双约束孤立子流的可积形变

提出了基于Lax矩阵的构造双约束孤立子流的可积形变的新方法.作为应用,导出了双约束KdV流和双约束mKdV流的可积形变,并给出了这些形变的Lax表示、r-矩阵和守恒积分.     

关于AKNS族和一个耦合的MKdV族的规范等价可积系统与r-矩阵

得到了一个新的耦合的MKdV族．通过规范变换，首次从AKNS族中得到耦合的MKdV族的Lax表示、可积系与约束流；利用Lax表示，构造了耦合MKdV族的约束流的γ-矩阵，同时也给出了该方程族约束流的第二组守恒积分与对合性．     

Neumann型约束Tu流的r矩阵和分离变量

本文发展了适用于Neumann型约束流的r矩阵方法．研究了Neumann型约束Tu流．获得了Neumann型约束Tu流的Lax表示；证明了在Dirac括号下这个Lax算子满足r矩阵关系，从而再次证明了Neumann型约束Tu流是Liouville完全可积的．另外，还利用r矩阵关系，找到了这个Neumann型约束流的一组正则变量，从而把Neumann型约束Tu流线性化．     

由伴随坐标得到的Dirac族的可积约束流

引入伴随坐标建立了Dirac族的某些非正则高阶约束流及其对应的Lax表示和r-矩阵,并证明这些约束流在Liouville意义下是完全可积的．     

超Broer-Kaup-Kupershmidt族的双非线性化

得出了超Broer-Kaup- Kupershmidt族Lax对的对称约束及其双非线性化.在得到的对称约束F,把超Broer- Kaup-Kupershmidt族的n阶流分解成定义在对应于动力变量x和tn的超对称流形上的两种超有限维可积Hamilton系统.此外,显式给出了Liouville可积性所需的运动积分.     

一个3×3谱问题产生的约束系统的动力r-矩阵

给出一个3×3谱问题产生的Harry-Dym型方程族的约束系统的Lax表示,动力r-矩阵及Poisson结构,并给出3N个守恒积分.从而利用一般r-矩阵理论证明了该约束系统在Liouville意义下的完全可积性.     

超Li系统的Bargmann对称约束和双非线性化

给出超Li系统的一个显式对称约束和关于其Lax对的双非线性化.在此对称约束下,超Li系统的时间部分和空间部分分别被约化为有限维Liouville可积超Li系统.     

关于广义Toda方程的r-矩阵方法

本文利用 r-矩阵及 Lie 双代数方法研究广义 Toda 方程的 Lax 表示,完全可积性以及解曲线的性质.     

一个3×3谱问题产生的约束系统的动力r-矩阵

给出一个3×3谱问题产生的Harry—Dym型方程族的约束系统的Lax表示，动力r矩阵及Poisson结构，并给出3N个守恒积分．从而利用一般，r-矩阵理论证明了该约束系统在Liouville意义下的完全可积性．     

Q-mKP系列流方程的等价形式

Q-形变的modified Kadomtsev-Petviashvili(q-mKP)系列是经典mKP系列的量子化推广,其流方程包括无穷多个微分方程簇,流方程的等价形式是一个广为关注的问题.类似于对mKP系列的研究,尝试沿着Sato理论框架,基于其Lax算子、Lax方程,给出该可积系列流方程的等价形式,这些结果显示出q-mKP系列与mKP系列的不同,并不是mKP系列的简单,是进一步探讨其递归算子、代数约束等可积性质的基础.     

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.     

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.     

Shock waves in the modified Korteweg-de Vries-Burgers equation

Summary The asymptotics ast + of shock waves of the modified Korteweg-de Vries-Burgers (MKdV-B) equation is investigated. An attractor interpretation of shock problems for integrable systems is presented and some problems of nonlinear stability are discussed. The MKdV-B equation is considered as a nonconservative perturbation of the integrable modified Korteweg-de Vries (MKdV) equation. The MKdV equation considered here has anon-self-adjoint Lax pair. In spite of this difficulty, acomplex Whitham deformation is constructed.     

New hierarchy of integrable system bi-Hamiltonian structure and constrained flows

We have considered the hierarchy of integrable systems associated with the unstable nonlinear Schrodinger equation. The spectral gradient approach and the trace identity are used to derive the bi-Hamiltonian structure of the system. The bi-Hamiltonian property and the square eigenfunctions determined via the spectral gradient approach are then used to construct constrained flows, which is also proved to be derivable from a rational Lax operator. This new Lax operator of the constrained flows is seen to generate the classical r-matrix. Lastly it is also explicitly demonstrated that the different integrals of motion of the constrained flows Poisson commute.     

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.     

The discrete Korteweg-de Vries equation

We review the different aspects of integrable discretizations in space and time of the Korteweg-de Vries equation, including Miura transformations to related integrable difference equations, connections to integrable mappings, similarity reductions and discrete versions of Painlevé equations as well as connections to Volterra systems.     

Tu方程族的高阶双约束流的分离变量

本文给出了Tu方程族的高阶双约束流,其自由度为2N＋l。根据通常的办法,利用Lax矩阵仅能引入N＋l对标准分离变量和N＋l个分离变量方程。本文构造出另外N对分离变量及N个分离变量方程。此外,还建立了双约束流和Tu方程族的Jacobi反演问题。     

Five constructions of integrable dynamical systems connected with the Korteweg-de Vries equation

Two countable sets of integrable dynamical systems which turn into the Korteweg-de Vries equation in a continous limit are constructed. The integrability of the dynamics of the scattering matrix entries for these systems is proved and an integrable reduction in the finitedimensional case is pointed out. A construction of the integrable dynamical systems connected with the simple Lie algebras and generalizing the discrete kdV equation is presented. Two general constructions of differential and integro-differential equations (with respect to time t) possessing a countable set of first integrals are found. These equations admit the Lax representation in some infinite-dimensional subalgebras of the Lie algebra of integral operators on an arbitrary manifold M ⁿ with measure . A construction of matrix equations having a set of attractors in the space of all matrix entries is given.