具有Lenard递推结构之发展方程族的Lax表示

该文对具有Ｌｅｎａｒｄ递推结构的发展方程族，通过转换速推结构为一个算子方程给出了Ｌａｘ表示的一种理论描述．这种描述可应用于各种１＋１维可积系统族Ｌａｘ表示的寻求之中，本文仅就ＫｄＶ可积族和Ａｎｔｏｎｏｗｉｃｚ—Ｆｏｒｄｙ可积族详细阐述了Ｌａｘ表示的具体构造过程．     

4×4AKNS方程族的Lax表示及其对合解

本文获得了一族４×４ＡＫＮＳ方程的Ｌａｘ表示．用Ｌａｘ组非线性化的方法得到了辛流形中的一组对合系，从而可以从两个非线性常微分方程组的解得到４×４ＡＫＮＳ非线性偏微分方程组的解．     

谱问题、发展方程族及Lax表示的等价性

讨论了四对谱问题、相应的保谱方程族以及它们Ｌａｘ表示的等价住，并利用这种等价关系可导出一些方程的Ｌａｘ表示．     

D—AKNS族的换位表示

本文根据曹策问教授的想法,求得了与D-AKNS族发展方程相联系的特征值之泛函梯度与Lenard算子对;并由此得到了D-AKNS族非线性发展方程的换位表示。文末还讨论了换位表示与定态D-AKNS方程之间的关系.     

一个有限维完全可积系统和MDWW方程解的对合表示

本文讨论高阶ＭＤＷＷ方程的Ｌａｘ对，在位势与特征函数之间的约束条件下，Ｌａｘ系统被非线性化成为有限维Ｌｉｏｕｖｉｌｌｅ完全可积系统．并且获得了高阶ＭＤＷＷ方程解的对合表示．     

Lax方程组的求解公式

本文给出了一些著名的非线性演化方程的Ｌａｘ方程组求解公式．利用这些公式，只需算积分就能不断求得这些方程的解．从而提供了利用Ｌａｘ方程组求解非线性方程的新途径．     

4＊4AKNS方程族的Lax表示及其对合解

本文获得了一族４＊４ＡＫＮＳ方程的Ｌａｘ表示，用Ｌａｘ组非线性化的方法得到了辛流形中的一组对合系，从而可以从两个非线性常微分方程组的解得到４＊４ＡＫＮＳ非线性偏微分方程组的解。     

一族4×4AKNS方程的对合解

本文获得了一族ＡＫＮＳ方程及其Ｌａｘ表示．用文献［２］和［３］的方法证明了可以从两个方面相容的常微分方程组的相容解构造出４×４ＡＫＮＳ非线性偏微分方程的解。     

Lax表示的变形与Hamilton方程族的Lax表示

本文首先给出了构造演化方程族的Ｌａｘ表示的马文秀方法的一种变形，后对这一方法作了改进，使之适用于Ｈａｍｉｌｔｏｎ形式的方程族．作为应用，得到了具有非等谱Ｌａｘ表示的杨方程族．     

带附加项的AKNS方程族的Darboux变换

本文了带附加项的ＡＫＮＳ方程族，给出了其Ｌａｘ表示，守恒律和Ｄａｒｂｏｕｘ变换。     

Dynamics and interpretation of some integrable systems via multiple orthogonal polynomials

High-order non-symmetric difference operators with complex coefficients are considered. The correspondence between dynamics of the coefficients of the operator defined by a Lax pair and its resolvent function is established. The method of investigation is based on the analysis of the moments for the operator. The solution of a discrete dynamical system is studied. We give explicit expressions for the resolvent function and, under some conditions, the representation of the vector of functionals, associated with the solution for the integrable systems.     

Integrable Systems Obtained by Puncture Fusion from Rational and Elliptic Gaudin Systems

Using the procedure for puncture fusion, we obtain new integrable systems with poles of orders higher than one in the Lax operator matrix and consider the Hamiltonians, symplectic structure, and symmetries of these systems. Using the Inozemtsev limit procedure, we find a Toda-like system in the elliptic case having nontrivial commutation relations between the phase-space variables.     

