Robert Knapp, "Transmission of solitons through random media", Physica D 85 (1995) 496-508
Abstract: The one dimensional nonlinear Schrödinger (NLS)
equation with a random potential function is used to model pulses in a
very long low dispersion optical fiber with random fluctuations. When
the initial pulse is a soliton solution of the unperturbed NLS
equation, we show that an approximate theory can be developed for the
effects of the fluctuations by using an equivalent particle method.
Results of direct numerical simulations are presented. Comparisons of
the approximate theory and the direct numerical solutions indicate
that the theory is accurate enough to use for finding statistics about
the transmission of the soliton-like pulses through the fluctuations.
Computations provide evidence for the existence of nonlinear
localization, though with a large localization length.
Roberto Camassa, M Gregory Forest, and Robert Knapp, "Enhancement of optical bistability by periodic layering", Nonlinearity 5 (1992) 721-742
Abstract: We study analytically and numerically the effect of a
periodic perturbation in the index of refraction of a nonlinear
optical medium. Using a one-dimensional model for monochromatic waves
incident on the medium, we find that near resonance between the
perturbation and the modulation of the electric field there can be
significant enhancement of optical bistability in the low intensity
regime. In particular, we discuss in detail the case of the
primary resonance, which is responsible for the most significant
effects, and show how one can optimize the bistability curve for the
implementation of an optical switch, based on the information provided
by the resonance structure.
James M. Hyman, Robert Knapp, and James C. Scovel, "High order finite volume approximations of differential operators on nonuniform grids", Physica D 60 (1992) p112-138
Abstract: When computing numerical solutions to partial differential
equations, difference operators that mimic the crucial properties of
the differential operators are usually more accurate than those that
do not. Properties such as symmetry, conservation, stability, and the
duality relationships between the gradient, curl, and divergence
operators are all important. Using the finite volume method, we have
derived local, accurate, reliable , and efficient difference methods
that mimic these properties on nonuniform rectangular and cuboid
grids. In a finite volume method, the divergence, gradient, and curl
operators are defined using discrete versions of the divergence
theorem and Stokes' theorem. These methods are especially powerful on
coarse nonuniform grids and in calculations where the mesh moves to
track interfaces or shocks. Numerical examples comparing local second
and fourth-order finite volume approximations to conservation laws on
very rough grids are used to demonstrate the advantages of the higher
order methods.
R. Knapp, G. Papanicolaou, and B. White "Transmission of waves by a nonlinear
random medium", Journal of Statistical Physics 63 (1991) 567-583
Abstract: We study analytically and numerically the effect of nonlinearity on
transmission of waves through a random medium. We introduce and
analyze quantities associated with the scattering problem that clarify
the lack of uniqueness due to the nonlinearity as well as the
localization of waves due to the random inhomogeneities. We show that
nonlinearity tends to delocalize the waves and that for very large
scattering regions the average transmitted energy is small.
R. Knapp, G. Papanicolaou, and B. White "Nonlinearity and localization
in one dimensional random media", Disorder and Nonlinearity
A.R. Bishop, D.K. Campbell, and S. Pnevmaticos, eds. (Springer 1989) 2-26
Abstract:
A nonlinear Fabry-Perot etalon with random inhomogeneities is modeled
by a one-dimensional stochastic Helmholtz equation. An asymptotic
estimate of the threshold intensity needed for optical bistability is
found for homogeneous media as a function of length. In the random
case localization is affected and estimates of the energy growth are
derived for large lengths with fixed output. Comparisons of the
theory and numerical simulations are presented.
R. Knapp, "Nonlinearity and localization
in one dimensional random media", Ph.D. thesis, New York University (1988)
Abstract:
A nonlinear Fabry-Perot etalon with random inhomogeneities is modeled
by a one-dimensional stochastic Helmholtz equation. An asymptotic
estimate of the threshold intensity needed for optical bistability is
found for homogeneous media as a function of length. In the random
case localization is affected and estimates of the energy growth are
derived for large lengths with fixed output. FOr fixed input and
length the effect of nonlinearity in the zero intensity limit is
shown to be a decrease in the transmittivity of the etalon.
Comparisons of the theory and numerical simulations are presented.