Riccati微分方程的可积条件

Ｉｎ１９９８,ＺｈａｏＬｉｎｌｏｎｇ［１］ｏｂｔａｉｎｅｄｔｈｅｉｎｔｅｇｒａｂｌｅｃｏｎｄｉｔｉｏｎ：Ｒ＝１αγＰｅ２∫（Ｑ－βＤ）ｄｘ　　　（α,β,γｉｓｃｏｎｓｔ）．（１）ＦｏｒＲｉｃｃａｔｉｅｑｕａｔｉｏｎ：ｙ′＝ｐ（ｘ）ｙ２＋Ｑ（ｘ）ｙ＋Ｒ（ｘ）　　（ＰＲ≠０）．（２）　　Ｈｅｒｅｔｈｅｎｅｗｉｎｔｅｇｒａｂｌｅｃｏｎｄｉｔｉｏｎｓｉｓｇｉｖｅｎ：Ｌ［ｙ０］＝１αγＰｅ２∫（Ｑ＋２ｙ０ｐ－βＤ）ｄｘ．（３）Ｌ［ＡＢ＋ｙ０］＝１αγ（ＡＢ）２Ｌ［ｙ０］ｅ２∫（２ＢＡＬ［ｙ０］＋Ｑ＋２ｙ…     

Lenard Chains for Classical Integrable Systems

A simple example shows the unexpected role of Lenard chains in the theory of classical integrable systems.     

一类广义Riccati方程的三个可积判据

考虑一类广义Riccati方程,通过函数变换,在所给条件下,将这类方程等价地化为变量分离方程,从而得到了该方程可积的三个充分性判据,并给出方程通解的参数表达形式,扩大了Riccati方程的可解性范围.     

关于一类广义Riccati方程三个可积判据的注记

通过变量变换的方法,将文[1]中一类广义Riccati方程的三个充分性判据统一起来,并加以推广.充分利用参数λ,κ的可变性揭示该结果与现有Riccati方程可积性间的关系,扩大了Riccati方程的可积性范围.     

Pritchard-Salamon系统的可稳定化性和代数Riccati方程

在本文中,我们给出了光滑Pritchard－Salamon系统（简称PS系统）能以紧算子为可容反馈可稳定化的充分必要条件．应用此结果,我们给出了光滑PS系统中的代数Riccati方程的所有非负自伴解的参数化表示,把[1]中的主要结果推广到了光滑PS系统．     

Integrable Systems and Rank-One Conditions for Rectangular Matrices

We give a determinantal formula for tau functions of the KP hierarchy in terms of rectangular constant matrices A, B, and C satisfying a rank-one condition. This result is shown to generalize and unify many previous results of different authors on constructions of tau functions for differential and difference integrable systems from square matrices satisfying rank-one conditions. In particular, its explicit special cases include Wilson's formula for tau functions of the rational KP solutions in terms of Calogero–Moser Lax matrices and our previous formula for the KP tau functions in terms of almost-intertwining matrices.     

Construction of Separation Variables for Finite-Dimensional Integrable Systems

We consider a class of integrable systems such that solutions of the corresponding Hamilton–Jacobi equation depend on n+m arbitrary parameters and are represented as products of flat curves. The first n parameters are identified with the values of the integrals of motion. The remaining parameters enter the definition of the integrals of motion as arbitrary constants (charges) and can be used to find separation variables. We show that on the coadjoint orbits of Lie groups, the Casimir operators not only generate a family of integrals but also allow constructing separation variables.     

两个可积系统及其约化

该文引入了一个李代数,然后定义了其相应的两个圈代数,利用圈代数构造了两个等谱问题,其相容性条件导出了两个可积动力系统.通过约化这样的系统,得到了某些有趣的非线性方程,如Burgers方程、组合KdV-MKdV方程和Kuramoto-Sivashinsky方程以及KdV方程的一种推广形式.最后,利用贝尔多项式讨论了广义KdV方程的可积性质,包括双线性形式、Lax对、贝克隆变换和无穷守恒律等.     

