Inverse Spectral Problem and Peakons of an Integrable Two-component Camassa-Holm System

In this paper, we are concerned with the explicit construction of peakon solutions of the integrable twocomponent system with cubic non-linearity. We establish the spectral and inverse spectral problem associated to the Lax pairs of the system. The inverse problem is solved by the classical results of Stieltjes continued fractions, which also contributes a lot to the spectral problem. The explicit formulas are obtained from solutions of the inverse problem. The positivity of the spectral measures is implied by J. Moser's work on the Jacobi spectral problem.     

超对称系统的边界条件与可积性

通过研究反射方程的解,构造了一类具有不同边界条件的超对称系统,同时证明了在一维情况下,这类系统是完全可积的.     

Thermodynamic Properties of the One-Dimensional Two-Component Log-Gas

We consider a one-dimensional continuum gas of pointlike positive and negative unit charges interacting via a logarithmic potential. The mapping onto a two-dimensional boundary sine-Gordon field theory with zero bulk mass provides the full thermodynamics (density-fugacity relationship, specific heat, etc.) of the log-gas in the whole stability range of inverse temperatures <1. An exact formula for the excess chemical potential of a foreign particle of an arbitrary charge, put into the log-gas, is derived. The results are checked by a small- expansion and at the collapse =1 point. The possibility to go beyond the collapse temperature is discussed.     

Reciprocal Transformations of Two Camassa-Holm Type Equations

The relation between the Camassa-Holm equation and the Olver-Rosenau-Qiao equation is obtained, and we connect a new Camassa-Holm type equation proposed by Qiao etc. with the first negative flow of the KdV hierarchy by a reciprocal transformation.     

Two-Component Solitary Waves in Hydrogen-Bonded Molecular Systems

The dynamics of two-component solitary waves in hydrogen-bonded chains in an external force and damping is investigated. The influence of the motion and the optical mode of the heavy ion sublattice on the portion sublattice is discussed. It will increase the soliton width and decrease the soliton mobility. The general expression for the kink soliton soliton is obtained. The velocity, the mobility and conductivity of the kink soliton are calculated. The results are in good agreement with the experimental data.     

A Note on the Integrability of Hamiltonian Systems

We consider a family of Hamiltonian systems

Analytical Cartesian solutions of the multi-component Camassa-Holm equations

Here, we give the existence of analytical Cartesian solutions of the multi-component Camassa-Holm (MCCH) equations. Such solutions can be explicitly expressed, in which the velocity function is given by u = b(t)+A(t)x and no extra constraint on the dimension N is required. The advantage of our method is that we turn the process of analytically solving MCCH equations into algebraically constructing the suitable matrix A(t). As the applications, we obtain some interesting results: 1) If u is a linear transformation on , then p takes a quadratic form of x. 2) If A = f (t)I + D with D^T = ?D, we obtain the spiral solutions. When N = 2, the solution can be used to describe “breather-type” oscillating motions of upper free surfaces. 3) If we obtain the generalized elliptically symmetric solutions. When N = 2, the solution can be used to describe the drifting phenomena of the shallow water flow.     

Degenerate Integrability of the Spin Calogero–Moser Systems and the Duality with the Spin Ruijsenaars Systems

It is shown that spin Calogero–Moser systems are completely integrable in a sense of degenerate integrability. Their Liouville tori have dimension less than half of the dimension of the phase space. It is also shown that rational spin Ruijsenaars systems are degenerately integrable and dual to spin Calogero–Moser systems in a sense that action-angle variables of one are angle-action variables of the other.     

广义色散Camassa-Holm模型的奇异孤子

引进一类广义色散Camassa-Holm模型，对其做奇异性分析.通过改进的WTC-Kruskal算法，证明该模型在Painlevé意义下可积，得到了它的一组Painlevé-Bcklund系统和Bcklund变换.应用Maple进行代数运算，得到了丰富的规则(regular)孤子和一类奇异(singular)孤子，扭结(kink)孤子,紧孤子(compacton)和反紧孤子(anti-compacton).特别地，推导出一类在扭结孤子的中间区域包含有一列周期尖点(cuspon)波的奇异结构.在这些规则的孤子系统的基础上，对可积广义系统应用Bcklund变换，得到三类奇异孤子，分别是具有驼峰结构的周期爆破波，具有爆破波结构的扭结孤子和紧孤子. 关键词： 广义Camassa-Holm 模型 周期尖点波 紧孤子 周期爆破波     

A three-component Camassa-Holm system with cubic nonlinearity and peakons

In this paper, we propose a three-component Camassa-Holm (3CH) system with cubic nonlinearity and peaked solitons (peakons). The 3CH model is proven to be integrable in the sense of Lax pair, Hamiltonian structure, and conservation laws. We show that this system admits peakons and multi-peakon solutions. Additionally, reductions of the 3CH system are investigated so that a new integrable perturbed CH equation with cubic nonlinearity is generated to possess peakon solutions.     

