114 research outputs found
An asymptotic expansion of the hyberbolic umbilic catastrophe integral
We obtain an asymptotic expansion of the hyperbolic umbilic catastrophe integral Ψ(H) (x,y,z) := ∫∞−∞∫∞−∞exp(i(s3+t3+zst +yt+xs))ds dt
for large values of |x| and bounded values of |y| and |z|. The expansion is given in terms of Airy functions and inverse powers of x. There is only one Stokes ray at argx=Ï€
. We use the modified saddle point method introduced in (López et al. J Math Anal Appl 354(1):347–359, 2009). The accuracy and the asymptotic character of the approximations are illustrated with numerical experiments.This research was supported by the Universidad Pública de Navarra, research grant PRO-UPNA (6158) 01/01/2022
Correlation Functions for a Chain of Short Range Oscillators
We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t-13 for position and momentum correlations and as t-23 for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t-14 for position and momentum correlators and with rate t-12 for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly
Correlation Functions for a Chain of Short Range Oscillators
We consider a system of harmonic oscillators with short range interactions
and we study their correlation functions when the initial data is sampled with
respect to the Gibbs measure. Such correlation functions display rapid
oscillations that travel through the chain. We show that the correlation
functions always have two fastest peaks which move in opposite directions and
decay at rate for position and momentum correlations and as
for energy correlations. The shape of these peaks is
asymptotically described by the Airy function. Furthermore, the correlation
functions have some non generic peaks with lower decay rates. In particular,
there are peaks which decay at rate for position and
momentum correlators and with rate for energy correlators.
The shape of these peaks is described by the Pearcey integral. Crucial for our
analysis is an appropriate generalisation of spacings, i.e. differences of the
positions of neighbouring particles, that are used as spatial variables in the
case of nearest neighbour interactions. Using the theory of circulant matrices
we are able to introduce a quantity that retains both localisation and analytic
viability. This also allows us to define and analyse some additional quantities
used for nearest neighbour chains. Finally, we study numerically the evolution
of the correlation functions after adding nonlinear perturbations to our model.
Within the time range of our numerical simulations the asymptotic description
of the linear case seems to persist for small nonlinear perturbations while
stronger nonlinearities change shape and decay rates of the peaks
significantly.Comment: 25 pages, 6 figure
- …