Invariant manifolds and Lax pairs for integrable nonlinear chains

We continue the previously started study of the development of a direct method for constructing the Lax pair for a given integrable equation. This approach does not require any addition assumptions about the properties of the equation. As one equation of the Lax pair, we take the linearization of the considered nonlinear equation, and the second equation of the pair is related to its generalized invariant manifold. The problem of seeking the second equation reduces to simple but rather cumbersome calculations and, as examples show, is effectively solvable. It is remarkable that the second equation of this pair allows easily finding a recursion operator describing the hierarchy of higher symmetries of the equation. At first glance, the Lax pairs thus obtained differ from usual ones in having a higher order or a higher matrix dimensionality. We show with examples that they reduce to the usual pairs by reducing their order. As an example, we consider an integrable double discrete system of exponential type and its higher symmetry for which we give the Lax pair and construct the conservation laws.     

Factorization of the $$ \mathcal{R} $$-matrix for the quantum algebra Uq( sℓ3)

The Yang-Baxter operator is obtained as a product of operators that permute representation parameters in the Lax operators. The construction relies on a factorization of the Lax operator into triangular matrices. Bibliography: 13 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 88–106.     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

微分特征列法在拟微分算子和Lax表示中的应用

微分特征列法用于拟微分算子和非线性发展方程Lax表示的计算.首先,利用微分特征列法和微分带余除法计算拟微分算子的逆和方根,由于不必求解常微分方程组,并将解代入,因此,使得计算得以简化.其次,利用微分特征列法,约化从广义Lax方程和Zakharov-Shabat推出的非线性偏微分方程,并得到相应的非线性发展方程.在Mathematica计算机代数系统上,编写了相关程序,从而可以利用计算机辅助完成一些非线性发展方程Lax表示的计算.     

Construction of a new class of quantum integrable inhomogeneous models

Integrable inhomogenous or impurity models are usually constructed by either shifting the spectral parameter in the Lax operator or using another representation of the spin algebra. We propose a more involved general method for such construction in which the Lax operator contains generators of a novel quadratic algebra, a generalization of the known quantum algebra. In forming the monodromy matrix, we can replace any number of the local Lax operators with different realizations of the underlying algebra, which can result in spin chains with nonspin impurities causing changed coupling across the impurity sites, as well as with impurities in the form of bosonic operators. Following the same idea, we can also generate integrable inhomogeneous versions of the generalized lattice sine-Gordon model, nonlinear Schrödinger equation, Liouville model, relativistic and nonrelativistic Toda chains, etc.     

Isospectral deformation of the reduced quasi-classical self-dual Yang–Mills equation

We derive a new four-dimensional partial differential equation with the isospectral Lax representation by shrinking the symmetry algebra of the reduced quasi-classical self-dual Yang–Mills equation and applying the technique of twisted extensions to the obtained Lie algebra. Then we find a recursion operator for symmetries of the new equation and construct a Bäcklund transformation between this equation and the four-dimensional Martínez Alonso–Shabat equation. Finally, we construct extensions of the integrable hierarchies associated to the hyper-CR equation for Einstein–Weyl structures, the reduced quasi-classical self-dual Yang–Mills equation, the four-dimensional universal hierarchy equation, and the four-dimensional Martínez Alonso–Shabat equation.     

On symmetries of the heat equation

Two aims are pursued in this article. The first one is methodological and consists of demonstrating the symmetry calculation techniques based on the commutation relation on the example of the quasi-linear heat equation. The second one consists of investigating the following open question: is the algebra of higher symmetries of the linear heat equation exhausted by those which may be obtained by means of the classical symmetries and well-known recursion operators? A positive solution of this question is given.     

An integrable nonlinear diffusion hierarchy with a two-dimensional arbitrary function

By introducing the hereditary condition to a general first order differential strong symmetry operator, we obtain general 1+1 and 2+1 dimensional integrable nonlinear diffusion hierarchies with infinitely many symmetries and Lax pairs. For a special example the infinitely many nonlocal conservation laws and some explicit and implicit exact solutions are also given.