Integrability and Solutions of the(2+1)-dimensional Hunter–Saxton Equation

In this paper,the(2+1)-dimensional Hunter-Saxton equation is proposed and studied.It is shown that the(2+1)-dimensional Hunter–Saxton equation can be transformed to the Calogero–Bogoyavlenskii–Schiff equation by reciprocal transformations.Based on the Lax-pair of the Calogero–Bogoyavlenskii–Schiff equation,a non-isospectral Lax-pair of the(2+1)-dimensional Hunter–Saxton equation is derived.In addition,exact singular solutions with a finite number of corners are obtained.Furthermore,the(2+1)-dimensional μ-Hunter–Saxton equation is presented,and its exact peaked traveling wave solutions are derived.     

Liouville Integrability of Conservative Peakons for a Modified CH equation

The modified Camassa-Holm (also called FORQ) equation is one of numerous cousins of the Camassa-Holm equation possessing non-smoth solitons (peakons) as special solutions. The peakon sector of solutions is not uniquely defined: in one peakon sector (dissipative^a) the Sobolev H¹ norm is not preserved, in the other sector (conservative), introduced in [2], the time evolution of peakons leaves the H¹ norm invariant. In this Letter, it is shown that the conservative peakon equations of the modified Camassa-Holm can be given an appropriate Poisson structure relative to which the equations are Hamiltonian and, in fact, Liouville integrable. The latter is proved directly by exploiting the inverse spectral techniques, especially asymptotic analysis of solutions, developed elsewhere [3].     

Generalized Lie Symmetries and the Integrability of Generalized Emden-Fowler Equations

Generalized Lie symmetries and the integrability of generalized Emden-Fowler equations (GEFEs) are considered. It is shown that the constraint which the variable-coefficient functions must satisfy for the GEFEs to have infinite-dimensional symmetry algebras is precisely the same as this in order that the equation may be transformed into the integrable Emden-Fowler equation. fiom the nature of the symmetry vector fields one can write down the integrals of motion for the above systems. The structure of the symmetry algebras is also presented.     

On a spectral analysis of scattering data for the Camassa-Holm equation

Physical details of the Camassa–Holm (CH) equation that are difficult to obtain in space-time simulation can be explored by solving the Lax pair equations within the direct and inverse scattering analysis context. In this spectral analysis of the completely integrable CH equation we focus solely on the direct scattering analysis of the initial condition defined in the physical space coordinate through the time-independent Lax equation. Both of the continuous and discrete spectrum cases for the initial condition under current investigation are analytically derived. The scattering data derived from the direct scattering transform for non-reflectionless case are also discussed in detail in spectral domain from the physical viewpoint.     

Multi-soliton solutions of a two-component Camassa–Holm system:Darboux transformation approach

We propose and develop another approach to constructing multi-soliton solutions of an integrable two-component Camassa–Holm(CH2)system.With the help of a reciprocal transformation and a gauge transformation,we relate the CH2 system to a negative flow of the Broer–Kaup or twoboson hierarchy.The solutions of this negative flow are given in terms of Wronskians via Darboux transformation.Then the multi-soliton solutions of the CH2 system are recovered in parametric form by inverting the reciprocal transformation and the gauge transformation.     

Lax Integrability of the Modified Camassa-Holm Equation and the Concept of Peakons

In this Letter we propose that for Lax integrable nonlinear partial differential equations the natural concept of weak solutions is implied by the compatibility condition for the respective distributional Lax pairs. We illustrate our proposal by comparing two concepts of weak solutions of the modified Camassa-Holm equation pointing out that in the peakon sector (a family of non-smooth solitons) only one of them, namely the one obtained from the distributional compatibility condition, supports the time invariance of the Sobolev H¹ norm.     

Addition Formulae of Discrete KP,q-KP and Two-Component BKP Systems

In this paper,we construct the addition formulae for several integrable hierarchies,including the discrete KP,the q-deformed KP,the two-component BKP and the D type Drinfeld–Sokolov hierarchies.With the help of the Hirota bilinear equations and τ functions of different kinds of KP hierarchies,we prove that these addition formulae are equivalent to these hierarchies.These studies show that the addition formula in the research of the integrable systems has good universality.     

Geometric Phase of a Coupled System

In this contribution, my main attention is paid to the study of the geometric phase ora two-level system.Asymptotic expansion of the eigenvalue and analytic information are obtained. The results are illustrated with an application to a specific atomic system.     

多波长系统孤子耦合方程的可积性

多波长系统孤子耦合方程存在Lax对，具有可积性，利用Hirota双线性方法求出了孤子耦合方程的单孤子解和双孤子解. 关键词： Lax对 Hirota双线性变换 可积性     

On the Integrability of the One-Dimensional Open XYZ Spin Chain

The Lax pair for the one-dimensional open XYZ spin chain is constructed, this shows that the system is completely integrable.