Loewner Transformations: Adjoint and Binary Darboux Connections

It is shown that a class of finite Bäcklund transformations introduced by Loewner in 1950 in a gasdynamics context may be represented as compound gauge and Darboux-type transformations. This result is used to construct iterated versions of the Loewner transformations based on established procedures in soliton theory.     

Binary Darboux Transformations of The Supersymmetric Heisenberg Magnet Model

Theoretical and Mathematical Physics - We investigate the standard binary Darboux transformation of the supersymmetric Heisenberg model and calculate multisoliton solutions of the quasi...     

Binary Representations of Finite Fields and Their Application to Complexity Theory

Binary representations of finite fields are defined as an injective mapping from a finite field tol-tuples with components in {0, 1} where 0 and 1 are elements of the field itself. This permits one to study the algebraic complexity of a particular binary representation, i.e., the minimum number of additions and multiplications in the field needed to compute the binary representation. The two-way complexity of a binary representation is defined as the sum of the algebraic complexities of the binary representation and of its inverse mapping. Two particular binary representations are studied: the standard representation and the logarithmic representation. A method of surrogate computation is developed and used to deduce relationships between the algebraic complexities of certain functions. The standard representation of a finite field is shown to be among the two-way easiest representations of this field. In particular, the standard representation of a finite field with characteristicpis two-way easy wheneverp− 1 has only small prime factors. For any finite field having a two-way easy binary representation, the algebraic complexity in this field is shown to be essentially equivalent to Boolean circuit complexity. For any finite field, the Boolean circuit complexity of Zech's (or Jacobi's) logarithm is shown to be closely related to the Boolean circuit complexity of the discrete logarithm problem that is used in public-key cryptography.     

Darboux Transformations from n-KdV to KP

The iterated Darboux transformations of an ordinary differential operator are constructively parametrized by an infinite-dimensional Grassmannian of finitely supported distributions. In the case that the operator depends on time parameters so that it is a solution to the n-KdV hierarchy, it is shown that the transformation produces a solution of the KP hierarchy. The standard definitions of the theory of -functions are applied to this Grassmannian and it is shown that these new -functions are quotients of KP -functions. The application of this procedure for the construction of higher rank KP solutions is discussed.     

Elementary Darboux Transformations and Factorization

We prove a general theorem on factorization of polynomial matrices and use it to reduce polynomial Darboux matrices to linear ones. We discuss some new examples of linear Darboux matrices.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 143–152, July, 2005.     

扩展的可积模型及其约化与达布变换

In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras?H and?T, respectively. By using the loop algebra?H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra?T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra?H to obtain an integrable hierarchy with variable coefficients.     

Bcklund变换与n孤子解

本文首先证明了KdV方程与sine-Gordon方程不同形式的Backlund变换是相互等价的;其次从双线性导数形式的Backlund变换出发给出多孤子解的Hirota表示与Wronski行列式表示,并利用Vandermonde行列式说明这两种孤子解的表示是一致的.     

Lax Pairs and Darboux Transformations for Euler Equations

In this article, we will report the recent developments on Lax pairs and Darboux transformations for Euler equations of inviscid fluids.     

Structure of Binary Darboux-Type Transformations for Hermitian Adjoint Differential Operators

For Hermitian adjoint differential operators, we consider the structure of Darboux–Bäcklund-type transformations in the class of parametrically dependent Hilbert spaces. By using the proposed new method, we obtain the corresponding integro-differential symbols of the operators of transformations in explicit form and consider the problem of their application to the construction of two-dimensional Lax-integrable nonlinear evolution equations and their Darboux–Bäcklund-type transformations.     

Ghost Symmetries and Multi-fold Darboux Transformations of Extended Toda Hierarchy

In this paper, the author constructs ghost symmetries of the extended Toda hierarchy with their spectral representations. After this, two kinds of Darboux transformations in different directions and their mixed Darboux transformations of this hierarchy are constructed. These symmetries and Darboux transformations might be useful in GromovWitten theory of CP¹.     

B?cklund变换与n孤子解

本文首先证明了KdV方程与sine-Gordon方程不同形式的B?cklund变换是相互等价的;其次从双线性导数形式的B?cklund变换出发给出多孤子解的Hirota表示与Wronski行列式表示,并利用Vandermonde行列式说明这两种孤子解的表示是一致的.     

广义耦合KdV孤子方程的达布变换及其偶孤子解

根据广义耦合KdV孤子方程的Lax对, 借助谱问题的规范变换, 一个包含多参数的达布变换被构造出来. 利用达布变换来产生广义耦合KdV孤子方程的偶孤子解, 并且用行列式的形式来表达广义耦合KdV孤子方程的偶孤子解. 作为应用, 广义耦合KdV孤子方程的偶孤子解的前两个例子被给出.     

一类变系数Boussinesq型方程的Darboux变换及多孤子解

主要讨论在某种约束下,变系数Boussinesq型方程和变系数Broer-KaupKupershmidt方程之间的联系,构造变系数Broer-Kaup-Kupershmidt方程的另外一种Darboux变换,且应用Darboux变换得到变系数Boussinesq型方程的孤子解.     

广义耦合KdV孤子方程的达布变换及其奇孤子解

借助谱问题的规范变换, 给出广义耦合KdV孤子方程的达布变换,利用达布变换来产生广义耦合KdV孤子方程的奇孤子解,并且用行列式的形式来表达广义耦合KdV孤子方程的奇孤子解．作为应用,广义耦合KdV孤子方程奇孤子解的前两个例子被给出．     

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.     

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.     

演化方程的Darboux变换的一类求法

本文给出了与文 [1 ]中的一个特征值问题相关的一类演化方程的三类 Darboux变换 ,并且也是 Backlund变换     

B(a)cklund变换与n孤子解

本文首先证明了KdV方程与sine-Gordon方程不同形式的B(a)cklund变换是相互等价的;其次从双线性导数形式的B(a)cklund变换出发给出多孤子解的Hirota表示与Wronski行列式表示,并利用Vandermonde行列式说明这两种孤子解的表示是一致的.     

Integrable Potentials by Darboux Transformations in Rings and Quantum and Classical Problems

We study a problem in associative rings of left and right factorization of a polynomial differential operator regarded as an evolution operator. In a direct sum of rings, the polynomial arising in the problem of dividing an operator by an operator for two commuting operators leads to a time-dependent left/right Darboux transformation based on an intertwining relation and either Miura maps or generalized Burgers equations. The intertwining relations lead to a differential equation including differentiations in a weak sense. In view of applications to operator problems in quantum and classical mechanics, we derive the direct quasideterminant or dressing chain formulas. We consider the transformation of creation and annihilation operators for specified matrix rings and study an example of the Dicke model.