In this talk we will discuss the study of the linear and non-linear stability of polygonal configuration of vortices. The non-linear stability analysis is carried out by applying a sufficient criterion due to Dirichlet. Linear and non-linear stability ranges coincide both in the plane and on the sphere. Furthermore, in both geometries the addition of a central vortex of strength K strongly modifies the stability ranges (Joint work with Hildeberto Cabral).

If the PDE also has a reflection symmetry, it can support spatially periodic breathers (SPBs). At the linear level, these states consist of a superposition of two travelling waves of the same amplitude travelling in opposite directions. At the nonlinear level, they are spatially periodic states which are also time-periodic. What is the appropriate generalisation of the sideband instability for these states?

Since the NLS equation is the appropriate model for the one-dimensional case, one might guess that the appropriate model for analysing the sideband instability of SPBs is coupled NLS equations. This has been tried and gives incorrect results. Edgar Knobloch and collaborators have proposed coupled NLS (or cGL in the non-conservative case) equations with additional integral terms, resulting in a partial differential delay equation. These equations predict the correct form of the sideband instability, but are for weakly nonlinear SPBs, and do not give a geometric picture of the nature of the instability. Surprisingly, these are the only results in the literature about instability of SPBs!

We approach this problem in a new way which gives more complete information about the local properties, and extends to finite-amplitude SPBs. The backbone of the argument is to use the multi-symplectic structures framework. In this setting such states can be characterised abstractly. Even in the MSS framework, a straightforward approach fails. Curiously, the crucial idea is to embed the SPBs in a family of a general quasiperiodic two-wave interactions, and then take the limit as the two-waves synchronise as an SPB. This might appear to be a dubious limit! But, the two-wave interaction has a toral (T^2) structure, and the limiting SPB also has a toral (T^2) structure (because it is time periodic, and there is an S^1 orbit of such waves). This last point becomes to sound plausible when you think about the analogous situation in finite-dimensions.

Although this analogy can not be stretched too far, an analogue in finite dimensions is O(2) equivariant Hamiltonian systems, with the spherical pendulum as prototype. The travelling waves correspond to conical pendulum solutions and the standing waves correspond to planar pendulum solutions. There is a torus of planar pendulum solutions obtained by acting with the circle group on a single planar pendulum solution. Note however, that there is no analogue of the sideband instability in finite dimensions.

Back to the PDE: the decomposition of SPBs also sheds light on the nature of the instability: it turns out that the usual sideband instability carries over, but it is doubled. The instability can be associated with a Hamiltonian Hopf bifurcation with double eigenvalues (i.e. 8 dimensions associated with the instability rather than 4), and the instability is in some sense associated with a breakup of the two synchronised waves.

The theory is not rigorous at present, although one is hopeful that parts of it can be fully confirmed, but other parts may require special consideration due to small divisors. It will also be shown how the theory can be applied to the instability of standing waves of the water-wave problem.

The above results can be described in some detail, and give a geometrical picture of spatially-extended O(2)-equivariant Hamiltonian PDEs. However, one of the most interesting outcomes is that the theory points the way towards a framework for more general wave interactions. For example, how do generalized SPBs, such as spatially periodic patterns on a hexagonal lattice, become unstable and/or breakup? Some preliminary results on this problem and their application will be described.

Travelling waves can be detected using a centre-manifold reduction technique by which the problem is reduced to a locally equivalent Hamiltonian system with a finite number of degrees of freedom. I will show how virtually any bifurcation or resonance known in Hamiltonian systems theory is included in the bifurcation catalogue for the reduced Hamiltonian system and carry out nonlinear bifurcation theory for a representative selection of bifurcation scenarios. In particular, I will present existence results for doubly periodic travelling waves, multi-pulse line solitary waves and further, rather exotic three-dimensional waves.

