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## Publications of Stephanos Venakides    :chronological  alphabetical  combined listing:

%% Papers Published
@article{fds341952,
Author = {Perez-Arancibia, C and Shipman, SP and Turc, C and Venakides,
S},
Title = {Domain Decomposition for Quasi-Periodic Scattering by
Layered Media via Robust Boundary-Integral Equations at All
Frequencies},
Journal = {Communications in Computational Physics},
Volume = {26},
Number = {1},
Pages = {265-310},
Publisher = {Global Science Press},
Year = {2019},
Month = {July},
url = {http://dx.doi.org/10.4208/cicp.OA-2018-0021},
Doi = {10.4208/cicp.OA-2018-0021},
Key = {fds341952}
}

@article{fds335545,
Author = {Aristotelous, AC and Crawford, JM and Edwards, GS and Kiehart, DP and Venakides, S},
Title = {Mathematical models of dorsal closure.},
Journal = {Progress in Biophysics and Molecular Biology},
Volume = {137},
Pages = {111-131},
Year = {2018},
Month = {September},
url = {http://dx.doi.org/10.1016/j.pbiomolbio.2018.05.009},
Abstract = {Dorsal closure is a model cell sheet movement that occurs
midway through Drosophila embryogenesis. A dorsal hole,
filled with amnioserosa, closes through the dorsalward
elongation of lateral epidermal cell sheets. Closure
requires contributions from 5 distinct tissues and well over
140 genes (see Mortensen et al., 2018, reviewed in Kiehart
et al., 2017 and Hayes and Solon, 2017). In spite of this
biological complexity, the movements (kinematics) of closure
are geometrically simple at tissue, and in certain cases, at
cellular scales. This simplicity has made closure the target
of a number of mathematical models that seek to explain and
quantify the processes that underlie closure's kinematics.
The first (purely kinematic) modeling approach recapitulated
well the time-evolving geometry of closure even though the
underlying physical principles were not known. Almost all
subsequent models delve into the forces of closure (i.e. the
dynamics of closure). Models assign elastic, contractile and
viscous forces which impact tissue and/or cell mechanics.
They write rate equations which relate the forces to one
another and to other variables, including those which
represent geometric, kinematic, and or signaling
characteristics. The time evolution of the variables is
obtained by computing the solution of the model's system of
equations, with optimized model parameters. The basis of the
equations range from the phenomenological to biophysical
first principles. We review various models and present their
contribution to our understanding of the molecular
mechanisms and biophysics of closure. Models of closure will
contribute to our understanding of similar movements that
characterize vertebrate morphogenesis.},
Doi = {10.1016/j.pbiomolbio.2018.05.009},
Key = {fds335545}
}

@article{fds340300,
Author = {Venakides, S and Komineas, S and Melcher, C},
Title = {Traveling domain walls in chiral ferromagnets},
Journal = {Nonlinearity},
Publisher = {London Mathematical Society},
Year = {2018},
Month = {June},
Key = {fds340300}
}

@article{fds330525,
Author = {Perez-Arancibia, C and Shipman, S and Turc, C and Venakides,
S},
Title = {DDM solutions of quasiperiodic transmission problems in
layered media via robust boundary integral equations at all
frequencies},
Journal = {Communications in Computational Physics},
Publisher = {Global Science Press},
Year = {2018},
Month = {May},
Key = {fds330525}
}

@article{fds330399,
Author = {Bruno, OP and Shipman, SP and Turc, C and Stephanos,
V},
Title = {Three-dimensional quasi-periodic shifted Green function
throughout the spectrum, including Wood anomalies.},
Journal = {Proceedings. Mathematical, Physical, and Engineering
Sciences},
Volume = {473},
Number = {2207},
Pages = {20170242},
Publisher = {The Royal Society},
Year = {2017},
Month = {November},
url = {http://dx.doi.org/10.1098/rspa.2017.0242},
Abstract = {This work, part II in a series, presents an efficient method
for evaluation of wave scattering by doubly periodic
diffraction gratings at or near what are commonly called
'Wood anomaly frequencies'. At these frequencies, there is a
grazing Rayleigh wave, and the quasi-periodic Green function
ceases to exist. We present a modification of the Green
function by adding two types of terms to its lattice sum.
The first type are transversely shifted Green functions with
coefficients that annihilate the growth in the original
lattice sum and yield algebraic convergence. The second type
are quasi-periodic plane wave solutions of the Helmholtz
equation which reinstate certain necessary grazing modes
without leading to blow-up at Wood anomalies. Using the new
quasi-periodic Green function, we establish, for the first
time, that the Dirichlet problem of scattering by a smooth
doubly periodic scattering surface at a Wood frequency is
uniquely solvable. We also present an efficient high-order
numerical method based on this new Green function for
scattering by doubly periodic surfaces at and around Wood
frequencies. We believe this is the first solver able to
handle Wood frequencies for doubly periodic scattering
problems in three dimensions. We demonstrate the method by
applying it to acoustic scattering.},
Doi = {10.1098/rspa.2017.0242},
Key = {fds330399}
}

