19,943 research outputs found
On the Newtonian Limit of General Relativity
We find a choice of variables for the 3+1 formulation of general relativity
which casts the evolution equations into (flux-conservative)
symmetric-hyperbolic first order form for arbitrary lapse and shift, for the
first time. We redefine the lapse function in terms of the determinant of the
3-metric and a free function U which embodies the lapse freedom. By rescaling
the variables with appropriate factors of 1/c, the system is shown to have a
smooth Newtonian limit when the redefined lapse U and the shift are fixed by
means of elliptic equations to be satisfied on each time slice. We give a
prescription for the choice of appropriate initial data with controlled
extra-radiation content, based on the theory of problems with different
time-scales. Our results are local, in the sense that we are not concerned with
the treatment of asymptotic regions. On the other hand, this local theory is
all what is needed for most problems of practical numerical computation.Comment: 16 pages, uses REVTe
Some implications of signature-change in cosmological models of loop quantum gravity
Signature change at high density has been obtained as a possible consequence
of deformed space-time structures in models of loop quantum gravity. This
article provides a conceptual discussion of implications for cosmological
scenarios, based on an application of mathematical results for mixed-type
partial differential equations (the Tricomi problem). While the effective
equations from which signature change has been derived are shown to be locally
regular and therefore reliable, the underlying theory of loop quantum gravity
may face several global problems in its semiclassical solutions.Comment: 35 pages, 5 figure
Renormalization Group Analysis of Boundary Conditions in Potential Scattering
We analyze how a short distance boundary condition for the Schrodinger
equation must change as a function of the boundary radius by imposing the
physical requirement of phase shift independence on the boundary condition. The
resulting equation can be interpreted as a variable phase equation of a
complementary boundary value problem. We discuss the corresponding infrared
fixed points and the perturbative expansion around them generating a short
distance modified effective range theory. We also discuss ultraviolet fixed
points, limit cycles and attractors with a given fractality which take place
for singular attractive potentials at the origin. The scaling behaviour of
scattering observables can analytically be determined and is studied with some
emphasis on the low energy nucleon-nucleon interaction via singular pion
exchange potentials. The generalization to coupled channels is also studied.Comment: 31 pages, 8 figure
Asymptotic methods for delay equations.
Asymptotic methods for singularly perturbed delay differential equations are in many ways more challenging to implement than for ordinary differential equations. In this paper, four examples of delayed systems which occur in practical models are considered: the delayed recruitment equation, relaxation oscillations in stem cell control, the delayed logistic equation, and density wave oscillations in boilers, the last of these being a problem of concern in engineering two-phase flows. The ways in which asymptotic methods can be used vary from the straightforward to the perverse, and illustrate the general technical difficulties that delay equations provide for the central technique of the applied mathematician. © Springer 2006
The Semiclassical Modified Nonlinear Schroedinger Equation I: Modulation Theory and Spectral Analysis
We study an integrable modification of the focusing nonlinear
Schroedinger equation from the point of view of semiclassical asymptotics. In
particular, (i) we establish several important consequences of the mixed-type
limiting quasilinear system including the existence of maps that embed the
limiting forms of both the focusing and defocusing nonlinear Schroedinger
equations into the framework of a single limiting system for the modified
equation, (ii) we obtain bounds for the location of discrete spectrum for the
associated spectral problem that are particularly suited to the semiclassical
limit and that generalize known results for the spectrum of the nonselfadjoint
Zakharov-Shabat spectral problem, and (iii) we present a multiparameter family
of initial data for which we solve the associated spectral problem in terms of
special functions for all values of the semiclassical scaling parameter. We
view our results as part of a broader project to analyze the semiclassical
limit of the modified nonlinear Schroedinger equation via the noncommutative
steepest descent procedure of Deift and Zhou, and we also present a
self-contained development of a Riemann-Hilbert problem of inverse scattering
that differs from those given in the literature and that is well-adapted to
semiclassical asymptotics.Comment: 56 Pages, 21 Figure
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