Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞, respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.     

Slow manifolds for dissipative dynamical systems

A class of infinite-dimensional dissipative dynamical systems is defined for which the slow invariant manifolds can be calculated. Large-time behavior of the evolution of such systems is studied.     

Unitary representations of unimodular Lie groups in Bergman spaces

For an arbitrary unimodular Lie group G, we construct strongly continuous unitary representations in the Bergman space of a strongly pseudoconvex neighborhood of G in the complexification of its underlying manifold. These representation spaces are infinite-dimensional and have compact kernels. In particular, the Bergman spaces of these natural manifolds are infinite-dimensional.     

Multiple homoclinic solutions for singular differential equations

The homoclinic bifurcations of ordinary differential equation under singular perturbations are considered. We use exponential dichotomy, Fredholm alternative and scales of Banach spaces to obtain various bifurcation manifolds with finite codimension in an appropriate infinite-dimensional space. When the perturbative term is taken from these bifurcation manifolds, the perturbed system has various coexistence of homoclinic solutions which are linearly independent.     

Frobenius manifolds from regular classical W-algebras

We obtain polynomial Frobenius manifolds from classical W-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras.     

Banach流形上映射度

本文讨论一种无限维流形上拓扑不变量，即具有可定向Ｆｒｅｄｈｏｌｍ结构的实Ｂａｎａｃｈ流形上　　Ｃｒ　　映射度．它是通常有限维流形上光滑映射度的一种自然推广．     

Gromov-Hausdorff convergence to nonmanifolds

We construct sequences of S⁷’s in Gromov-Hausdorff space converging to nonmanifolds. The manifolds in these sequences have a common contractibility function. There are two main classes of examples—examples for which the limit is an infinite-dimensional space of finite cohomological dimension and examples for which the limit is a finite-dimensional ANR homology manifold. Communicated by Karsten Grove     

Observations about equicontinuity and related concepts

The concepts of equicontinuity, even continuity, topological equicontinuity and the newly defined concepts of compact equicontinuity and compact topological equicontinuity are compared. It is shown that for a set of group homomorphisms from a semitopological group to a topological group all these notions of equicontinuities coincide. It is shown also that an infinite-dimensional normed space endowed with its weak topology is an example of a space which does not satisfy the Ascoli theorem in Noble's sense.     

Transgression on hyperkähler manifolds and generalized higher torsion forms

We propose a generalization of the Hodge dd_c-lemma to the case of hyperk?hler manifolds. As an application we derive a global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperk?hler manifolds. In Section 3 we consider the fourth order transgression for the infinite-dimensional bundle arising from local families of hyperk?hler manifolds. We propose a local construction of the fourth order transgression of the Chern character form. We derive an explicit expression for the arising hypertorsion differential form. Its zero-degree part may be expressed in terms of the Laplace operators defined on the fibres of the local family.     

Almost periodic dynamics of perturbed infinite-dimensional dynamical systems

This paper is concerned with the dynamics of an infinite-dimensional gradient system under small almost periodic perturbations. Under the assumption that the original autonomous system has a global attractor given as the union of unstable manifolds of a finite number of hyperbolic equilibrium solutions, we prove that the perturbed non-autonomous system has exactly the same number of almost periodic solutions. As a consequence, the pullback attractor of the perturbed system is given by the union of unstable manifolds of these finitely many almost periodic solutions. An application of the result to the Chafee–Infante equation is discussed.     

Computation of Smooth Manifolds Via Rigorous Multi-parameter Continuation in Infinite Dimensions

In this paper, we introduce a constructive rigorous numerical method to compute smooth manifolds implicitly defined by infinite-dimensional nonlinear operators. We compute a simplicial triangulation of the manifold using a multi-parameter continuation method on a finite-dimensional projection. The triangulation is then used to construct local charts and an atlas of the manifold in the infinite-dimensional domain of the operator. The idea behind the construction of the smooth charts is to use the radii polynomial approach to verify the hypotheses of the uniform contraction principle over a simplex. The construction of the manifold is globalized by proving smoothness along the edge of adjacent simplices. We apply the method to compute portions of a two-dimensional manifold of equilibria of the Cahn–Hilliard equation.     

Horizontal Lift of an Infinite Dimensional Diffusion

We prove existence of the horizontal lift to a line bundle of certain diffusion processes on some infinite-dimensional manifolds. We provide three classes of finite-dimensional manifolds for which the corresponding loop spaces have a line bundle and thus provide three classes of loop manifolds on which certain diffusion processes admit a horizontal lift. Applications to Quantum Field Theory are indicated.     