I will also present some examples in which a complete reduction to a finite-dimensional reduced Hamiltonian system is not possible. In particular, existence theories for modulating pulses yield a reduction of the water-wave problem to a nonlinear wave equation and existence theories for `periodically modulated' solitary waves (which have a solitary-wave profile in the direction of propagation and are periodic in the transverse direction) require an infinite-dimensional version of the Lyapunov centre theorem. The question of the existence of a `fully localised solitary wave' (which decays to zero in all spatial directions) will also be addressed.

m_t + u m_x + b u_x m = \nu m_{xx}

with momentum $m = u - \alpha^2 u_{xx}$ for fluid velocity $u(x,t)$. This equation contains three constant parameters: lengthscale $\alpha$, dimensionless balance parameter $b$ and Navier-Stokes viscosity $\nu$. As we vary these parameters, we find numerically and analytically that:

* The solutions of this 1+1 evolutionary PDE that emerge in different parameter regimes are pulsons, ramps-and-cliffs, peakons, compactons, leftons and solitons that act like billiard balls, with no internal degrees of freedom below the length scale $\alpha$.

* These solutions change stability as we vary the parameter $b$. They change their behavior at the special values $b=0,\pm1,\pm2,\pm3$.

* This family of equations admits the classic Burgers ``ramps-and-cliffs'' solutions which are stable for $-1 < b < 1$ with small viscosity. For $b=0$ additional special solutions are available.

* For $b<-1$, the Burgers ramps-and-cliffs are unstable. The stable pulse solutions that emerge for $b<-1$ move leftward instead of rightward and they tend to a stationary pulse train whose individual pulses are each given when $\nu=0$ by $u(x)=u_0sech^2(x/{2\alpha})$ for the case $b=-2$ and by $u(x)=u_0sech(x/\alpha)$ for the case $b=-3$.

* For $b>1$, the Burgers ramps-and-cliffs are again unstable. The stable solitary traveling wave that develops for $b>1$ and $\nu=0$ is the ``peakon'' solution $u(x,t)=ce^{-|x-ct|/\alpha}$. Nonlinear interactions among these pulsons or peakons are governed by the superposition of solutions for $b>1$ and $\nu=0$,

u(x,t) = \sum_{i=1}^N p_i(t) e^{-|x-q_i(t)|/\alpha}

These peakon solutions obey a finite dimensional dynamical system for the time-dependent speeds $p_i(t)$ and positions $q_i(t)$. We study the peakon interactions analytically, and we determine their fate numerically under adding viscosity.

Many-body Hamiltonians have such singularity at singular configurations as two-body Hamiltonians do at the origin (ie, two-body collision), when expressed in terms spherical polar coordinates. However, the singularity is not essential, so that the energy integral has finite value. This can be proved by using the boundary conditions of wave functions at singular configurations.

This yields an isomorphism between (Q\times Q)/G and \tilde{G}\oplus(S\times S). Both the discrete connection and the associated continuous connection are necessary to express the Discrete Lagrange-Poincare operator in coordinates.

There are a number of well known models for geostrophic flow which differ in the assumed parent model, in the precise scaling assumptions, and the choice of coordinate system. It is often not well understood how these different models relate to each other.

In this talk I will demonstrate an ansatz based on variational asymptotics---involving an infinitesimal change of coordinates in the low-Rossby number expansion of the variational principle---which can reproduce a number of known models as well as new ones. This formulation makes it easy distinguish models based on their analytical properties (their well-posedness and regularity) and choose those most consistent with the model setting: Vortical motion at large scales.

When we look at the the movement of a small body in the gravitational potential of a planet and the sun, we encounter a system that is not even properly degenerate. Usually, this system is approximated by a perturbed Keplerian system and denoted as the spatial Lunar problem. It turns out to be a three degree of freedom system. Because of the superintegrability of the Kepler Hamiltonian its unperturbed part does depend on one of three actions only. The first order term of its perturbation does not depend on all actions either. Thus, motion takes place according to three different time scales.

In this talk we will show that there exist crucial similarities between this kind of system and properly degenerate ones. This leads to the formulation of an appropriate KAM Theorem. A short description of the problems that have to be overcome when applying the Theorem to the lunar problem finishes this talk.

Joint work with Marcel Oliver (Tubingen/Bremen) and Matt West (CalTech).