Characterization of the Darboux point for particular classes of problems

A minimal sufficient condition for global optimality involving the Darboux point, analogous to the minimal sufficient condition of local optimality involving the conjugate point, is presented. The Darboux point is then characterized for optimal control problems with linear dynamics, cost functionals with a general terminal state term and an integrand quadratic in the state and control, and general terminal conditions. The Darboux point is shown to be the supremum of a sequence of conjugate points. If the terminal state term is quadratic, along with a scalar quadratic boundary condition, then the Darboux point is also the time at which the Riccati matrix becomes unbounded, giving a characterization of the unboundedness of the Riccati matrix at points which are not in general conjugate points.This research was supported by the National Science Foundation under Grant No. GK-30115.This is Definition 2.1 of Ref. 1.     

About <Emphasis Type="Italic">I</Emphasis>(<Emphasis Type="Italic">J</Emphasis>)-approximately continuous functions

A generalization of density point with respect to category and I(J)-density topology has been presented in [5]. This papers deals with some basic properties of the corresponding I(J)-approximately continuous functions and gives answer when these functions are of the first Baire class and when they are Darboux functions.     

Darboux连续函数在不连续点的局部性态

通过分析一个具体的函数实例，发现Darboux连续函数在不连续点附近具有无限震荡的特性，并给出严格证明，从而得到Darboux连续函数在不连续点处独有的局部性态．     

Some Remarks on Ishimori's Magnet Model

Gauge transformation properties for an associated linear system of the Ishimori's magnet model are discussed. Explicit formulas for the gauge transformation matrix are obtained. A Darboux transformation is suggested and the appropriate dressing relations are found. Bibliography: 15 titles.     

Darboux transforms and spectral curves Darboux transforms and spectral curves of constant mean curvature surfaces revisited

We study the geometric properties of Darboux transforms of constant mean curvature (CMC) surfaces and use these transforms to obtain an algebro-geometric representation of constant mean curvature tori. We find that the space of all Darboux transforms of a CMC torus has a natural subset which is an algebraic curve (called the spectral curve) and that all Darboux transforms represented by points on the spectral curve are themselves CMC tori. The spectral curve obtained using Darboux transforms is not bi-rational to, but has the same normalisation as, the spectral curve obtained using a more traditional integrable systems approach.     

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.     

A semidiscrete Gardner equation

We construct the Darboux transformations, exact solutions, and infinite number of conservation laws for a semidiscrete Gardner equation. A special class of solutions of the semidiscrete equation, called table-top solitons, are given. The dynamical properties of these solutions are also discussed.     

THE CALCULUS OF GENERATING FUNCTIONS AND THE FORMAL ENERGY FOR HAMILTONIAN ALGORITHMS

1.DarbouxTransformationConsidercotangentbundleT*R"acRZnwithnaturalsymplecticstructureandtheproductofcotangentbundles(T*R")x(T*R")=R4nwithnaturalproductsymplecticstructureCorrespondingly,weconsidertheproductspaceR"xR"rsRZn.ItscotangentbunT*(R"xR")=T*Rzn=R'nhasnaturalsymplecticstructure')PreparedbyQinMengzhaoChoosesymplecticcoordinatesz~(p,q)onthesymplecticmanifold,thenforsymplectictransformationg:T*R"~T*R",wehaveitisaLagrangiansubmanifoldofT*R"xT*RninR4n~(R'",J4.).NotethatonR4nth…     

Darboux transformations for a matrix long-wave–short-wave equation and higher-order rational rogue-wave solutions

A new matrix long-wave–short-wave equation is proposed with the of help of the zero-curvature equation. Based on the gauge transformation between Lax pairs, both onefold and multifold classical Darboux transformations are constructed for the matrix long-wave–short-wave equation. Resorting to the classical Darboux transformation, a multifold generalized Darboux transformation of the matrix long-wave–short-wave equation is derived by utilizing the limit technique, from which rogue wave solutions, in particular, can be obtained by employing the generalized Darboux transformation. As applications, we obtain rogue-wave solutions of the long-wave–short-wave equation and some explicit solutions of the three-component long-wave–short-wave model, including soliton solutions, breather solutions, the first-order and higher-order rogue-wave solutions, and others by using the generalized Darboux transformation.     

Asymptotics of pseudodifferential operators acting on functions with corner singularities

A new definition of the expansion in smoothness of a distribution at a point is introduced. Some properties of the new expansion are studied and a uniqueness result for the coefficients of the expansion is proved. The main application of the definition is for obtaining the asymptotics in smooth ness of pseudodifferential operators acting on a function near a corner point of singsupp f. An example of usage of the obtained results in computed tomog raphy is presented.     

An integrable lattice hierarchy, associated integrable coupling, Darboux transformation and conservation laws

A new integrable lattice hierarchy is constructed from a discrete matrix spectral problem, some related properties of the new hierarchy are discussed. The Hamiltonian structures and Liouville integrability of the new hierarchy are established by using the discrete trace identity. A kind of integrable coupling for the new hierarchy is constructed through enlarging spectral problems. A Darboux transformation (DT) with two variable parameters and the infinitely many conservation laws for a typical lattice equation in the new hierarchy are constructed based on its Lax representation, the explicit solutions are obtained via the DT, the structures for those solutions are graphically investigated. All these properties might be helpful to understanding some physical phenomena.     

