75 research outputs found
Phase Shift in the Whitham Zone for the Gurevich-Pitaevskii Special Solution of the Korteweg-de Vries Equation
We get the leading term of the Gurevich-Pitaevskii special solution to the
KdV equation in the oscillation zone without using averaging methods.Comment: 13 pages, 3 figure
Autoresonance in a Dissipative System
We study the autoresonant solution of Duffing's equation in the presence of
dissipation. This solution is proved to be an attracting set. We evaluate the
maximal amplitude of the autoresonant solution and the time of transition from
autoresonant growth of the amplitude to the mode of fast oscillations.
Analytical results are illustrated by numerical simulations.Comment: 22 pages, 3 figure
Asymptotics for a special solution to the second member of the Painleve I hierarchy
We study the asymptotic behavior of a special smooth solution y(x,t) to the
second member of the Painleve I hierarchy. This solution arises in random
matrix theory and in the study of Hamiltonian perturbations of hyperbolic
equations. The asymptotic behavior of y(x,t) if x\to \pm\infty (for fixed t) is
known and relatively simple, but it turns out to be more subtle when x and t
tend to infinity simultaneously. We distinguish a region of algebraic
asymptotic behavior and a region of elliptic asymptotic behavior, and we obtain
rigorous asymptotics in both regions. We also discuss two critical transitional
asymptotic regimes.Comment: 19 page
Properties of field functionals and characterization of local functionals
Functionals (i.e. functions of functions) are widely used in quantum field
theory and solid-state physics. In this paper, functionals are given a rigorous
mathematical framework and their main properties are described. The choice of
the proper space of test functions (smooth functions) and of the relevant
concept of differential (Bastiani differential) are discussed.
The relation between the multiple derivatives of a functional and the
corresponding distributions is described in detail. It is proved that, in a
neighborhood of every test function, the support of a smooth functional is
uniformly compactly supported and the order of the corresponding distribution
is uniformly bounded. Relying on a recent work by Yoann Dabrowski, several
spaces of functionals are furnished with a complete and nuclear topology. In
view of physical applications, it is shown that most formal manipulations can
be given a rigorous meaning.
A new concept of local functionals is proposed and two characterizations of
them are given: the first one uses the additivity (or Hammerstein) property,
the second one is a variant of Peetre's theorem. Finally, the first step of a
cohomological approach to quantum field theory is carried out by proving a
global Poincar\'e lemma and defining multi-vector fields and graded functionals
within our framework.Comment: 32 pages, no figur
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