120,464 research outputs found
Interacting String Multi-verses and Holographic Instabilities of Massive Gravity
Products of large-N conformal field theories coupled by multi-trace
interactions in diverse dimensions are used to define quantum multi-gravity
(multi-string theory) on a union of (asymptotically) AdS spaces. One-loop
effects generate a small O(1/N) mass for some of the gravitons. The boundary
gauge theory and the AdS/CFT correspondence are used as guiding principles to
study and draw conclusions on some of the well known problems of massive
gravity - classical instabilities and strong coupling effects. We find examples
of stable multi-graviton theories where the usual strong coupling effects of
the scalar mode of the graviton are suppressed. Our examples require a fine
tuning of the boundary conditions in AdS. Without it, the spacetime background
backreacts in order to erase the effects of the graviton mass.Comment: 51 pages, 3 figures; v2 typos corrected, version published in NPB; v3
added appendix E on general class of fixed points in multi-trace deformation
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
Global stability of steady states in the classical Stefan problem
The classical one-phase Stefan problem (without surface tension) allows for a
continuum of steady state solutions, given by an arbitrary (but sufficiently
smooth) domain together with zero temperature. We prove global-in-time
stability of such steady states, assuming a sufficient degree of smoothness on
the initial domain, but without any a priori restriction on the convexity
properties of the initial shape. This is an extension of our previous result
[28] in which we studied nearly spherical shapes.Comment: 14 pages. arXiv admin note: substantial text overlap with
arXiv:1212.142
Defect-Mediated Stability: An Effective Hydrodynamic Theory of Spatio-Temporal Chaos
Spatiotemporal chaos (STC) exhibited by the Kuramoto-Sivashinsky (KS)
equation is investigated analytically and numerically. An effective stochastic
equation belonging to the KPZ universality class is constructed by
incorporating the chaotic dynamics of the small KS system in a coarse-graining
procedure. The bare parameters of the effective theory are computed
approximately. Stability of the system is shown to be mediated by space-time
defects that are accompanied by stochasticity. The method of analysis and the
mechanism of stability may be relevant to a class of STC problems.Comment: 34 pages + 9 figure
Pattern formation driven by cross--diffusion in a 2D domain
In this work we investigate the process of pattern formation in a two
dimensional domain for a reaction-diffusion system with nonlinear diffusion
terms and the competitive Lotka-Volterra kinetics. The linear stability
analysis shows that cross-diffusion, through Turing bifurcation, is the key
mechanism for the formation of spatial patterns. We show that the bifurcation
can be regular, degenerate non-resonant and resonant. We use multiple scales
expansions to derive the amplitude equations appropriate for each case and show
that the system supports patterns like rolls, squares, mixed-mode patterns,
supersquares, hexagonal patterns
The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions
Fundamental global similarity solutions of the standard form
u_\g(x,t)=t^{-\a_\g} f_\g(y), with the rescaled variable y= x/{t^{\b_\g}},
\b_\g= \frac {1-n \a_\g}{10}, where \a_\g>0 are real nonlinear eigenvalues (\g
is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} =
\nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The
present paper continues the study began by the authors in the previous paper
P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem
for a tenth-order thin film equation I. Bifurcation of self-similar oscillatory
fundamental solutions, Mediterranean Journal of Mathematics, No. 4, Vol. 10
(2013), 1759-1790.
Thus, the following questions are also under scrutiny:
(I) Further study of the limit n \to 0, where the behaviour of finite
interfaces and solutions as y \to infinity are described. In particular, for
N=1, the interfaces are shown to diverge as follows: |x_0(t)| \sim 10 \left(
\frac{1}{n}\sec\left( \frac{4\pi}{9} \right) \right)^{\frac 9{10}} t^{\frac
1{10}} \to \infty as n \to 0^+.
(II) For a fixed n \in (0, \frac 98), oscillatory structures of solutions
near interfaces.
(III) Again, for a fixed n \in (0, \frac 98), global structures of some
nonlinear eigenfunctions \{f_\g\}_{|\g| \ge 0} by a combination of numerical
and analytical methods
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