Prolongation structures of a generalized coupled Korteweg-de Vries equation and Miura transformation

In this paper, the prolongation structures of a generalized coupled Korteweg-de Vries (KdV) equation are investigated and two integrable coupled KdV equations associated with their Lax pairs are derived. Furthermore, a Miura transformation related to a integrable coupled KdV equation is derived, from which a new coupled modified KdV equations is obtained.     

Complete integrability and the Miura transformation of a coupled KdV equation

In this letter, a Painlevé integrable coupled KdV equation is proved to be also Lax integrable by a prolongation technique. The Miura transformation and the corresponding coupled modified KdV equation associated with this equation are derived.     

Some new integrable systems constructed from the bi-Hamiltonian systems with pure differential Hamiltonian operators

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in [10] offers an useful way to construct new integrable system starting from the bi-Hamiltonian system. We construct some new integrable systems by means of the generalized Kupershmidt deformation in the cases of Harry Dym hierarchy, classical Boussinesq hierarchy and coupled KdV hierarchy. We show that the GKD of Harry Dym equation, GKD of classical Boussinesq equation and GKD of coupled KdV equation are equivalent to the new integrable Rosochatius deformations of these soliton equations with self-consistent sources. We present the Lax pair for these new systems. Therefore the generalized Kupershmidt deformation provides a new way to construct new integrable systems from bi-Hamiltonian systems and also offers a new approach to obtain the Rosochatius deformation of soliton equation with self-consistent sources.     

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.     

Integrable twofold hierarchy of perturbed equations and application to optical soliton dynamics

We construct well-known integrable equations with their Lax pairs from the corresponding linear equations using our nonlinearization scheme. Using negative powers in the spectral flow to deform the time Lax operator, we find a class of perturbations that unlike the usual perturbations, which spoil the system integrability, exhibit a twofold integrable hierarchy, including those for the KdV, modified KdV, sine-Gordon, nonlinear Schrödinger (NLS), and derivative NLS equations. We discover hidden possibilities of using the perturbed hierarchy of the NLS equations to amplify and control optical solitons propagating through a fiber in a doped nonlinear resonant medium.     

Integrability aspects of some two-component KdV systems

Some two-component Korteweg–de Vries systems are studied by prolongation technique and Painlevé analysis. Especially, the two-component KdV system conjectured to be integrable by Foursov is proved to be both Lax integrable and P-integrable. Its conservation laws are investigated based on the obtained Lax pair. Furthermore, it is shown that the three two-component Korteweg–de Vries systems are identical under certain invertible linear transformations. Finally, the auto-Bäcklund transformation and some exact solutions for the two-component Korteweg–de Vries system are derived explicitly.     

Darboux transformation and explicit solutions for the integrable sixth-order KdV equation for nonlinear waves

Korteweg-de Vries (KdV)-type equations can describe some physical phenomena in fluids, nonlinear optics, quantum mechanics, plasmas, etc. In this paper, with the aid of symbolic computation, the integrable sixth-order KdV equation is investigated. Darboux transformation (DT) with an arbitrary parameter is presented. Explicit solutions are derived with the DT. Relevant properties are graphically illustrated, which might be helpful to understand some physical processes in fluids, plasmas, optics and quantum mechanics.     

INTEGRABLE COUPLING OF THE TC HIERARCHY OF EQUATIONS

A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the integrable coupling of the famous KdV equation is presented.     

Discrete and continuous coupled nonlinear integrable systems via the dressing method

A discrete analog of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear Schrödinger type. First, a demonstration is given of how discrete nonlinear integrable equations can be derived starting from their linear counterparts. Then, starting from two uncoupled, discrete one‐directional linear wave equations, an appropriate matrix Riemann‐Hilbert problem is constructed, and a discrete matrix nonlinear Schrödinger system of equations is derived, together with its Lax pair. The corresponding compatible vector reductions admitted by these systems are also discussed, as well as their continuum limits. Finally, by increasing the size of the problem, three‐component discrete and continuous integrable discrete systems are derived, as well as their generalizations to systems with an arbitrary number of components.     

新的变系数可积耦合非线性Schr(o)dinger方程及其孤子解

基于延拓结构和Hirota双线性方法研究了广义的变系数耦合非线性Schrdinger方程.首先导出了3组新的变系数可积耦合非线性Schrdinger方程及其线性谱问题(Lax对),然后利用Hirota双线性方法给出了它们的单、双向量孤子解.这些向量孤子解在光孤子通讯中有重要的应用.     

