Multi-symplectic structures and

 symplectic pattern formation

 Department of Mathematics
 University of Surrey

Multi-symplectic structures and wave propagation
  A Hamiltonian structure is presented, which generalizes classical Hamiltonian structure, by assigning a distinct symplectic operator for each unbounded space direction and time, of a Hamiltonian evolution equation on one or more space dimensions. This generalization, called multi-symplectic structures, is shown to be natural for dispersive wave propagation problems. Application of the abstract properties of the multi-symplectic structures framework leads to a new variational principle for space-time periodic states reminiscent of the variational principle for invariant tori, a geometric reformulation of the concepts of action and action flux, a rigorous proof of the instability criterion predicted by the Whitham modulation equations, a new symplectic decomposition of the Noether theory, generalization of the concept of reversibility to space-time and a proof of Lighthill's geometric criterion for instability of periodic waves travelling in one space dimension. The nonlinear Schrödinger equation and the water-wave problem are characterized as Hamiltonian systems on a multi-symplectic structure for example. Further ramifications of the generalized symplectic structure of theoretical and practical interest are also discussed.
  T.J. Bridges. Multi-symplectic structures and wave propagation, Math. Proc. Camb. Phil. Soc. 121 147-190 (1997)
MPCPS website  

Toral-equivariant PDEs and quasiperiodic patterns
  Spatially quasiperiodic patterns with q (q>1) independent wavenumbers in each spatial direction, which satisfy a nonlinear partial differential equation (PDE), are considered. For a class of toral-equivariant PDEs, a framework for the existence and linear stability analysis of such patterns is developed. The idea is to reformulate the PDE on the jet space, which for a large class of PDEs is finite dimensional. This reformulation induces an orthogonal action of the toral group on the finite-dimensional jet space. A non-trivial generalization of relative equilibria is introduced and used to characterize quasiperiodic patterns. The geometric framework associated with this generalization of relative equilibria on the jet space leads to a rigourous analysis of the stability equation, for quasiperiodic patterns of arbitrary amplitude, and a general instability criterion for a class of quasiperiodic patterns. We show that this framework is a natural setting for a geometric analysis of patterns, and show how the theory extends to pattern formation PDEs without explicit toral symmetry. The theory is applied to the two-toral pattern solutions of a system of Ginzburg-Landau equations in one and two space dimensions, and to the Swift-Hohenberg equation and its generalizations.
  T.J. Bridges. Toral-equivariant partial differential equations and quasiperiodic patterns, Nonlinearity 11 467-500 (1998)   IoP website    

Multi-symplectic Dirac operators and the TEA bundle
  The aim of this paper is to construct multi-symplectic structures starting with the geometry of an oriented Riemannian manifold, independent of a Lagrangian or a particular partial differential equation (PDE). The principal observation is that on an n-dimensional orientable manifold M there is a canonical quadratic form Θ associated with the total exterior algebra bundle on M. On the fibre, which has dimension 2n, the form Θ can be locally decomposed into n classical symplectic structures. When concatenated, these n-symplectic structures define a partial differential operator, J∂, which turns out to be a Dirac operator with multi-symplectic structure. The operator J∂ generalizes the product operator J(d/dt) in classical symplectic geometry, and M is a generalization of the base manifold (i.e. time) in classical Hamiltonian dynamics. The structure generated by Θ provides a natural setting for analysing a class of covariant nonlinear gradient elliptic operators. The operator J∂ is elliptic, and the generalization of Hamiltonian systems, J∂Z=∇S(Z), for a section Z of the total exterior algebra bundle, is also an elliptic PDE. The inverse problem—find S(Z) for a given elliptic PDE—is shown to be related to a variant of the Legendre transform on k-forms. The theory is developed for flat base manifolds, but the constructions are coordinate free and generalize to Riemannian manifolds with non-trivial curvature. Some applications and implications of the theory are also discussed.
  T.J. Bridges. Canonical multi-symplectic structure on the total exterior algebra bundle. Proc. Roy. Soc. London A 462 1531-1551 (2006)    Royal Society website    
Wave action, signature and instabilities
  Action, symplecticity, signature and complex instability are fundamental concepts in Hamiltonian dynamics which can be characterised in terms of the symplectic structure. In this paper, Hamiltonian PDEs on unbounded domains are characterised in terms of a multi-symplectic structure where a distinct differential two-form is assigned to each space direction and time. This leads to a new geometric formulation of the conservation of wave action for linear and nonlinear Hamiltonian PDEs, and, via Stokes' Theorem, a conservation law for symplecticity. Each symplectic structure is used to define a signature invariant on the eigenspace of a normal mode. The first invariant in this family is classical Krein signature (or energy sign, when the energy is time independent) and the other (spatial) signatures are energy flux signs, leading to a classification of instabilities that includes information about directional spatial spreading of an instability. The theory is applied to several examples: the Boussinesq equation, the water-wave equations linearised about an arbitrary Stokes' wave, rotating shallow water flow and flow past a compliant surface. Some implications for non-conservative systems are also discussed.
  T. J. Bridges. A geometric formulation of the conservation of wave action and its implications for signature and the classification of istabilities, Proc. Roy. Soc. London A 453 1365-1395 (1997)     JSTOR website    

