In the paper, we derive a multi-symplectic Fourier pseudospectral method for Zhiber-Shabat equation. The Zhiber-Shabat equation, which describes many important physical phenomena, has been investigated widely in last several decades. The multi-symplectic geometry and multi-symplectic Fourier pseudospectral method for the Zhiber-Shabat equation is presented. The numerical experiments are given, ...

In this note we apply a 4-fold sum operation to develop an associativity rule for the pairwise symplectic sum. This allows us to show that certain diieomorphic symplectic 4-manifolds made out of elliptic surfaces are in fact symplectically deformation equivalent. We also show that blow-up points can be traded from one side of a symplectic sum to another without changing the symplectic deformati...

Long-term stability studies of nonlinear Hamiltonian systems require symplectic integration algorithms which are both fast and accurate. In this paper, we study a symplectic integration method wherein the symplectic map representing the Hamiltonian system is refactorized using polynomial symplectic maps. This method is analyzed in detail for the three degree of freedom case. We obtain explicit ...

Symplectic integrators are numerical integration schemes for hamiltonian systems. The integration step is an explicit symplectic map. We find symplectic integrators using universal exponential identities or relations among formal Lie series. We give here general methods to compute such identities in a free Lie algebra. We recover by these methods all the previously known symplectic integrators ...

This paper is a survey of results which have brought techniques from the theory of complex surfaces to bear on symplectic 4-manifolds. Lefschetz fibrations are defined and some basic examples from complex surfaces discussed. Two results on the relationship between admitting a symplectic structure and admitting a Lefschetz fibration are explained. We also review the question of geography: which ...

The packing stability in symplectic geometry was first noticed by Biran [Bir97]: the symplectic obstructions to embed several balls into a manifold disappear when their size is small enough. This phenomenon is known to hold for all closed manifolds with rational symplectic class (see [Bir99] for the 4-dimensional case, and [BH11, BH13] for higher dimensions), as well as ellipsoids [BH13]. In th...

In this paper, we demonstrate a relation among Seiberg-Witten invariants which arises from embedded surfaces in four-manifolds whose self-intersection number is negative. These relations, together with Taubes’ basic theorems on the Seiberg-Witten invariants of symplectic manifolds, are then used to prove the symplectic Thom conjecture: a symplectic surface in a symplectic fourmanifold is genus-...

We give a simple construction yielding homology classes in (non-simply-connected) symplectic four-manifolds which admit infinitely many pairwise non-isotopic symplectic representatives. Examples are constructed in which the symplectic curves can have arbitrarily large genus. The examples are built from surface bundles over surfaces and involve only elementary techniques. As a corollary we see t...

A n-dimensional Lie group G equipped with a left invariant symplectic form ω+ is called a symplectic Lie group. It is well-known that ω+ induces a left invariant affine structure on G. Relatively to this affine structure we show that the left invariant Poisson tensor π+ corresponding to ω+ is polynomial of degree 1 and any right invariant k-multivector field on G is polynomial of degree at most...

Classical mechanical systems are modeled by a symplectic manifold (M,ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called symplectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on ...