Poisson brackets of mappings obtained as (<Emphasis Type="Italic">q</Emphasis>,−<Emphasis Type="Italic">p</Emphasis>) reductions of lattice equations

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The (q,?p) reductions are (p + q)-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the (3,?2) reductions of the integrable partial difference equations are Liouville integrable in their own right.     

ON GENERATING EQUATIONS FOR THE KAUP-NEWELL HIERARCHY

It is shown that the Kanp-Newell hierarchy can be derived from the so-called gen- erating equations which are Lax integrable.Positive and negative flows in the hierarchy are derived simultaneously.The generating equations and mutual commutativity of these flows en- able us to construct new Lax integrable equations.     

Darboux transformation and N-soliton solutions for a more general set of coupled integrable dispersionless system

The Darboux transformation and Lax pair of a more general set of coupled integrable dispersionless system are derived. By Darboux transformation, N-soliton solutions for the coupled integrable dispersionless system are obtained. In particular, the multi-soliton solutions are shown through some figures.     

Integrable symplectic maps associated with discrete Korteweg‐de Vries‐type equations

In this paper, we present novel integrable symplectic maps, associated with ordinary difference equations, and show how they determine, in a remarkably diverse manner, the integrability, including Lax pairs and the explicit solutions, for integrable partial difference equations which are the discrete counterparts of integrable partial differential equations of Korteweg‐de Vries‐type (KdV‐type). As a consequence it is demonstrated that several distinct Hamiltonian systems lead to one and the same difference equation by means of the Liouville integrability framework. Thus, these integrable symplectic maps may provide an efficient tool for characterizing, and determining the integrability of, partial difference equations.     

A note on “A study on an integrable system of coupled KdV equations”

We analyze the paper by Wazwaz [Wazwaz AM. A study on an integrable system of coupled KdV equations. Commun Nonlinear Sci Numer Simul, 2010;15:2846–2850]. Author tried to show that the system of coupled KdV equations is completely integrable but he has used the curious approach for the proof. We demonstrate that, author has taken the relation between dependent variables and has obtained the well known result by Hirota for the KdV equation.     

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.     

Effect of inhomogeneity in energy transfer through alpha helical proteins with interspine coupling

The dynamics of homogeneous and inhomogeneous alpha helical proteins with interspine coupling is under investigation in this paper by proposing a suitable model Hamiltonian. For specific choice of parameters, the dynamics of homogeneous alpha helical proteins is found to be governed by a set of completely integrable three coupled derivative nonlinear Schrödinger (NLS) equations (Chen–Lee–Liu equations). The effect of inhomogeneity is understood by performing a perturbation analysis on the resulting perturbed three coupled NLS equation. An equivalent set of integrable discrete three coupled derivative NLS equations is derived through an appropriate generalization of the Lax pair of the original Ablowitz–Ladik lattice and the nature of the energy transfer along the lattice is studied.     

On the Integrability of a Generalized Variable‐Coefficient Forced Korteweg‐de Vries Equation in Fluids

With the inhomogeneities of media taken into account, under investigation is hereby a generalized variable‐coefficient forced Korteweg‐de Vries (vc‐fKdV) equation, which describes shallow‐water waves, internal gravity waves, etc. Under an integrable constraint condition on the variable coefficients, in this paper, the complete integrability of the generalized vc‐fKdV equation is proposed. By virtue of a generalization of Bells polynomials, we systematically present its bilinear representations, Bäcklund transformations, Lax pairs and Darboux covariant Lax pairs, which can be reduced to the ones of some integrable models, such as vcKdV model, cylindrical KdV equation, and an analytical model of tsunami generation. It is very interesting that its bilinear formulism is free for the integrable constraint condition. Besides, researching the Lax equations yield its infinitely conservation laws, all conserved densities and fluxes of them are obtained by explicit recursion formulas. Furthermore, by considering its bilinear formulism with an extra auxiliary variable, we present the soliton solutions and Riemann theta function periodic wave solutions of the equation. According to the constraint among the nonlinear, dispersive, and line‐damping coefficients, we further discuss the solitonic structures and interaction properties by some graphic analysis. Finally, the relationships between the periodic wave solutions and soliton solutions are presented in detail by a limiting procedure.     

一个求孤子方程有限带势解的方法

基于Lax对非线性化方法，我们以KdV方程为例给出了一个构造孤子方程的有限带势解的方法．通过Lax对非线性化KdV方程被分解成两个有限维可积系统，进而找到这些有限维可积系统公共的角-作用坐标，最终我们获得了KdV方程的有限带势解．