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．     

Relation among nonlinear evolution equations and their reductions

The LCZ soliton hierarchy is presented, and their generalized Hamiltonian structures are deduced. From the compatibility of soliton equations, it is shown that this soliton hierarchy is closely related to the Burger equation, the mKP equation and a new (2 + 1)-dimensional nonlinear evolution equation (NEE). Resorting to the nonlinearization of Lax pairs (NLP), all the resulting NEEs are reduced into integrable Hamiltonian systems of ordinary differential equations (ODEs). As a concrete application, the solutions for NEEs can be derived via solving the corresponding ODEs.     

A family of integrable differential–difference equations,its bi-Hamiltonian structure and binary nonlinearization of the Lax pairs and adjoint Lax pairs

A family of integrable differential–difference equations is derived by the method of Lax pairs. A discrete Hamiltonian operator involving two arbitrary real parameters is introduced. When the parameters are suitably selected, a pair of discrete Hamiltonian operators is presented. Bi-Hamiltonian structure of obtained family is established by discrete trace identity. Then, Liouville integrability for the obtained family is proved. Ultimately, through the binary nonlinearization of the Lax pairs and adjoint Lax pairs, every differential–difference equation in obtained family is factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense.     

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.     

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.     

WEAK CONVERGENCE FOR NON-UNIFORM φ-MIXING RANDOM FIELDS

ＷＥＡＫＣＯＮＶＥＲＧＥＮＣＥＦＯＲＮＯＮＵＮＩＦＯＲＭφＭＩＸＩＮＧＲＡＮＤＯＭＦＩＥＬＤＳＬＵＣＨＵＡＮＲＯＮＧＡｂｓｔｒａｃｔＬｅｔ｛ξｔ，ｔ∈Ｚｄ｝ｂｅａｎｏｎｕｎｉｆｏｒｍφｍｉｘｉｎｇｓｔｒｉｃｔｌｙｓｔａｔｉｏｎａｒｙｒｅａ...     

Symbolic-computation study of integrable properties for the (2 + 1)-dimensional Gardner equation with the two-singular manifold method

The singular manifold method from the Painlevé analysiscan be used to investigate many important integrable propertiesfor the non-linear partial differential equations. In this paper,the two-singular manifold method is applied to the (2 + 1)-dimensionalGardner equation with two Painlevé expansion branchesto determine the Hirota bilinear form, Bäcklund transformation,Lax pairs and Darboux transformation. Based on the obtainedLax pairs, the binary Darboux transformation is constructedand the N x N Grammian solution is also derived by performingthe iterative algorithm N times with symbolic computation.     

The Application of Neumann Type Systems for Solving Integrable Nonlinear Evolution Equations

An algorithm to obtain finite‐gap solutions of integrable nonlinear evolution equations (INLEEs) is provided by using the Neumann type systems in the framework of algebraic geometry. From the nonlinearization of Lax pairs, some INLEEs in 1+1 and 2+1 dimensions are reduced into a class of new Neumann type systems separating the spatial and temporal variables of INLEEs over a symplectic submanifold (M, ω²) . Based on the Lax representations of INLEEs, we deduce the Lax–Moser matrix for those Neumann type systems that yield the integrals of motion, elliptic variables, and a hyperelliptic curve of Riemann surface. Then, we attain the Liouville integrability for a hierarchy of Neumann type systems in view of a Lax equation on (M, ω²) and a set of quasi‐Abel–Jacobi variables. We also specify the relationship between Neumann type systems and INLEEs, where the involutive solutions of Neumann type systems give rise to the finite parametric solutions of INLEEs and the Neumann map cuts out a finite dimensional invariant subspace for INLEEs. Under the Abel–Jacobi variables, the Neumann type flows, the 1+1, and 2+1 dimensional flows are integrated with Abel–Jacobi solutions; as a result, the finite‐gap solutions expressed by Riemann theta functions for some 1+1 and 2+1 dimensional INLEEs are achieved through the Jacobi inversion with the aid of the Riemann theorem.     

两类(2+1)维可积系统

利用一个广义等谱问题的相容性得到了一个广义零曲率方程.作为其应用,首先利用loop代数设计了两个广义等谱问题,然后利用零曲率方程导出了两类(2+1)维Lax可积系统.     

Integrable decomposition of a hierarchy of soliton equations and integrable coupling system by semidirect sums of Lie algebras

Staring from a new spectral problem, a hierarchy of the soliton equations is derived. It is shown that the associated hierarchies are infinite-dimensional integrable Hamiltonian systems. By the procedure of nonlinearization of the Lax pairs, the integrable decomposition of the whole soliton hierarchy is given. Further, we construct two integrable coupling systems for the hierarchy by the conception of semidirect sums of Lie algebras.     

SYMMETRY CONSTRAINT OF THE LEVI EQUATIONS BY BINARY NONLINEARIZATION

In this paper, the translation of the Lax pairs of the Levi equations is presented. Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems. Finally, the involutive solutions of the Levi equations are presented.     

A quest for the integrable equation in 3+1 dimensions

The L and T operators of the Korteweg-de Vries equation are modified to seek a (3+1)-dimensional integrable equation. However, the Lax equation in this case is eventually reduced to a (2+1)-dimensional equation. We also propose other modified equations and their Lax pairs. A similar attempt is made to derive a higher-dimensional Harry Dym (HD) equation. As a result, a new (2+1)-dimensional HD equation is presented. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 305–309, February, 2000.     