色散长波方程的Darboux变换及多孤子解

根据色散长波方程的可积性,首先借助符号计算构造出该方程的Lax对,接着构建一个包含多参数的Darboux变换,通过应用Darboux变换,得到色散长波方程的2N-孤子解,最后通过图像研究了孤子解的性质,这些解和图像可能对解释色散长波方程所描述的水波现象有所帮助.     

Generalized cofactors and nonlinear superposition principles

It is known from Lie's works that the only ordinary differential equation of first order in which the knowledge of a certain number of particular solutions allows the construction of a fundamental set of solutions is, excepting changes of variables, the Riccati equation. For planar complex polynomial differential systems, the classical Darboux integrability theory exists based on the fact that a sufficient number of invariant algebraic curves permits the construction of a first integral or an inverse integrating factor. In this paper, we present a generalization of the Darboux integrability theory based on the definition of generalized cofactors.     

On a vector long wave-short wave-type model

A new vector long wave-short wave-type model is proposed by resorting to the zero-curvature equation. Based on the resulting Riccati equations related to the Lax pair and the gauge transformations between the Lax pairs, multifold Darboux transformations are constructed for the vector long wave-short wave-type model. This method is general and is suitable for constructing the Darboux transformations of other soliton equations, especially in the absence of symmetric conditions for Lax pairs. As an illustrative example of the application of the Darboux transformations, exact solutions of the two-component long wave-short wave-type model are obtained, including solitons, breathers, and rogue waves of the first, second, third, and fourth orders. All the solutions derived by the Darboux transformations involve square roots of functions, which is not observed in the investigation of other nonlinear integrable equations. This model describes new nonlinear phenomena.     

Symplectic Geometry of Manifolds with Almost Nowhere Vanishing Closed 2-Form

We study local geometric properties of manifolds equipped with a closed 2-form nondegenerate at all points of a dense proper subset. We introduce the natural notion of tame singular point, at which the matrix of the 2-form degenerates in a regular way. We find a condition for Hamiltonian dynamical systems to be extended smoothly to tame singular points, generalize the Darboux theorem about the local reduction of the matrix of the 2-form to canonical form, and study the singular behavior of directional gradients.     

Pseudo-spherical Darboux images and lightcone images of principal-directional curves of nonlightlike curves in Minkowski 3-space

Choosing an alternative frame, which is the Frenet frame of the principal-directional curve along a nonlightlike Frenet curve γ , we define de Sitter Darboux images, hyperbolic Darboux images, and lightcone images generated by the principal directional curves of nonlightlike Frenet curves and investigate geometric properties of these associated curves under considerations of singularity theory, contact, and Legendrian duality. It is shown that pseudo-spherical Darboux images and lightcone images can occur singularities (ordinary cusp) characterized by some important invariants. More interestingly, the cusp is closely related to the contact between nonlightlike Frenet curve γ and a slant helix, the principal-directional curve ψ of γ and a helix or the principal-directional curve ψ and a slant helix. In addition, some relations of Legendrian dualities between C-curves and pseudo-spherical Darboux images or lightcone images are shown. Some concrete examples are provided to illustrate our results.     

Generalized Lattice Heisenberg Magnet Model and Its Quasideterminant Soliton Solutions

We consider a Darboux transformation of a generalized lattice (or semidiscrete) Heisenberg magnet (GLHM) model. We define a Darboux transformation on solutions of the Lax pair and on solutions of the spin evolution equation of the GLHM model. The solutions are expressed in terms of quasideterminants. We give a general expression for K-soliton solutions in terms of quasideterminants. Finally, we obtain one- and two-soliton solutions of the GLHM model using quasideterminant properties.     

Negaton, positon and complexiton solutions of the nonisospectral KdV equations with self-consistent sources

The negaton, positon, and complexiton solutions of the nonisospectral KdV equations with self-consistent sources (KdVESCSs) are obtained by the generalized binary Darboux transformation (GBDT) with N arbitrary t-functions. Taking the special initial seed solution for auxiliary linear problems, the negaton, positon, and complexiton solutions of the nonisospectral KdVESCSs are considered through the GBDT by selecting the negative, positive and complex spectral parameters. It is important to point out that these solutions of the nonisospectral KdVESCSs are analytical and singular. We also show differences between these solutions with singularities. Moreover, the detailed characteristics of these solutions with nonisospectral properties and sources effects are described through some figures.     

Darboux transformation and explicit solutions for two integrable equations

A new N-fold Darboux transformation for two integrable equations is constructed with the help of a gauge transformation for the spectral problem proposed by Qiao [Z.J. Qiao, Phys. Lett. A 192 (1994) 316-322]. By the Darboux transformation, explicit soliton and multi-soliton solutions for the two equations are obtained. In particular, soliton and complexiton solutions are shown through some figures.     

On the Darboux theorem for weak symplectic manifolds

A new tool to study reducibility of a weak symplectic form to a constant one is introduced and used to prove a version of the Darboux theorem more general than previous ones. More precisely, at each point of the considered manifold a Banach space is associated to the symplectic form (dual of the phase space with respect to the symplectic form), and it is shown that the Darboux theorem holds if such a space is locally constant. The following application is given. Consider a weak symplectic manifold on which the Darboux theorem is assumed to hold (e.g. a symplectic vector space). It is proved that the Darboux theorem holds also for any finite codimension symplectic submanifolds of , and for symplectic manifolds obtained from by the Marsden-Weinstein reduction procedure.