从具体例子看惯性流形概念的推广

惯性流形的概念要求所有轨道指数收敛于唯一吸引子[5],这对于很多物理问题,例如sine-Gordon方程是很难满足的[4],本文中给出的人工例子建议了惯性流形的推广形式,这个推广形式去掉了整体吸引子是唯一的预先要求,该推广概念使用于sine-Gordon方程。     

A sharp condition for existence of an inertial manifold

It is shown that a perturbation argument that guarantees persistence of inertial (invariant and exponentially attracting) manifolds for linear perturbations of linear evolution equations applies also when the perturbation is nonlinear. This gives a simple but sharp condition for existence of inertial manifolds for semi-linear parabolic as well as for some nonlinear hyperbolic equations. Fourier transform of the explicitly given equation for the tracking solution together with the Plancherel's theorem for Banach valued functions are used.     

Hyperconifold transitions,mirror symmetry,and string theory

Multiply-connected Calabi–Yau threefolds are of particular interest for both string theorists and mathematicians. Recently it was pointed out that one of the generic degenerations of these spaces (occurring at codimension one in moduli space) is an isolated singularity which is a finite cyclic quotient of the conifold; these were called hyperconifolds. It was also shown that if the order of the quotient group is even, such singular varieties have projective crepant resolutions, which are therefore smooth Calabi–Yau manifolds. The resulting topological transitions were called hyperconifold transitions, and change the fundamental group as well as the Hodge numbers. Here Batyrev?s construction of Calabi–Yau hypersurfaces in toric fourfolds is used to demonstrate that certain compact examples containing the remaining hyperconifolds — the Z₃${\mathbb{Z}}_3$ and Z₅${\mathbb{Z}}_5$ cases — also have Calabi–Yau resolutions. The mirrors of the resulting transitions are studied and it is found, surprisingly, that they are ordinary conifold transitions. These are the first examples of conifold transitions with mirrors which are more exotic extremal transitions. The new hyperconifold transitions are also used to construct a small number of new Calabi–Yau manifolds, with small Hodge numbers and fundamental group Z₃${\mathbb{Z}}_3$ or Z₅${\mathbb{Z}}_5$. Finally, it is demonstrated that a hyperconifold is a physically sensible background in Type IIB string theory. In analogy to the conifold case, non-perturbative dynamics smooth the physical moduli space, such that hyperconifold transitions correspond to non-singular processes in the full theory.     

Linearization of vector fields and embedding of diffeomorphisms in flows via Nash–Moser theorem

In the first part of this paper we give suitable spectral properties of the adjoint operators induced by appropriate perturbations of some hyperbolic linear vector fields. These properties are useful to prove general facts based on the Nash–Moser inverse function theorem. In the second part of this work we study circumstances where a global linearization of a vector field X$X$ in a real numerical space is feasible and where some diffeomorphisms which are close to exp(X)$exp\left(X\right)$ can be embedded in a flow.     

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non-zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.     

THE FRACTIONAL QUADRATIC-FORM IDENTITY AND BI-HAMILTONIAN STRUCTURES OF AN INTEGRABLE COUPLING OF THE FRACTIONAL COUPLED BURGERS HIERARCHY

Based on a general isospectral problem of fractional order and the fractional quadratic-form identity by Yue and Xia, the new integrable coupling of fractional coupled Burgers hierarchy and its fractional bi-Hamiltonian structures are obtained.     

A super hierarchy of coupled derivative nonlinear Schrödinger equations and conservation laws

By solving the zero-curvature equation associated with a 3 × 3 matrix spectral problem, a super hierarchy of coupled derivative nonlinear Schrödinger equations is proposed. The corresponding super bi-Hamiltonian structures are established by means of the super trace identity. Then, we derive infinite conservation laws of the super coupled derivative nonlinear Schrödinger equation with the aid of spectral parameter expansions.     

Integral manifolds of impulsive fractional functional differential systems

In this paper, a class of impulsive Caputo fractional functional differential systems with variable impulsive perturbations is investigated. Sufficient conditions for the existence of integral manifolds are obtained. The main results are proved by means of piecewise continuous Lyapunov functions and the fractional comparison principle. The demonstrated techniques can be applied in studying properties of many applied problems of diverse interest.