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 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.     

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     

Rank 2 commuting ordinary differential operators and Darboux conjugates of KdV

By considering the factorizations (flags) and associated (simultaneous) second order Darboux transformations of the square and cube of an arbitrary second order Schrödinger operator, we generate commuting ordinary differential operators of orders four and six with a singular elliptic spectrum. This procedure generates true rank 2 commutative algebras. Under the KdV flow, each such factorization (flag) leads to an integrable equation for which the corresponding Darboux transformation generates a Lax-type operator as one of a commuting pair of orders four and six with singular elliptic spectrum. Hence, these integrable equations are Darboux conjugates of KdV.     

Constructive factorizations of unitons via singular Darboux transformations

Some new factorizations of unitons into Lie groups are established via singular Darboux transformations. The factorization processes for Grassmannian unitons are also considered. Furthermore, a purely algebraic method for constructing Grassmannian unitons is presented.     

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.     

Darboux maps andD-modules

We give some results concerning Laplace transformations of linear systems of partial differential equations and show how the theory of such transformations is related to D-modules. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 163–169, February, 2000.     

Contact Relative Differential Invariants for Non Generic Parabolic Monge-Ampère Equations

We find relative differential invariants of different orders for non generic parabolic Monge-Ampère equations (MAE’s). They are constructed in terms of some tensors associated with the derived flag of the characteristic distribution. The vanishing of such invariants allows one to determine the classes of each non generic parabolic MAE with respect to contact transformations.     

一族Liouville可积系及其约束流的Lax表示、Darboux变换

利用屠规彰格式求出了一族Liouville可积系,通过高阶位势特征函数约束将可积系分解成x部分和t_n部分可积Hamilton系统,求出了该系统的Lax表示及三类Darboux变换。     

Binary Darboux transformations andN-wave systems in rings

The covariance theorems for elementary and binary Darboux transformations in rings are formulated and proved for generalized Zakharov-Shabat problems. The definition of the elementary Darboux transformation is extended to an arbitrary number of orthogonal idempotents. The binary transformation is defined as a sequence of elementary transformations for direct and conjugate problems. The heredity property for the reduction constraints is established for some UV pairs in rings; hence, the transformation generates solutions and infinitesimal symmetries of the corresponding zero-curvature equations. The explicit expressions for the transformations, solitons, and infinitesimals are given in the general case and in physically significant cases of extended non-AbelianN-wave equations (with linear terms added). Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 239–250, February, 2000.     

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.     

Positive and negative integrable hierarchies, associated conservation laws and Darboux transformation

Two hierarchies of integrable positive and negative lattice equations in connection with a new discrete isospectral problem are derived. It is shown that they correspond to positive and negative power expansions respectively of Lax operators with respect to the spectral parameter, and each equation in the resulting hierarchies is Liouville integrable. Moreover, infinitely many conservation laws of corresponding positive lattice equations are obtained in a direct way. Finally, a Darboux transformation is established with the help of gauge transformations of Lax pairs for the typical lattice soliton equations, by means of which the exact solutions are given.     

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.     

Darboux invariants of laplace transforms of normal congruences

In a biaxial space of the hyperbolic type in the case when Laplace transformations of a normal congruence are normal congruences the following theorem is proved: a focal net of lines on all focal surfaces of a sequence of Laplace transformations is a net R and the point and tangential Darboux invariants on all focal surfaces of the sequence of Laplace transformations, respectively equal to one another.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 11, pp. 1551–1557, November, 1991.     

Darboux transformations for the strict KP hierarchy

Theoretical and Mathematical Physics - We introduce the notion of Darboux transformations for the strict KP hierarchy. We previously showed that solutions of this integrable hierarchy can be...     

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.     

Vectorial Darboux Transformations for the Kadomtsev-Petviashvili Hierarchy

Summary. We consider the vectorial approach to the binary Darboux transformations for the Kadomtsev-Petviashvili hierarchy in its Zakharov-Shabat formulation. We obtain explicit formulae for the Darboux transformed potentials in terms of Grammian type determinants. We also study the n -th Gel'fand-Dickey hierarchy introducing spectral operators and obtaining similar results. We reduce the above-mentioned results to the Kadomtsev-Petviashvili I and II real forms, obtaining corresponding vectorial Darboux transformations. In particular for the Kadomtsev-Petviashvili I hierarchy, we get the line soliton, the lump solution, and the Johnson-Thompson lump, and the corresponding determinant formulae for the nonlinear superposition of several of them. For Kadomtsev-Petviashvili II apart from the line solitons, we get singular rational solutions with its singularity set describing the motion of strings in the plane. We also consider the I and II real forms for the Gel'fand-Dickey hierarchies obtaining the vectorial Darboux transformation in both cases. Received June 4, 1997; final revision received March 6, 1998; accepted March 23, 1998     

Darboux Moutard Transformations and Poincaré—Steklov Operators

Formulas relating Poincaré–Steklov operators for Schrödinger equations related by Darboux–Moutard transformations are derived. They can be used for testing algorithms of reconstruction of the potential from measurements at the boundary.     

Darboux Transformations Associated with Boiti-Tu Eigenvalue Problem

To the hierarchy of nonlinear evolution equations which associate with Boiti-Tu eigenvalue problem, an explicit and universal form of Baeklund transformations and auniversal proof are presented. It is called Darboux transformation. By this method, to ask for a new solution of every system of equations of the hierarchy, it is sufficient to solve some linear problems. Here the constraints at the boundary for the potentials (for example, at x=\pm \infinity) are removed.