Integrability of the coupled KdV equations derived from two-layer fluids: Prolongation structures and Miura transformations

The Lax integrability of the coupled KdV equations derived from two-layer fluids [S.Y. Lou, B. Tong, H.C. Hu, X.Y. Tang, Coupled KdV equations derived from two-layer fluids, J. Phys. A: Math. Gen. 39 (2006) 513-527] is investigated by means of prolongation technique. As a result, the Lax pairs of some Painlevé integrable coupled KdV equations and several new coupled KdV equations are obtained. Finally, the Miura transformations and some coupled modified KdV equations associated with the Lax integrable coupled KdV equations are derived by an easy way.     

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.     

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.     

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.     

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.     

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 study on an integrable system of coupled KdV equations

In this work, we study a system of coupled KdV equations. The Hirota’s bilinear method is applied to show that this system is completely integrable. Multiple-soliton solutions and multiple singular soliton solutions are derived for this system. The resonance phenomenon is examined as well.     

Solitons of curvature

An intristic geometry of surfaces is discussed. In geodesic coordinates the Gauss equation is reduced to the Schrödinger equation where the Gaussian curvature plays the role of a potential. The use of this fact provides an infinite set of explicit expressions for the curvature and metric of a surface. A special case is governed by the KdV equation for the Gaussian curvature. We consider the integrable dynamics of curvature via the KdV equation, higher KdV equations and (2+1)-dimensional integrable equations with breaking solitons.     

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.     

Time-space integrable decompositions of nonlinear evolution equations

Separation of the time and space variables of evolution equations is analyzed, without using any structure associated with evolution equations. The resulting theory provides techniques for constructing time-space integrable decompositions of evolution equations, which transform an evolution equation into two compatible Liouville integrable ordinary differential equations in the time and space variables. The techniques are applied to the KdV, MKdV and diffusion equations, thereby yielding several new time-space integrable decompositions of these equations.     

Four (2 + 1)-dimensional integrable extensions of the KdV equation: Multiple-soliton and multiple singular soliton solutions

In this work, four (2 + 1)-dimensional nonlinear completely integrable equations, generated by extending the KdV equation are developed. The necessary condition for the complete integrability of these equation are formally derived. Multiple-soliton solutions and multiple singular soliton solutions are determined to emphasize the compatability of these models. The dispersion relations of these models are characterized by distinct physical structures. The resonance phenomenon for these equations does not exist for any model.     

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.     

The dressing chain of discrete symmetries and proliferation of nonlinear equations

In the examples of sine-Gordon and Korteweg-de Vries (KdV) equations, we propose a direct method for using dressing chains (discrete symmetries) to proliferate integrable equations. We give a recurrent procedure (with a finite number of steps in general) that allows the step-by-step production of an integrable system and its L-A pair from the known L-A pair of an integrable equation. Using this algorithm, we reproduce a number of known results for integrable systems of the KdV type. We also find a new integrable equation of the sine-Gordon series and investigate its simplest soliton solution of the double π-kink type. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. 115. No. 2. pp. 199–214. May. 1998.     

Soliton interaction for the extended Korteweg-de Vries equation

Soliton interactions for the extended Korteweg-de Vries (KdV)equation are examined. It is shown that the extended KdV equationcan be transformed (to its order of approximation) to a higher-ordermember of the KdV hierarchy of integrable equations. This transformationis used to derive the higher-order, two-soliton solution forthe extended KdV equation. Hence it follows that the higher-ordersolitary-wave collisions are elastic, to the order of approximationof the extended KdV equation. In addition, the higher-ordercorrections to the phase shifts are found. To examine the exactnature of higher-order, solitary-wave collisions, numericalresults for various special cases (including surface waves onshallow water) of the extended KdV equation are presented. Thenumerical results show evidence of inelastic behaviour wellbeyond the order of approximation of the extended KdV equation;after collision, a dispersive wavetrain of extremely small amplitudeis found behind the smaller, higher-order solitary wave.     

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.     

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.     

耦合KdV方程组的对称,精确解和守恒律

通过利用修正的CK直接方法建立了耦合KdV方程组的对称群理论.利用对称群理论和耦合KdV方程组的旧解得到了它们的新的精确解.基于上述理论和耦合KdV方程组的共轭方程组的理论,得到了耦合KdV方程组的守恒律.     

A note on “The integrable KdV6 equation: Multiple soliton solutions and multiple singular soliton solutions”

The goal of this short note is to provide another kind soliton solutions with Hirota form, which is different from what Wazwaz obtained in [A.M. Wazwaz, The integrable KdV6 equations: Multiple soliton solutions and multiple singular soliton solutions, Appl. Math. Comput. 204 (2008) 963-972]. Meanwhile we newly construct the MKdV6 equation and derive a Miura transformation between KdV6 equation and MKdV6 equation.