1 : 3 Resonance is a possible cause of nonlinear panel flutter

A nonlinear boundary value problem modeling oscillations of a plate in a supersonic gas flow is considered. Using the normal forms method, the method of integral manifolds for dynamical systems with infinite-dimensional phase space, and asymptotic methods combined with numerical techniques, it is shown that the 1: 3 resonance of eigenfrequencies of the linearized boundary value problem can be a cause of subcritical bifurcations and hard excitation of oscillations.     

Approximate inertial manifolds of Burgers equation approached by nonlinear Galerkin’s procedure and its application

The approximate inertial manifolds (AIMs) of Burgers equation is approached by nonlinear Galerkin methods, and it can be used to capture and study the shock wave numerically in a reduced system with low dimension. Following inertial manifolds, the asymptotic behavior of Burgers equation, an infinite dimensional dissipative dynamic systems, will evolve to a compact set known as a global attractor, which is finite-dimensional, and the nonlinear phenomena are included and captured in such global attractor. In the application, nonlinear Galerkin methods is introduced to approach such inertial manifolds. By this method, the solution of the original system is projected onto the complete space spanned by the eigenfunctions or the modes of the linear operator of Burgers equation, and nonlinear Galerkin method splits the infinite-dimensional phase space into two complementary subspaces: a finite-dimensional one and its infinite-dimensional complement. Then, the post-processed Galerkin’s procedure is used to approximate the solution of the reduced system, with the introduction of the interaction between lower and higher modes. Additionally, some numerical examples are presented to make a comparison between the traditional Galerkin method and nonlinear Galerkin method, in particular, some sharp jumping phenomena, which are related to the shock wave, have been captured by the numerical method presented. As the conclusion, it can be drawn that it is possible to completely describe the dynamics on the attractor of a nonlinear partial differential equation (PDE) with a finite-dimensional dynamical system, and the study can provide a numerical method for the analysis of the nonlinear continuous dynamic systems and complicated nonlinear phenomena in finite-dimensional dynamic system, whose nonlinear dynamics has been developed completely compared with infinite-dimensional dynamic system.     

The exterior algebra for Wiemann manifolds

We discuss the theory of infinite-dimensional manifolds from the point of view of establishing a widely applicable framework for generalization of the finite-dimensional Hodge theory. The principal result is the development of an exterior algebra based on a weakened definition of differentiation, so that “C^∞” partitions of unity are available for paracompact manifolds modelled on arbitrary real separable Banach spaces. We prove a Poincaré lemma for our new notion of exterior differentiation, and go on to discuss the relationship of the exterior derivative with current research efforts toward the definition of an infinite-dimensional Laplace-Beltrami operator.     

Flux Homomorphisms and Principal Bundles over Infinite Dimensional Manifolds

Flux homomorphisms for closed vector-valued differential forms on infinite dimensional manifolds are defined. We extend the relation between the kernel of the flux for a closed 2-form and Kostants exact sequence associated to a principal bundle with curvature to the context of infinite-dimensional fiber and base space. We then use these results to construct central extensions of infinite dimensional Lie groups.Received April 19, 2002; in revised form October 15, 2002 Published online June 23, 2003     

Admissible vector fields and related diffusions on infinite-dimensional manifolds

A variation on the notion of “admissibility” for vector fields on certain infinite-dimensional manifolds with measures on them is described. It leads to the construction of associated diffusions and Markov semigroups on these manifolds via Dirichlet forms. Some classes of concrete examples are given.     

Normal systems over ANR’s, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di-mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U ) = w(X ) (where "w"is the topological weight) for each open nonempty subset U of X , then X itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W (X, ) = {(x n ) ∞ n=1 ∈ X ω : x n = for almost all n} is homeomorphic to a pre-Hilbert space E with E ～ = ΣE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.     

The chaotic linear map in the function space defined on a planar lattices

In this paper, we study the chaoticity and dynamics of shift maps operating on an infinite-dimensional function space defined on a planar lattices.     

Fibred Frobenius theorem

We give a version of Frobenius Theorem for fibred manifolds whose proof is shorter than the “short proofs” of the classical Frobenius Theorem. In fact, what shortens the proof is the fibred form of the statement, since it permits an inductive process which is not possible from the standard statement.