symmetry approach in symbolic representation. Applying it, we classify the homogeneous integrable equations of fourth and sixth order (in the space derivative) equations, as well as we have found three new tenth-order integrable equations. To prove the integrability we provide the corresponding bi-Hamiltonian structures and recursion operators.     

A new integrable equation that combines the KdV equation with the negative‐order KdV equation

In this work, we develop a new integrable equation by combining the KdV equation and the negative‐order KdV equation. We use concurrently the KdV recursion operator and the inverse KdV recursion operator to construct this new integrable equation. We show that this equation nicely passes the Painlevé test. As a result, multiple soliton solutions and other soliton and periodic solutions are guaranteed and formally derived.     

A new Lie algebra along with its induced Lie algebra and associated with applications

A 3 × 3 Lie algebra H is introduced whose induced Lie algebra by decomposition and linear combinations is obtained, which may reduce to the Lie algebra given by AP Fordy and J Gibbons. By employing the induced Lie algebra and the zero curvature equation, a kind of enlarged Boussinesq soliton hierarchy is produced. Again making use of a subalgebra of the induced Lie algebra leads to the well-known KdV hierarchy whose expanding integrable system is also worked out. As an applied example of the Lie algebra H, we obtain a new integrable coupling of the well-known AKNS hierarchy.     

Constructing super D-Kaup-Newell hierarchy and its nonlinear integrable coupling with self-consistent sources

How to construct new super integrable equation hierarchy is an important problem. In this paper, a new Lax pair is proposed and the super D-Kaup-Newell hierarchy is generated, then a nonlinear integrable coupling of the super D-Kaup-Newell hierarchy is constructed. The super Hamiltonian structures of coupling equation hierarchy is derived with the aid of the super variational identity. Finally, the self-consistent sources of super integrable coupling hierarchy is established. It is indicated that this method is a straight- forward and efficient way to construct the super integrable equation hierarchy.     

New hierarchy of integrable system bi-Hamiltonian structure and constrained flows

We have considered the hierarchy of integrable systems associated with the unstable nonlinear Schrodinger equation. The spectral gradient approach and the trace identity are used to derive the bi-Hamiltonian structure of the system. The bi-Hamiltonian property and the square eigenfunctions determined via the spectral gradient approach are then used to construct constrained flows, which is also proved to be derivable from a rational Lax operator. This new Lax operator of the constrained flows is seen to generate the classical r-matrix. Lastly it is also explicitly demonstrated that the different integrals of motion of the constrained flows Poisson commute.     

Construction of coupled Harry Dym hierarchy and its solutions from Stäckel systems

In this paper we show how to construct the coupled (multicomponent) Harry Dym (cHD) hierarchy from classical Stäckel separable systems. Both nonlocal and purely differential parts of hierarchies are obtained. We also construct various classes of solutions of cHD hierarchy from solutions of corresponding Stäckel systems.     

Conservation laws and self-consistent sources for a super integrable equation hierarchy

In this paper, a super integrable equation hierarchy is considered based on a Lie superalgebra and supertrace identity. Then, a super integrable equation hierarchy with self-consistent sources is established. Furthermore, we introduce two variables F and G to construct conservation laws of the super integrable equation hierarchy and the first two conserved densities and fluxes are listed. It would be specially mentioned that the Fermi variables play an important role in super integrable systems which is different from the ordinary integrable systems.     

广义耦合KdV孤子方程的达布变换及其偶孤子解

根据广义耦合KdV孤子方程的Lax对, 借助谱问题的规范变换, 一个包含多参数的达布变换被构造出来. 利用达布变换来产生广义耦合KdV孤子方程的偶孤子解, 并且用行列式的形式来表达广义耦合KdV孤子方程的偶孤子解. 作为应用, 广义耦合KdV孤子方程的偶孤子解的前两个例子被给出.     

广义耦合KdV孤子方程的达布变换及其奇孤子解

借助谱问题的规范变换, 给出广义耦合KdV孤子方程的达布变换,利用达布变换来产生广义耦合KdV孤子方程的奇孤子解,并且用行列式的形式来表达广义耦合KdV孤子方程的奇孤子解．作为应用,广义耦合KdV孤子方程奇孤子解的前两个例子被给出．