@article{fds329310,
Author = {Kiehart, DP and Crawford, JM and Aristotelous, A and Venakides, S and Edwards, GS},
Title = {Cell Sheet Morphogenesis: Dorsal Closure in Drosophila
melanogaster as a Model System.},
Journal = {Annual Review of Cell and Developmental Biology},
Volume = {33},
Pages = {169-202},
Year = {2017},
Month = {October},
url = {http://dx.doi.org/10.1146/annurev-cellbio-111315-125357},
Abstract = {Dorsal closure is a key process during Drosophila
morphogenesis that models cell sheet movements in chordates,
including neural tube closure, palate formation, and wound
healing. Closure occurs midway through embryogenesis and
entails circumferential elongation of lateral epidermal cell
sheets that close a dorsal hole filled with amnioserosa
cells. Signaling pathways regulate the function of cellular
structures and processes, including Actomyosin and
complexes, and endocytosis/vesicle trafficking. These
orchestrate complex shape changes and movements that entail
interactions between five distinct cell types. Genetic and
laser perturbation studies establish that closure is robust,
resilient, and the consequence of redundancy that
contributes to four distinct biophysical processes:
contraction of the amnioserosa, contraction of supracellular
Actomyosin cables, elongation (stretching?) of the lateral
epidermis, and zipping together of two converging cell
sheets. What triggers closure and what the emergent
properties are that give rise to its extraordinary
resilience and fidelity remain key, extant
questions.},
Doi = {10.1146/annurev-cellbio-111315-125357},
Key = {fds329310}
}

@article{fds320539,
Author = {Bruno, OP and Shipman, SP and Turc, C and Venakides,
S},
Title = {Superalgebraically convergent smoothly windowed lattice sums
for doubly periodic Green functions in three-dimensional
space},
Journal = {Proceedings. Mathematical, Physical, and Engineering
Sciences},
Volume = {472},
Number = {2191},
Pages = {20160255},
Year = {2016},
Month = {July},
url = {http://dx.doi.org/10.1098/rspa.2016.0255},
work, part I in a two-part series, presents: (i) a simple
and highly efficient algorithm for evaluation of
quasi-periodic Green functions, as well as (ii) an
associated boundary-integral equation method for the
numerical solution of problems of scattering of waves by
doubly periodic arrays of scatterers in three-dimensional
space. Except for certain 'Wood frequencies' at which the
quasi-periodic Green function ceases to exist, the proposed
approach, which is based on smooth windowing functions,
gives rise to tapered lattice sums which converge
superalgebraically fast to the Green function-that is,
faster than any power of the number of terms used. This is
in sharp contrast to the extremely slow convergence
exhibited by the lattice sums in the absence of smooth
windowing. (The Wood-frequency problem is treated in part
II.) This paper establishes rigorously the superalgebraic
convergence of the windowed lattice sums. A variety of
numerical results demonstrate the practical efficiency of
the proposed approach.},
Doi = {10.1098/rspa.2016.0255},
Key = {fds320539}
}

@article{fds320428,
Author = {Komineas, S and Shipman, SP and Venakides, S},
Title = {Lossless polariton solitons},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {316},
Pages = {43-56},
Publisher = {Elsevier BV},
Year = {2016},
Month = {February},
url = {http://dx.doi.org/10.1016/j.physd.2015.10.018},
excitons in a semiconductor microcavity interact to form
exciton-polariton condensates. These are governed by a
nonlinear quantum-mechanical system involving exciton and
photon wavefunctions. We calculate all non-traveling
harmonic soliton solutions for the one-dimensional lossless
system. There are two frequency bands of bright solitons
when the inter-exciton interactions produce an attractive
nonlinearity and two frequency bands of dark solitons when
the nonlinearity is repulsive. In addition, there are two
frequency bands for which the exciton wavefunction is
discontinuous at its symmetry point, where it undergoes a
phase jump of π. A band of continuous dark solitons merges
with a band of discontinuous dark solitons, forming a larger
band over which the soliton far-field amplitude varies from
0 to ∞ ; the discontinuity is initiated when the operating
frequency exceeds the free exciton frequency. The far fields
of the solitons in the lowest and highest frequency bands
(one discontinuous and one continuous dark) are linearly
unstable, whereas the other four bands have linearly stable
far fields, including the merged band of dark
solitons.},
Doi = {10.1016/j.physd.2015.10.018},
Key = {fds320428}
}

