1,330 research outputs found
Dissipative perturbations for the K(n,n) Rosenau-Hyman equation
Compactons are compactly supported solitary waves for nondissipative
evolution equations with nonlinear dispersion. In applications, these model
equations are accompanied by dissipative terms which can be treated as small
perturbations. We apply the method of adiabatic perturbations to compactons
governed by the K(n,n) Rosenau-Hyman equation in the presence of dissipative
terms preserving the "mass" of the compactons. The evolution equations for both
the velocity and the amplitude of the compactons are determined for some linear
and nonlinear dissipative terms: second-, fourth-, and sixth-order in the
former case, and second- and fourth-order in the latter one. The numerical
validation of the method is presented for a fourth-order, linear, dissipative
perturbation which corresponds to a singular perturbation term
Soliton approach to the noisy Burgers equation: Steepest descent method
The noisy Burgers equation in one spatial dimension is analyzed by means of
the Martin-Siggia-Rose technique in functional form. In a canonical formulation
the morphology and scaling behavior are accessed by mean of a principle of
least action in the asymptotic non-perturbative weak noise limit. The ensuing
coupled saddle point field equations for the local slope and noise fields,
replacing the noisy Burgers equation, are solved yielding nonlinear localized
soliton solutions and extended linear diffusive mode solutions, describing the
morphology of a growing interface. The canonical formalism and the principle of
least action also associate momentum, energy, and action with a
soliton-diffusive mode configuration and thus provides a selection criterion
for the noise-induced fluctuations. In a ``quantum mechanical'' representation
of the path integral the noise fluctuations, corresponding to different paths
in the path integral, are interpreted as ``quantum fluctuations'' and the
growth morphology represented by a Landau-type quasi-particle gas of ``quantum
solitons'' with gapless dispersion and ``quantum diffusive modes'' with a gap
in the spectrum. Finally, the scaling properties are dicussed from a heuristic
point of view in terms of a``quantum spectral representation'' for the slope
correlations. The dynamic eponent z=3/2 is given by the gapless soliton
dispersion law, whereas the roughness exponent zeta =1/2 follows from a
regularity property of the form factor in the spectral representation. A
heuristic expression for the scaling function is given by spectral
representation and has a form similar to the probability distribution for Levy
flights with index .Comment: 30 pages, Revtex file, 14 figures, to be submitted to Phys. Rev.
Cumulant expansion for studying damped quantum solitons
The quantum statistics of damped optical solitons is studied using
cumulant-expansion techniques. The effect of absorption is described in terms
of ordinary Markovian relaxation theory, by coupling the optical field to a
continuum of reservoir modes. After introduction of local bosonic field
operators and spatial discretization pseudo-Fokker-Planck equations for
multidimensional s-parameterized phase-space functions are derived. These
partial differential equations are equivalent to an infinite set of ordinary
differential equations for the cumulants of the phase-space functions.
Introducing an appropriate truncation condition, the resulting finite set of
cumulant evolution equations can be solved numerically. Solutions are presented
in Gaussian approximation and the quantum noise is calculated, with special
emphasis on squeezing and the recently measured spectral photon-number
correlations [Spaelter et al., Phys. Rev. Lett. 81, 786 (1998)].Comment: 17 pages, 13 figures, revtex, psfig, multicols, published in
Phys.Rev.
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