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Generalized Isospectral-Nonisospectral Modified Korteweg-de Vries Integrable Hierarchies and Related Properties

Annals of Applied Mathematics 2024年第2期40卷 105-138页

作者： Huanhuan Lu Xinan Ren Yufeng Zhang Hongyi Zhang School of Mathematics China University of Mining and TechnologyXuzhouJiangsu 221116China

In this article, a new technique for deriving integrable hierarchy is discussed, i.e., such that are derived by combining the Tu scheme with the vector product. Several classes of spectral problems are introduced by threedimensional loop algebra and six-dimensional loop algebra whose commutators are vector product, and the six-dimensional loop algebra is derived from the enlargement of the three-dimensional loop algebra. It is important that we make use of the variational method to create a new vector-product trace identity for which the Hamiltonian structure of the isospectral integrable hierarchy is worked out. The derived integrable hierarchies are reduced to the modified Korteweg-de Vries(mKdV) equation, generalized coupled mKdV integrable system and nonisospectral mKdV equation under specific parameter selection. Starting from a 3×3 matrix spectral problem, we subsequently construct an explicit N-fold Darboux transformation for integrable system(2.8) with the help of a gauge transformation of the corresponding spectral problem. At the same time, the determining equations of nonclassical symmetries associated with mKdV equation are presented in this paper. It follows that we investigate the coverings and the nonlocal symmetries of the nonisospectral mKdV equation by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

关键词： Darboux transformation nonlocal symmetry covering manifold contact structure

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DISTRIBUTIONS ON ALMOST contact MANIFOLDS

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Acta Mathematica Scientia 2017年第3期37卷 695-702页

作者： Hyunjoo CHO National Center for Theoretical Sciences Mathematics Division Taipei 10617 China

It is known that any hypersurface in an almost complex space admits an almost contact manifold [11, 14]. In this article we show that a hyperplane in an almost contact manifold has an almost complex structure. Along with this result, we explain how to deter- mine when an almost contact structure induces a contact structure, followed by examples of a manifold with a closed G2-structure.

关键词： Almost contact structure contact structure G2-structure foliation integra-bility

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The Conformally Symmetric K-contact Manifolds

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Chinese Quarterly Journal of Mathematics 1992年第1期7卷 5-10页

作者：郭震云南师范大学数学系昆明650092

In this paper, we establish a differential equation about scalar curvature of conformally flat K-contact manifolds, and prove that a conformally symmetric K-contact manifold is a Riemann manifold with constant curvature 1. At the same time, the results on Sasaki manifolds which are given by Miyazaawa and Yamagushi are generalized to K-contact manifolds.

关键词： Conformally symmetric contact structure Killing vector field

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Killing Operator for the Kerr Metric

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Journal of Modern Physics 2023年第1期14卷 31-59页

作者： J.-F. Pommaret CERMICS Ecole des Ponts ParisTech Paris France

When D: E →F is a linear differential operator of order q between the sections of vector bundles over a manifold X of dimension n, it is defined by a bundle map Φ: Jq(E) →F=F0 that may depend, explicitly or implicitly, on constant parameters a, b, c, ... . A “direct problem” is to find the generating compatibility conditions (CC) in the form of an operator D1: F0 →F1. When D is involutive, that is when the corresponding system Rq = ker (Φ) is involutive, this procedure provides successive first order involutive operators D1, ..., Dn. Though D1 οD = 0 implies ad (D) οad(D1) = 0 by taking the respective adjoint operators, then ad (D) may not generate the CC of ad (D1) and measuring such “gaps” led to introduce extension modules in differential homological algebra. They may also depend on the parameters and such a situation is well known in ordinary or partial control theory. When Rq is not involutive, a standard prolongation/projection (PP) procedure allows in general to find integers r, s such that the image of the projection at order q+r of the prolongation is involutive but it may highly depend on the parameters. However, sometimes the resulting system no longer depends on the parameters and the extension modules do not depend on the parameters because it is known that they do not depend on the differential sequence used for their definition. The purpose of this paper is to study the above problems for the Kerr (m, a), Schwarzschild (m, 0) and Minkowski (0, 0) parameters while computing the dimensions of the inclusions for the respective Killing operators. Other striking motivating examples are also presented.

关键词： Differential Operator Adjoint Operator Differential Sequence Einstein Equations Kerr Metric Differential Module Extension Module contact structure

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