Bihamiltonian geometry and separation of variables for Toda lattices

We discuss the bihamiltonian geometry of the Toda lattice (periodic and open). Using some recent results on the separation of variables for bihamiltonian manifolds, we show that these systems can be explicitly integrated via the classical Hamilton-Jacobi method in the so-called Darboux-Nijenhuis coordinates.Comment: 12 pages, Latex with amsmath and amssymb. Report of talks given at NEEDS9

On the Higher Poisson Structures of the Camassa–Holm Hierarchy

We find a generating series for the higher Poisson structures of the nonlocal Camassa–Holm hierarchy, following the method used by Enriques, Orlov, and third author for the KdV case

The Sato Grassmannian and the CH hierarchy

We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmannian.Comment: 10 pages, no figure

Simple two-layer dispersive models in the Hamiltonian reduction formalism

A Hamiltonian reduction approach is defined, studied, and finally used to derive asymptotic models of internal wave propagation in density stratified fluids in two-dimensional domains. Beginning with the general Hamiltonian formalism of Benjamin [1] for an ideal, stably stratified Euler fluid, the corresponding structure is systematically reduced to the setup of two homogeneous fluids under gravity, separated by an interface and confined between two infinite horizontal plates. A long-wave, small-amplitude asymptotics is then used to obtain a simplified model that encapsulates most of the known properties of the dynamics of such systems, such as bidirectional wave propagation and maximal amplitude travelling waves in the form of fronts. Further reductions, and in particular devising an asymptotic extension of Dirac's theory of Hamiltonian constraints, lead to the completely integrable evolution equations previously considered in the literature for limiting forms of the dynamics of stratified fluids. To assess the performance of the asymptotic models, special solutions are studied and compared with those of the parent equations.Comment: 29 pages, 4 figure

Multi-Hamiltonian structures for r-matrix systems

For the rational, elliptic and trigonometric r-matrices, we exhibit the links between three "levels" of Poisson spaces: (a) Some finite-dimensional spaces of matrix-valued holomorphic functions on the complex line; (b) Spaces of spectral curves and sheaves supported on them; (c) Symmetric products of a surface. We have, at each level, a linear space of compatible Poisson structures, and the maps relating the levels are Poisson. This leads in a natural way to Nijenhuis coordinates for these spaces. At level (b), there are Hamiltonian systems on these spaces which are integrable for each Poisson structure in the family, and which are such that the Lagrangian leaves are the intersections of the symplective leaves over the Poisson structures in the family. Specific examples include many of the well-known integrable systems.Comment: 26 pages, Plain Te

A 3-component extension of the Camassa-Holm hierarchy

We introduce a bi-Hamiltonian hierarchy on the loop-algebra of sl(2) endowed with a suitable Poisson pair. It gives rise to the usual CH hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a 3-component extension of the CH equation.Comment: 15 pages; minor changes; to appear in Letters in Mathematical Physic

Functional safety assessment of a liquid metal divertor for the European demo tokamak

A reliable strategy for the heat exhaust problem for fusion reactors is among the milestones indicated in EUROfusion (2018). In a fusion reactor, the divertor targets are subject to extremely large heat and particle fluxes. For fusion to be economically feasible, these conditions must be withstood without damage for long time. The “baseline” strategy will be employed for the ITER experiment (which is being built in France) and is based on actively cooled tungsten monoblocks. It is unclear whether this strategy will extrapolate to a future fusion reactor (such as the EU-DEMO, whose pre-conceptual design is ongoing within the EUROfusion consortium). For this reason, alternative solutions are under study, which will eventually be tested in a dedicated experiment in Italy, namely the Divertor Tokamak Test (DTT). One possibility is to employ liquid metal divertors (LMDs), for which the plasma-facing surface is inherently self-healing and immune to thermo-mechanical stresses. Within the framework of the pre-conceptual design of an LMD for the EU-DEMO, safety issues need to be considered at an early stage. In this work we present a preliminary but systematic safety analysis for this system, by means of the Functional Failure Mode and Effect Analysis (FFMEA). The FFMEA allows to identify possible accident initiators for systems undergoing pre-conceptual design, when more specific safety evaluations (e.g. at the component level) are not possible, US Nuclear Regulatory Commission (2009). This is done by postulating the loss of a system function rather than a specific component failure, thus compensating for the lack of detailed design information. For each function, the potential causes of its loss, a plausible evolution and preventive and mitigative measures are investigated, possibly specifying the need for further information. The initiating events are grouped according to consequences and the plant response. For each group, the Postulated Initiating Event (PIE) is chosen. The PIEs list drives and limits the set of accidental scenarios which will undergo deterministic analysis in a successive phase of the work, in order to evaluate the capacity of the system to withstand/mitigate its consequences. This will assess whether safety limits are respected or whether additional safety provisions are required. From the PIEs list, the design basis accident (DBA) and beyond design basis accident (BDBA) will eventually be selected