Analytic model for a frictional shallow-water undular bore

We use the integrable Kaup-Boussinesq shallow water system, modified by a small viscous term, to model the formation of an undular bore with a steady profile. The description is made in terms of the corresponding integrable Whitham system, also appropriately modified by friction. This is derived in Riemann variables using a modified finite-gap integration technique for the AKNS scheme. The Whitham system is then reduced to a simple first-order differential equation which is integrated numerically to obtain an asymptotic profile of the undular bore, with the local oscillatory structure described by the periodic solution of the unperturbed Kaup-Boussinesq system. This solution of the Whitham equations is shown to be consistent with certain jump conditions following directly from conservation laws for the original system. A comparison is made with the recently studied dissipationless case for the same system, where the undular bore is unsteady.Comment: 24 page

Dynamics of Macroscopic Tunneling in Elongated BEC

We investigate macroscopic tunneling from an elongated quasi 1-d trap, forming a 'cigar shaped' BEC. Using recently developed formalism we get the leading analytical approximation for the right hand side of the potential wall, i.e. outside the trap, and a formalism based on Wigner functions, for the left side of the potential wall, i.e. inside the BEC. We then present accomplished results of numerical calculations, which show a 'blip' in the particle density traveling with an asymptotic shock velocity, as resulted from previous works on a dot-like trap, but with significant differences from the latter. Inside the BEC a pattern of a traveling dispersive shock wave is revealed. In the attractive case, we find trains of bright solitons frozen near the boundary.Comment: 6 pages, 15 figure

Remnants of dark matter clumps

What happened to the central cores of tidally destructed dark matter clumps in the Galactic halo? We calculate the probability of surviving of the remnants of dark matter clumps in the Galaxy by modelling the tidal destruction of the small-scale clumps. It is demonstrated that a substantial fraction of clump remnants may survive through the tidal destruction during the lifetime of the Galaxy if the radius of a core is rather small. The resulting mass spectrum of survived clumps is extended down to the mass of the core of the cosmologically produced clumps with a minimal mass. Since the annihilation signal is dominated by the dense part of the core, destruction of the outer part of the clump affects the annihilation rate relatively weakly and the survived dense remnants of tidally destructed clumps provide a large contribution to the annihilation signal in the Galaxy. The uncertainties in minimal clump mass resulting from the uncertainties in neutralino models are discussed.Comment: 13 pages, 6 figures, added reference

Evolution of initial discontinuities in the Riemann problem for the Kaup-Boussinesq equation with positive dispersion

We consider the space-time evolution of initial discontinuities of depth and flow velocity for an integrable version of the shallow water Boussinesq system introduced by Kaup. We focus on a specific version of this "Kaup-Boussinesq model" for which a flat water surface is modulationally stable, we speak below of "positive dispersion" model. This model also appears as an approximation to the equations governing the dynamics of polarisation waves in two-component Bose-Einstein condensates. We describe its periodic solutions and the corresponding Whitham modulation equations. The self-similar, one-phase wave structures are composed of different building blocks which are studied in detail. This makes it possible to establish a classification of all the possible wave configurations evolving from initial discontinuities. The analytic results are confirmed by numerical simulations