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Transverse intersection of invariant manifolds in perturbed multi-symplectic systems. (English) Zbl 1151.37048

A multi-symplectic system is a PDE with a Hamiltonian or (pre)-symplectic structure in both spatial and temporal variables. Although existence of a multi-symplectic structure might seem to be overly restrictive, it turns out that many Hamiltonian PDEs have such structure. Many multi-symplectic systems have symmetries, that means that such systems usually possess families of special solutions such as (travelling) solitary waves or fronts. Perturbations can be lead to the break-up of those solutions and subsequent formation of spatio-temporal chaos.
In the reviewed article the main attention is payed to the effects of spatially periodic perturbations on the invariant manifold associated with solitary waves/fronts of multi-symplectic system with symmetry. Sufficient criteria are derived for persistence of this manifold, or parts of one, under spatially periodic perturbations both equivariant and symmetry-breaking ones. Also intersections of the persisting manifolds are investigated, as these can signify the existence of chaotic dynamics through a suitable modification of the Smale-Birkhoff theorem.
As illustration of the general theory the effect of various spatially periodic perturbations of the nonlinear generalized Schrödinger equation is analyzed.

MSC:

37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
53D05 Symplectic manifolds (general theory)
35Q51 Soliton equations
53D20 Momentum maps; symplectic reduction
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
37J20 Bifurcation problems for finite-dimensional Hamiltonian and Lagrangian systems
37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
35Q55 NLS equations (nonlinear Schrödinger equations)
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