KP系统Lax算子及主对称的换位公式

本文讨论的是ＫＰ系统Ｌａｘ算子及主对称的换位公式．通过拓广速降函数空间及对 ＫＰ方程 Ｌａｘ算子的讨论,找到了 Ｌａｘ算子的表示向量;并通过对 Ｌａｘ算子、 Ｌａｘ流、 Ｌａｘ算子表示向量之间联系的讨论,得出了计算 Ｌａｘ算子李括号的表示向量的方法,从而解决了 ＫＰ方程主对称的换位公式问题．最后本文还利用伴随算子给出了从ＫＰ方程任一主对称得到其一个对称的公式．     

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．     

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．     

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．     

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．     

一个演化方程族约束系统的r-矩阵

本文给出一个演化方程约束系统的Ｌａｘ表示,ｒ－矩阵,经典Ｐｏｉｓ－ｓｏｎ结构与可积性的证明。     

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。     

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。     

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．     

积分型Darboux变换

Ｄａｒｂｏｕｘ变换是求解孤立子方程的有力工具,针对方程的特殊性（如与方程相联系的Ｌａｘ对受到某种限制）需建立相应的Ｄａｒｂｏｕｘ变换本文通过引进一种新的积分型Ｄａｒｂｏｕｘ变换,用来求解一类Ｌａｘ对为反自共轭算子的孤立子方程,并给出了这种新的Ｄａｒｂｏｕｘ变换的迭代式     

发展方程族Lax表示的广义结构*

本文给出非线性发展方程族的一个生成格式(该格式包含了保谱族与非保谱族作为其两个特殊情况),并提供该格式下发展方程族Lax表示的广义结构.最后,作为应用,我们讨论了Levi族发展方程.     

Involutive Solutions and Commutator Representations of the Dirac Hierarchy

ConsidertheDiracspectralproblemwherep,qaretwopotentials,Aisaspectralparameter.L*isaninjectivehomomorphism.ThefunctionalgradientVA=(2RR,ri-of)TofeigenvalueAwithrespecttop,qsatisfiesarecalledtheLenard'soperatorpairof(1).Theorem1LetG(1)(x),G(z)(x)betwoarbitarysmoothfunctions,G=(G(1),G(2))".ThenthefollowingoperatorequationwithrespecttoV=V(G),possessestheoperatorsolutionwhereL.'-jisthecommutator;L=L(p,q),K,Jaredefinedby(1)l(4)respectively.ProofSubstitute(6)into(5),directlycalculate.Defin…     

Complete integrability,gauge equivalence and Lax representation of inhomogeneous nonlinear evolution equations

The gauge equivalence between the inhomogeneous versions of the nonlinear Schrödinger and the Heisenberg ferromagnet equations is studied. An unexplicit criterion for integrability is proposed. Examples of gauge equivalent inhomogeneous nonlinear evolution equations are presented. It is shown that in the nonintegrable cases the M-operators in their Lax representations possess unremovable pole singularities lying on the spectrum of the L-operators.To the memory of Michail Constantinovich PolivanovInstitute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria. Published in Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 3, pp. 374–386, September, 1992.     

Jacobi polynomials and Lax representation for completely integrable dynamical systems

We consider a method aimed at the investigation of completely integrable dynamical systems by using the Lax representation of their equations of motion. The Lax representations are found for the integrable case of the Henon-Heiles system and for an anisotropic oscillator.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1088–1091, August, 1994.     

Cyclic Bases of Zero-Curvature Representations: Five Illustrations to One Concept

The paper contains five examples of using cyclic bases of zero-curvature representations in studies of weak and strong Lax pairs, hierarchies of evolution systems, and recursion operators.     

Lie algebras and Lax equations with spectral parameter on an elliptic curve

Lax equations with elliptic spectral parameter are considered in the framework of the general scheme using affine Lie algebras and the r-matrix formalism. A number of examples of elliptic Lax representations for systems of interacting tops are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 150, pp. 104–118, 1986.     

Darboux transformations and recursion operators for differential-difference equations

We review two concepts directly related to the Lax representations of integrable systems: Darboux transformations and recursion operators. We present an extensive list of integrable differential-difference equations with their Hamiltonian structures, recursion operators, nontrivial generalized symmetries, and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra-type equations and integrable discretizations of derivative nonlinear Schrödinger equations such as the Kaup-Newell, Chen-Lee-Liu, and Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattices. We also compute the weakly nonlocal inverse recursion operators.     

Quasi‐Periodic Solutions of Nonlinear Evolution Equations Associated with a 3 × 3 Matrix Spectral Problem

A hierarchy of new nonlinear evolution equations associated with a 3 × 3 matrix spectral problem with three potentials is proposed. With the aid of the characteristic polynomial of Lax matrix for the hierarchy, we introduce an algebraic curve of arithmetic genus m ? 1 , and discuss in detail the properties of the associated Baker–Akhiezer function and meromorphic function. On the basis of the theory of algebraic curves, we obtain the explicit theta function representations of the Baker–Akhiezer function, the meromorphic function, and, in particular, that of solutions for the entire hierarchy of nonlinear evolution equations.