The variational bicomplex
  Multisymplecticity and the variational bicomplex are two subjects which have developed independently. Our main observation is that re-analysis of multisymplectic systems from the view of the variational bicomplex not only is natural but also generates new fundamental ideas about multisymplectic Hamiltonian PDEs. The variational bicomplex provides a natural grading of differential forms according to their base and fibre components, and this structure generates a new relation between the geometry of the base, covariant multisymplectic PDEs and the conservation of symplecticity. Our formulation also suggests a new view of Noether theory for multisymplectic systems, leading to a definition of multimomentum maps that we apply to give a coordinate-free description of multisymplectic relative equilibria. Our principal example is the class of multisymplectic systems on the total exterior algebra bundle over a Riemannian manifold.
  T.J. Bridges, P.E. Hydon & J.K. Lawson. Multi-symplectic structures and the variational bicomplex, Math. Proc. Camb. Phil. Soc. 148 159-178 (2010)     MPCPS website  

Geometric numerical discretization
  Recent results on numerical integration methods that exactly preserve the symplectic structure in both space and time for Hamiltonian PDEs are discussed. The Preissman box scheme is considered as an example, and it is shown that the method exactly preserves a multi-symplectic conservation law and any conservation law related to linear symmetries of the PDE. Local energy and momentum are not, in general, conserved exactly, but semi-discrete versions of these conservation laws are. Then, using Taylor series expansions, one obtains a modified multi-symplectic PDE and modified conservation laws that are preserved to higher order. These results are applied to the nonlinear Schrödinger (NLS) equation and the sine-Gordon equation in relation to the numerical approximation of solitary wave solutions. This strategy is introduced in Bridges & Reich (2001). The paper of Bridges & Reich (2006) provides an introduction and survey of conservative discretization methods for Hamiltonian PDEs. The emphasis is on variational, symplectic and multi-symplectic methods. The derivation of methods as well as some of their fundamental geometric properties are discussed. Basic principles are illustrated by means of examples from wave and fluid dynamics.
  T.J. Bridges & S. Reich. Numerical methods for Hamiltonian PDEs, J. Phys. A: Math. Gen. 39 5287-5320 (2006)   IoP website    
  T.J. Bridges & S. Reich. Multi-symplectic integrators: numerical schemes for Hamiltonian PDEs that conserve symplecticity, Phys. Lett. A 284 184-193 (2001)   PLA website    

Geometric theory for multi-periodic patterns
  Space and time periodic waves at the two-dimensional surface of an irrotational inviscid fluid of finite depth are considered. The governing equations are shown to have a new formulation as a generalized Hamiltonian system on a multisymplectic structure where there is a distinct symplectic operator corresponding to each unbounded space direction and time. The wave-generated mean flow in this framework has an interesting characterization as drift along a group orbit. The theory has interesting connections with, and generalizations of, the concepts of action, action flux, pseudofrequency and pseudowavenumber of the Whitham theory. The multisymplectic structure and novel characterization of mean flow lead to a new constrained variational principle for all space and time periodic patterns on the surface of a finite-depth fluid. With the additional structure of the equations, it is possible to give a direct formulation of the linear stability problem for three-dimensional travelling waves. The linear instability theory is valid for waves of arbitrary amplitude. For weakly nonlinear waves the linear instability criterion is shown to agree exactly with the previous results of Benney-Roskes, Hayes, Davey-Stewartson and Djordjevic-Redekopp obtained using modulation equations. Generalizations of the instability theory to study all periodic patterns on the ocean surface are also discussed.
  T. J. Bridges. Periodic patterns, linear instability, symplectic structure and mean-flow dynamics for three-dimensional surface waves, Phil. Trans. Roy. Soc. London A 354 533-574 (1996)   Royal Society website    

Multi-symplectic structures and wave interaction
  Standing waves are a fundamental class of solutions of nonlinear wave equations with a spatial reflection symmetry, and they routinely arise in optical and oceanographic applications. At the linear level they are composedof two synchronized counterpropagating periodic traveling waves. At the nonlinear level, they can be defined abstractly by their symmetry properties. In this paper, general aspects of the modulational instability of standing waves are considered. This problem has difficulties that do not arise in the modulational instability of traveling waves. Here we propose a new geometric formulation for the linear stability problem, based on embedding the standing wave in a four-parameter family of nonlinear counterpropagating waves. Multisymplectic geometry is shown to encode the stability properties in an essential way. At the weakly nonlinear level we obtain the surprising result that standing waves are modulationally unstable only if the component traveling waves are modulation unstable. Systems of nonlinear wave equations will be used for illustration, but general aspects will be presented, applicable to a wide range of Hamiltonian PDEs, including water waves.
  T.J. Bridges. Nonlinear counterpropagating waves, multisymplectic geometry, and the instability of standing waves, SIAM J. Appl. Math. 64 2096-2120 (2004)   SIAM JSTOR website    

Transverse instability of solitary waves
  Transverse instabilities correspond to a class of perturbations traveling in a direction transverse to the direction of the basic solitary wave. Solitary waves traveling in one space direction generally come in one-parameter families. We embed them in a two-parameter family and deduce a new geometric condition for transverse instability of solitary waves. This condition is universal in the sense that it does not require explicit properties of the solitary wave - or the governing equation. In this paper the basic idea is presented and applied to the Zakharov-Kuznetsov equation for illustration. An indication of how the theory applies to a large class of equations in physics and oceanography is also discussed.
  T. J. Bridges. Universal geometric condition for the transverse instability of solitary waves, Phys. Rev. Lett. 84 2614-1617 (2000)   PRL website    
  T. J. Bridges. Transverse instability of solitary-wave states of the water-wave problem, J. Fluid Mech. 439 255-278 (2001)   JFM website    
  T.J. Bridges. On the susceptibility of bright nonlinear Schrödinger solitons to long–wave transverse instability, Proc. Roy. Soc. London A 460 2605-2615 (2004)   Royal Society website    

  Department of Mathematics     University of Surrey