一个特征值问题的非线性化及其生成的经典可积系统

Under the Bargmann constrained condition, the spatial part of a new Lax pairof the higher order MkdV equation is nonlinearized to be a completely integrable system(R^2N,dp∧dq, H0=1/2F0)(F0=〈Aq,p〉 〈Ap, p〉 〈p,q〉^2). While the nonlinearization of the time part leads to its N-involutive system (Fm).     

Some types of solutions and generalized binary Darboux transformation for the mKP equation with self-consistent sources

In this paper, an mKP equation with self-consistent sources (mKPESCSs) is structured in the framework of the constrained mKP equation. Based on the conjugate Lax pairs, we construct the generalized binary Darboux transformation and the N-times repeated Darboux transformation with arbitrary functions at time t for the mKPESCSs which offers a non-auto-Bäcklund transformation between two mKPESCSs with different degrees of sources. With the help of these transformations, some new solutions for the mKPESCSs such as soliton solutions, rational solutions, breather type solutions and exponential solutions are found by taking the special initial solution for auxiliary linear problems and the special functions of t-time.     

Isothermic Surfaces in Spaces of Arbitrary Dimension: Integrability, Discretization, and Bäcklund Transformations—A Discrete Calapso Equation

A vector analog of the classical Calapso equation governing isothermic surfaces in R ^{n +2} is introduced. It is shown that this vector Calapso system admits a nonlocal) scalar Lax pair based on the classical Moutard equation. The analog of Darboux's Bäcklund transformation for isothermic surfaces in R³ is derived in a systematic manner and shown that it may be formulated in terms of the classical Moutard transformation acting on the scalar Lax pair. A permutability theorem for isothermic surfaces is set down that manifests itself in an explicit superposition principle for the vector Calapso system. This superposition principle in vectorial form is shown to constitute an integrable discretization of the vector Calapso system and, therefore, defines discrete isothermic surfaces in R ^{n +2}. The discrete Calapso equation is related to the discrete Korteweg–de Vries equation and discrete holomorphic functions. A matrix Lax pair based on Clifford algebras and a scalar Lax pair are derived for the discrete Calapso equation. A discrete Moutard-type transformation for the discrete Calapso equation is obtained, and it is shown that the discrete Calapso equation may be specialized to an integrable discrete version of the O( n +2) nonlinear σ-model.     

Multi-component Wronskian solution to the Kadomtsev-Petviashvili equation

It is known that the Kadomtsev-Petviashvili (KP) equation can be decomposed into the first two members of the coupled Ablowitz-Kaup-Newell-Segur (AKNS) hierarchy by the binary non-linearization of Lax pairs. In this paper, we construct the N-th iterated Darboux transformation (DT) for the second- and third-order m-coupled AKNS systems. By using together the N-th iterated DT and Cramer’s rule, we find that the KPII equation has the unreduced multi-component Wronskian solution and the KPI equation admits a reduced multi-component Wronskian solution. In particular, based on the unreduced and reduced two-component Wronskians, we obtain two families of fully-resonant line-soliton solutions which contain arbitrary numbers of asymptotic solitons as y → ?∞ to the KPII equation, and the ordinary N-soliton solution to the KPI equation. In addition, we find that the KPI line solitons propagating in parallel can exhibit the bound state at the moment of collision.     

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.     

Bäcklund transformation,infinite conservation laws and periodic wave solutions to a generalized (2+1)-dimensional Boussinesq equation

Under investigation in this paper is a generalized (2+1)-dimensional Boussinesq equation, which can be used to describe the water wave interaction. By using Bell polynomials, a lucid and systematic approach is proposed to systematically study the integrability of the equation, including its bilinear representation, soliton solutions, periodic wave solutions, Bäcklund transformation and Lax pairs, respectively. Furthermore, by virtue of its Lax equations, the infinite conservation laws of the equation are also derived with the recursion formulas. Finally, the asymptotic behavior of periodic wave solutions is shown with a limiting procedure.     

COUPLED NONLOCAL NONLINEAR SCHR?DINGER EQUATION AND N-SOLITON SOLUTION FORMULA WITH DARBOUX TRANSFORMATION

Ablowitz and Musslimani proposed some new nonlocal nonlinear integrable equations including the nonlocal integrable nonlinear Schr?dinger equation. In this paper, we investigate the Darboux transformation of coupled nonlocal nonlinear Schr?dinger(CNNLS) equation with a spectral problem. Starting from a special Lax pairs, the CNNLS equation is constructed. Then, we obtain the one-, two-and N-soliton solution formulas of the CNNLS equation with N-fold Darboux transformation. Based on the obtained solutions, the propagation and interaction structures of these multi-solitons are shown, the evolution structures of the one-dark and one-bright solitons are exhibited with N = 1,and the overtaking elastic interactions among the two-dark and two-bright solitons are considered with N = 2. The obtained results are different from those of the solutions of the local nonlinear equations. Some different propagation phenomena can also be produced through manipulating multi-soliton waves.The results in this paper might be helpful for understanding some physical phenomena described in plasmas.     

Necessary Covariance Conditions for a One-Field Lax Pair

We study the covariance with respect to Darboux transformations of polynomial differential and difference operators with coefficients given by functions of one basic field. In the scalar (Abelian) case, the functional dependence is established by equating the Frechet differential (the first term of the Taylor series on the prolonged space) to the Darboux transform; a Lax pair for the Boussinesq equation is considered. For a pair of generalized Zakharov-Shabat problems (with differential and shift operators) with operator coefficients, we construct a set of integrable nonlinear equations together with explicit dressing formulas. Non-Abelian special functions are fixed as the fields of the covariant pairs. We introduce a difference Lax pair, a combined gauge-Darboux transformation, and solutions of the Nahm equations.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 122–132, July, 2005.