Associative Cones and Integrable Systems

We identify R7 as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S6. It is known that a cone over a surface M in S6 is an associative submanifold of R7 if and only if M is almost complex in S6. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S6 are the equation for primitive maps associated to the 6-symmetric space G2/T2, and use this to explain some of the known results. Moreover, the equation for S1-symmetric almost complex curves in S6 is the periodic Toda lattice, and a discussion of periodic solutions is given.     

Associative Cones and Integrable Systems

Abstract We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S⁶. It is known that a cone over a surface M in S⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ are the equation for primitive maps associated to the 6-symmetric space G₂=T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0529756.     

Autonomous Planar Systems of Riccati Type

The role of Riccati type systems in the plane along with the related linear, second order differential equation is examined. If $x$ and $y$ are the variables of the Riccati differential equation, then any integrable Riccati system has two independent invariant curves dependent upon these variables whose nature is easily determined from the solution of the linear equation. Each of these curves has the same cofactor. Other invariant curves depend upon $x$ alone and are shown to be less important. The systems have both Liouvillian and non--Liouvillian solutions and are easily transformable to symmetric systems. However, systems derived from them may not be symmetric in their transformed variables. Several systems from the literature are discussed with regard to the forms of the invariant curves presented in the paper. The relation of certain Riccati type systems is considered with respect to Abel differential equations.     

Real Forms of Elliptic Integrable Systems

Theoretical and Mathematical Physics - We describe the real forms of classical elliptic integrable systems such as the elliptic Calogero-Moser system and the elliptic Euler-Arnold top in the...     

Noncommutative Integrable Systems of Hydrodynamic Type

On noncommutative spaces, integrable hierarchies of hydrodynamic type systems (1-order quasilinear PDE’s) do not, in general, exist. Nevertheless, an infinite-component hydrodynamic chain defined below is shown to be integrable. Its modified version is also constructed and it exhibits a new purely noncommutative phenomenon: the number of modified variables is either or .     

Kinetic Equations and Integrable Hamiltonian Systems

A survey of interrelations between kinetic equations and integrable systems is presented. We discuss common origin of special classes of solutions of the Boltzmann kinetic equation for Maxwellian particles and special solutions for integrable evolution equations. The thermodynamic limit and soliton kinetic equation for the integrable Korteweg-de Vries equation are considered. The existence of decaying and degenerate dispersion laws and the appearance of additional integrals of motion for the interacting waves is discussed. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 731–741, June, 2005.     

Integrable Systems in Three-Dimensional Riemannian Geometry

Summary. In this paper we introduce a new infinite-dimensional pencil of Hamiltonian structures. These Poisson tensors appear naturally as the ones governing the evolution of the curvatures of certain flows of curves in 3-dimensional Riemannian manifolds with constant curvature. The curves themselves are evolving following arclength-preserving geometric evolutions for which the variation of the curve is an invariant combination of the tangent, normal, and binormal vectors. Under very natural conditions, the evolution of the curvatures will be Hamiltonian and, in some instances, bi-Hamiltonian and completely integrable. Received May 31, 2001; accepted January 2, 2002 Online publication March 11, 2002 Communicated by A. Bloch Communicated by A. Bloch rid="     

The Mirror Systems of Integrable Equations

We provide an algorithm to convert integrable equations to regular systems near noncharacteristic, movable singularity manifolds of solutions. We illustrate how the algorithm is equivalent to the Painlevé test. We also use thealgorithm to prove the convergence of the Laurent series obtained from the Painlevé test.     

应用拓展的Riccati展开法求非线性发展方程的精确解

借助Mathematica符号计算软件,利用拓展的Riccati展开法和变量分离法,得到非线性发展方程的精确解.通过选择适当的函数,获得(2+1)维色散长波方程的亮暗dromion解.