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The gravitational interaction is generally considered to be too weak to be easily submitted to systematic experimental investigation in the quantum, microscopic, domain. In this paper we attempt to remedy this situation by considering the gravitational influence exerted by a crystalline nanosphere of mesoscopic size on itself, in the semi-classical, mean field, regime. We study in depth the self-localisation process induced by the corresponding non-linear potential of (gravitational) self-interaction. In particular, we characterize the stability of the associated self-collapsed ground state and estimate the magnitude of the corrections that are due to the internal structure of the object (this includes size-effects and corrections due to the discrete, atomic, structure of the sphere). Finally, we derive an approximated, gaussian, dynamics which mimics several essential features of the selfgravitating dynamics and, based on numerical results derived from this model, we propose a concrete experimental setting which we believe might, in the foreseeable future, reveal the existence of gravitational self-interaction effects.

We present a novel way to apply the singularity confinement property as a discrete integrability criterion.We shall use what we call a full deautonomisation approach, which consists in treating the free parameters in the mapping as functions of the independent variable, applied to a mapping complemented with terms that are absent in the original mapping but which do not change the singularity structure. We shall show, on a host of examples including the well-known mapping of Hietarinta-Viallet, that our approach offers a way to compute the algebraic entropy for these mappings exactly, thereby allowing one to distinguish between the integrable and non-integrable cases even when both have confined singularities.

A general symmetry of the bilinear BKP hierarchy is studied in terms of tau functions. We use this symmetry to define reductions of the BKP hierarchy, among which new integrable systems can be found. The reductions are connected to constraints on the Lax operator as well as on the bilinear formulation. A class of solutions for the reduced equations is derived.

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Abstract. The 'deautonomisation' of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage invariably leads to a nonintegrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painleve equations, how their regularisation through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.

Abstract. We present a systematic construction of the discrete KP hierarchy in terms of Sato-Wilson-type shift operators. Reductions of the equations in this hierarchy to 1+1-dimensional integrable lattice systems are considered, and the problems that arise with regard to the symmetry algebra underlying the reduced systems as well as the ultradiscretizability of these systems are discussed. A scheme for constructing ultradiscretizable reductions that give rise to Yang-Baxter maps is explained in two explicit examples.2000 Mathematics Subject Classification. 39A10, 39A12. Introduction.A problem widely recognized as one of the most fundamental open problems in the theory of integrable systems is that of the relation between classical and quantum integrable systems with infinitely many degrees of freedom. Recently a possible route emerged through which a firm connection between the classical and quantum settings might be established: the ultradiscretization of (classical) discrete integrable systems [8,22]. Crystal bases, arising in the zero-temperature limit of quantum enveloping algebras, have been shown to play an important rôle in the description of the dynamical properties of so-called box and ball systems (BBSs) [20], which in turn can be obtained from 1+1-dimensional discrete integrable systems through a special limiting procedure: the ultradiscrete limit [22]. On the other hand, geometric crystals [1] are classical analogues of such crystals. These are such that (quantum) crystal bases can be obtained from them through a limiting procedure very much like the ultradiscrete limit. Geometric crystals have been studied extensively in connection with the combinatorial properties of BBSs [10, 11], but, most importantly, they also offer prime examples of Yang-Baxter (YB) maps [7], i.e. of set-theoretical solutions to the YB equation [6,26]. In particular, it can be shown that the R-matrix associated to such a YB map (or with the tropical version thereof, obtained from the geometric crystal) in the ultradiscrete limit turns into the combinatorial R-matrix that governs the time evolution of the BBS [8]. This R-matrix, in turn, is directly related to the crystal associated with that BBS. The question that remains however,

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