857 research outputs found
Formation of Quantum Shock Waves by Merging and Splitting Bose-Einstein Condensates
The processes of merging and splitting dilute-gas Bose-Einstein condensates
are studied in the nonadiabatic, high-density regime. Rich dynamics are found.
Depending on the experimental parameters, uniform soliton trains containing
more than ten solitons or the formation of a high-density bulge as well as
quantum (or dispersive) shock waves are observed experimentally within merged
BECs. Our numerical simulations indicate the formation of many vortex rings. In
the case of splitting a BEC, the transition from sound-wave formation to
dispersive shock-wave formation is studied by use of increasingly stronger
splitting barriers. These experiments realize prototypical dispersive shock
situations.Comment: 10 pages, 8 figure
Dispersive and diffusive-dispersive shock waves for nonconvex conservation laws
We consider two physically and mathematically distinct regularization
mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the
combination of diffusion and dispersion are known to give rise to monotonic and
oscillatory traveling waves that approximate shock waves. The zero-diffusion
limits of these traveling waves are dynamically expanding dispersive shock
waves (DSWs). A richer set of wave solutions can be found when the flux is
non-convex. This review compares the structure of solutions of Riemann problems
for a conservation law with non-convex, cubic flux regularized by two different
mechanisms: 1) dispersion in the modified Korteweg--de Vries (mKdV) equation;
and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation.
In the first case, the possible dynamics involve two qualitatively different
types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the
second case, in addition to RWs, there are traveling wave solutions
approximating both classical (Lax) and non-classical (undercompressive) shock
waves. Despite the singular nature of the zero-diffusion limit and rather
differing analytical approaches employed in the descriptions of dispersive and
diffusive-dispersive regularization, the resulting comparison of the two cases
reveals a number of striking parallels. In contrast to the case of convex flux,
the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is
identified as an undercompressive DSW. Other prominent features, such as
shock-rarefactions, also find their purely dispersive counterparts involving
special contact DSWs, which exhibit features analogous to contact
discontinuities. This review describes an important link between two major
areas of applied mathematics, hyperbolic conservation laws and nonlinear
dispersive waves.Comment: Revision from v2; 57 pages, 19 figure
Defect Modes and Homogenization of Periodic Schr\"odinger Operators
We consider the discrete eigenvalues of the operator
H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x), where V(\x) is periodic and Q(\y)
is localized on . For \eps>0 and sufficiently small, discrete
eigenvalues may bifurcate (emerge) from spectral band edges of the periodic
Schr\"odinger operator, H_0 = -\Delta_\x+V(\x), into spectral gaps. The
nature of the bifurcation depends on the homogenized Schr\"odinger operator
L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y). Here, denotes the inverse
effective mass matrix, associated with the spectral band edge, which is the
site of the bifurcation.Comment: 26 pages, 3 figures, to appear SIAM J. Math. Ana
Socially stable matchings in the hospitals / residents problem
In the Hospitals/Residents (HR) problem, agents are partitioned into hospitals and residents. Each agent wishes to be matched to an agent in the other set and has a strict preference over these potential matches. A matching is stable if there are no blocking pairs, i.e., no pair of agents that prefer each other to their assigned matches. Such a situation is undesirable as it could lead to a deviation in which the blocking pair form a private arrangement outside the matching. This however assumes that the blocking pair have social ties or communication channels to facilitate the deviation. Relaxing the stability definition to take account of the potential lack of social ties between agents can yield larger stable matchings.
In this paper, we define the Hospitals/Residents problem under Social Stability (HRSS) which takes into account social ties between agents by introducing a social network graph to the HR problem. Edges in the social network graph correspond to resident-hospital pairs in the HR instance that know one another. Pairs that do not have corresponding edges in the social network graph can belong to a matching M but they can never block M. Relative to a relaxed stability definition for HRSS, called social stability, we show that socially stable matchings can have different sizes and the problem of finding a maximum socially stable matching is NP-hard, though approximable within 3/2. Furthermore we give polynomial time algorithms for three special cases of the problem
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