On Darboux and singular Darboux transformations ofG-unitons

We present a unified method for constructing generalized flag transformations of unitons into a Lie groupG via singular Darboux transformations. These flag transformations provide the possibility to establish some factorizations forG-unitons.     

On Darboux and singular Darboux transformations of G-unitons

We present a unified method for constructing generalized flag transformations of unitons intoa Lie group G via singular Darboux transformations. These flag transformations provide the possibility toestablish some factorizations for G-unitons.     

Gaston Darboux

Sans résumé

---

Gaston Darboux 1842–1917

---

Traduction du discours prononcé le 12 mai 1917 à la séance publique annuelle de l'Académie des Sciences de Goettingue (Nachrichten der K. Gesellschaft der Wissenschaften zu G?ttinge, 1917, p. 71).     

On Darboux and singular Darboux transformations ofG-unitons

We present a unified method for constructing generalized flag transformations of unitons into a Lie groupG via singular Darboux transformations. These flag transformations provide the possibility to establish some factorizations forG-unitons.     

Two-sided Darboux transformations

We refine the notion of a Darboux transformation for differential operators of an order higher than two and consider commutative rings of differential operators that are invariant under the group of dilations of the independent variable.     

The Darboux point

A theory of global optimality based upon the Darboux-point concept is developed. A definition is proposed for the Darboux point, and the Darboux point is shown to exist on nonglobally optimal trajectories under relatively general conditions. A mutually exclusive classification of Darboux points is noted, and several properties are proved for one of these classes (the Type-1 Darboux point). Numerous examples are included to illustrate the Darboux-point definition and properties.     

Characterizations of Darboux selections

A characterization of those set-valued mappings from a real interval into the family of non-void intervals which admit a Darboux selection is given.     

关于Darboux定理

本文细致地讨论了 Darboux定理及有关导函数的几种情形 ,并指出了文献 [1 ]中的一点不足之处 .     

Higher-order Darboux transformations with foreign auxiliary equations and equivalence with generalized Darboux transformations

We show that a recently developed modified Darboux transformation that uses foreign auxiliary equations, can be unified with the Darboux transformation for generalized Schrödinger equations. As a consequence of this unification, we obtain explicit Darboux transformations with foreign auxiliary equations of arbitrary order.     

Broer-Kaup系统的达布变换及其孤子解

根据Broer-Kaup系统的Lax对, 借助Broer-Kaup系统的谱问题的规范变换, 一个包含多参数的达布变换被构造出. 以一个平凡解作为种子解, 利用达布变换, 可以求得Broer-Kaup系统的非平凡解的一般表达式. 并且讨论了N=1和N=2两种孤子解的情形. 这是一种与2X2谱问题有关的孤子碰撞图像的新类型.     

On Darboux continuity and continuity

Lipinski's (Colloq. Math.19 (1968), 251–253) theorem for a Darboux function to be continuous is extended and a question raised by him is partially answered.     

Monomial Derivations without Darboux Polynomials

In this paper, it is proved that a monomial derivation d of k[x, y, z] has no Darboux polynomials if and only if d is a strict derivation with a trivial ring of constants, and we give the specific conditions when it has no Darboux polynomials.     

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.     

Pseudo-Abelian integrals along Darboux cycles

We study polynomial perturbations of integrable, non-Hamiltoniansystem with first integral of Darboux-type with positive exponents.We assume that the unperturbed system admits a period annulus.The linear part of the Poincaré return map is given bypseudo-Abelian integrals. In this paper we investigate analyticproperties of these integrals. We prove that iterated variationsof these integrals vanish identically. Using this relation weprove that the number of zeros of these integrals is locallyuniformly bounded under generic hypothesis. This is a genericanalog of the Varchenko-Khovanskii theorem for pseudo-Abelianintegrals. Finally, under some arithmetic properties of exponents,the pseudo-Abelian integrals are a sum over exponents a_j ofpolynomials in log h with meromorphic functions of h^1/a_j ascoefficients.     

用区间套定理证明Darboux定理

Darboux定理是数学分析中的一个重要定理.在已有文献的基础上,对该定理作了进一步的研究,利用区间套定理给出了它的新的证明方法.证明思路与现有的其它证明思路是不同的.     

Darboux transformation and two-dimensional Toda lattice

The Darboux transformation method is used to construct multisoliton solutions of an infinite two-dimensional Toda lattice, which depend on an arbitrary number of functional parameters.