@article{fds226701,
Author = {Sergey Belov and Stephanos Venakides},
Title = {Smooth parametric dependence of asymptotics of the
semiclassical focusing NLS},
Journal = {Analysis & PDE},
Volume = {8},
Number = {2},
Pages = {257-288},
Year = {2015},
Month = {April},
url = {http://dx.doi.org/10.2140/apde.2015.8.257},
Abstract = {We consider the one-dimensional focusing (cubic) nonlinear
Schrödinger equation (NLS) in the semiclassical limit with
exponentially decaying complex-valued initial data, whose
phase is multiplied by a real parameter. We prove smooth
dependence of the asymptotic solution on the parameter.
Numerical results supporting our estimates of important
quantities are presented.},
Doi = {10.2140/apde.2015.8.257},
Key = {fds226701}
}

@article{fds244138,
Author = {Komineas, S and Shipman, SP and Venakides, S},
Title = {Continuous and discontinuous dark solitons in polariton
condensates},
Journal = {Physical Review B},
Volume = {91},
Number = {13},
Publisher = {American Physical Society (APS)},
Year = {2015},
Month = {April},
ISSN = {1098-0121},
url = {http://dx.doi.org/10.1103/PhysRevB.91.134503},
Abstract = {© 2015 American Physical Society. Bose-Einstein condensates
of exciton-polaritons are described by a Schrödinger system
of two equations. Nonlinearity due to exciton interactions
gives rise to a frequency band of dark soliton solutions,
which are found analytically for the lossless zero-velocity
case. The soliton's far-field value varies from zero to
infinity as the operating frequency varies across the band.
For positive detuning (photon frequency higher than exciton
frequency), the exciton wave function becomes discontinuous
when the operating frequency exceeds the exciton frequency.
This phenomenon lies outside the parameter regime of
validity of the Gross-Pitaevskii (GP) model. Within its
regime of validity, we give a derivation of a single-mode GP
model from the initial Schrödinger system and compare the
continuous polariton solitons and GP solitons using the
healing length notion.},
Doi = {10.1103/PhysRevB.91.134503},
Key = {fds244138}
}

@article{fds303561,
Author = {Belov, S and Venakides, S},
Title = {Smooth parametric dependence of asymptotics of the
semiclassical focusing NLS},
Journal = {Analysis & Pde},
Volume = {8},
Number = {2},
Pages = {257-288},
Publisher = {Mathematical Sciences Publishers},
Year = {2015},
Month = {January},
url = {http://arxiv.org/abs/1211.7111v2},
Abstract = {© Mathematical Sciences Publishers. We consider the
one-dimensional focusing (cubic) nonlinear Schrödinger
equation (NLS) in the semiclassical limit with exponentially
decaying complex-valued initial data, whose phase is
multiplied by a real parameter. We prove smooth dependence
of the asymptotic solution on the parameter. Numerical
results supporting our estimates of important quantities are
presented.},
Doi = {10.2140/apde.2015.8.257},
Key = {fds303561}
}

@article{fds320540,
Author = {Belov, S and Venakides, S},
Title = {Long-time limit studies of an obstruction in the g-function
mechanism for semiclassical focusing NLS},
Year = {2015},
Abstract = {We consider the long-time properties of the an obstruction
in the Riemann-Hilbert approach to one dimensional focusing
Nonlinear Schr\"odinger equation in the semiclassical limit
for a one parameter family of initial conditions. For
certain values of the parameter a large number of solitons
in the system interfere with the $g$-function mechanism in
the steepest descent to oscillatory Riemann-Hilbert
problems. The obstruction prevents the Riemann-Hilbert
analysis in a region in $(x,t)$ plane. We obtain the long
time asymptotics of the boundary of the region (obstruction
curve). As $t\to\infty$ the obstruction curve has a vertical
asymptotes $x=\pm \ln 2$. The asymptotic analysis is
supported with numerical results.},
Key = {fds320540}
}

