Abstract. Bridges and Reich suggested the idea of multi-symplectic spectral discretization on Fourier space [4]. Based on their theory, we investigate the multi-symplectic Fourier pseudospectral discretization of the nonlinear Schrödinger equation (NLS) on real space. We show that the multi-symplectic semi-discretization of the nonlinear Schrödinger equation with periodic boundary conditions ha...

Let M be a smooth, closed oriented 4−manifold. An orientation-compatible symplectic structure on M is a closed two form ω such that ω ∧ ω is nowhere vanishing and gives the orientation. Two such symplectic structures ω 1 and ω 2 on M are said to be deformation equivalent if there is an orientation-preserving diffeomorphism φ such that φ * ω 1 and ω 2 are connected by a path of symplectic forms....

Let M be a smooth, closed oriented 4−manifold. An orientation-compatible symplectic structure on M is a closed two form ω such that ω ∧ ω is nowhere vanishing and gives the orientation. Two such symplectic structures ω 1 and ω 2 on M are said to be deformation equivalent if there is an orientation-preserving diffeomorphism φ such that φ * ω 1 and ω 2 are connected by a path of symplectic forms....

By a symplectic manifold (or a symplectic n-fold) we mean a compact Kaehler manifold of even dimension n with a non-degenerate holomorphic 2form ω, i.e. ω is a nowhere-vanishing n-form. This notion is generalized to a variety with singularities. We call X a projective symplectic variety if X is a normal projective variety with rational Gorenstein singularities and if the regular locus U of X ad...

In this paper, we investigate multi-symplectic Runge-Kutta-Nyström (RKN) methods for nonlinear Schrödinger equations. Concatenating symplecitc Nyström methods in spatial direction and symplecitc Runge-Kutta methods in temporal direction for nonlinear Schrödinger equations leads to multi-symplecitc integrators, i.e. to numerical methods which preserve the multi-symplectic conservation law (MSCL)...

We study germs of singular varieties in a symplectic space. In [A1], V. Arnol’d discovered so called ‘‘ghost’’ symplectic invariants which are induced purely by singularity. We introduce algebraic restrictions of di¤erential forms to singular varieties and show that this ghost is exactly the invariants of the algebraic restriction of the symplectic form. This follows from our generalization of ...

We study symplectic manifolds (M, ω) equipped with a symplectic torsion-free affine (also called Fedosov) connection ∇ and admitting a metaplectic structure. Let S be the so called symplectic spinor bundle and let R be the curvature tensor field of the symplectic spinor covariant derivative ∇ associated to the Fedosov connection ∇. It is known that the space of symplectic spinor valued exterior...

2 Local symplectic, complex and Kähler geometry: a quick review 10 2.1 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Symplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Complex manifolds . . . . . . . . . . . . . . ....

The construction of symplectic numerical schemes for stochastic Hamiltonian systems is studied. An approach based on generating functions method is proposed to generate the stochastic symplectic integration of any desired order. In general the proposed symplectic schemes are fully implicit, and they become computationally expensive for mean square orders greater than two. However, for stochasti...

We analyze the global structure of Lagrangian Grassmannian in the product symplectic space and investigate the local properties of generic symplectic relations. The cohomological symplectic invariant of discrete dynamical systems is generalized to the class of generalized canonical mappings. Lower bounds for the number of two-point and three-point symplectic invariants for billiard-type dynamic...