@article{fds220956,
Author = {Oscar P. Bruno and Stephen P. Shipman and Catalin Turc and Stephanos
Venakides},
Title = {Efficient Evaluation of Doubly Periodic Green Functions in
3D Scattering, Including Wood Anomaly Frequencies},
Journal = {ArXiv>Mathematics > Analysis of PDEs},
Year = {2013},
Month = {July},
url = {http://arxiv.org/abs/1307.1176},
Abstract = {We present efficient methods for computing wave scattering
by diffraction gratings that exhibit two-dimensional
periodicity in three dimensional (3D) space. Applications
include scattering in acoustics, electromagnetics and
elasticity. Our approach uses boundary-integral equations.
The quasi-periodic Green function is a doubly infinite sum
of scaled 3D free-space outgoing Helmholtz Green functions.
Their source points are located at the nodes of a
periodicity lattice of the grating. For efficient numerical
computation of the lattice sum, we employ a smooth
truncation. Super-algebraic convergence to the Green
function is achieved as the truncation radius increases,
except at frequency-wavenumber pairs at which a Rayleigh
wave is at exactly grazing incidence to the grating. At
these "Wood frequencies", the term in the Fourier series
representation of the Green function that corresponds to the
grazing Rayleigh wave acquires an infinite coefficient and
the lattice sum blows up. At Wood frequencies, we modify the
Green function by adding two types of terms to it. The first
type adds weighted spatial shifts of the Green function to
itself with singularities below the grating; this yields
algebraic convergence. The second-type terms are
quasi-periodic plane wave solutions of the Helmholtz
equation. They reinstate (with controlled coefficients) the
grazing modes, effectively eliminated by the terms of first
type. These modes are needed in the Green function for
guaranteeing the well-posedness of the boundary-integral
equation that yields the scattered field. We apply this
approach to acoustic scattering by a doubly periodic 2D
grating near and at Wood frequencies and scattering by a
doubly periodic array of scatterers away from Wood
frequencies.},
Key = {fds220956}
}

@article{fds244139,
Author = {Jackson, AD and Huang, D and Gauthier, DJ and Venakides,
S},
Title = {Destructive impact of imperfect beam collimation in
extraordinary optical transmission.},
Journal = {Journal of the Optical Society of America
A},
Volume = {30},
Number = {6},
Pages = {1281-1290},
Year = {2013},
Month = {June},
ISSN = {1084-7529},
url = {http://dx.doi.org/10.1364/josaa.30.001281},
Abstract = {We investigate the difference between analytic predictions,
numerical simulations, and experiments measuring the
transmission of energy through subwavelength, periodically
arranged holes in a metal film. At normal incidence, theory
predicts a sharp transmission minimum when the wavelength is
equal to the periodicity, and sharp transmission maxima at
one or more nearby wavelengths. In experiments, the sharpest
maximum from the theory is not observed, while the others
appear less sharp. In numerical simulations using commercial
electromagnetic field solvers, we find that the sharpest
maximum appears and approaches our predictions as the
computational resources are increased. To determine possible
origins of the destruction of the sharp maximum, we
incorporate additional features in our model. Incorporating
imperfect conductivity and imperfect periodicity in our
model leaves the sharp maximum intact. Imperfect
collimation, on the other hand, incorporated into the model
causes the destruction of the sharp maximum as happens in
experiments. We provide analytic support through an
asymptotic calculation for both the existence of the sharp
maximum and the destructive impact of imperfect
collimation.},
Doi = {10.1364/josaa.30.001281},
Key = {fds244139}
}

@article{fds244166,
Author = {Shipman, SP and Venakides, S},
Title = {An exactly solvable model for nonlinear resonant
scattering},
Journal = {Nonlinearity},
Volume = {25},
Number = {9},
Pages = {2473-2501},
Publisher = {IOP Publishing},
Year = {2012},
Month = {September},
ISSN = {0951-7715},
url = {http://dx.doi.org/10.1088/0951-7715/25/9/2473},
Abstract = {This work analyses the effects of cubic nonlinearities on
certain resonant scattering anomalies associated with the
dissolution of an embedded eigenvalue of a linear scattering
system. These sharp peak-dip anomalies in the frequency
domain are often called Fano resonances. We study a simple
model that incorporates the essential features of this kind
of resonance. It features a linear scatterer attached to a
transmission line with a point-mass defect and coupled to a
nonlinear oscillator. We prove two power laws in the small
coupling (γ→0) and small nonlinearity (μ→0) regime.
The asymptotic relation μ→Cγ4characterizes the emergence
of a small frequency interval of triple harmonic solutions
near the resonant frequency of the oscillator. As the
nonlinearity grows or the coupling diminishes, this interval
widens and, at the relation μ→Cγ2, merges with another
evolving frequency interval of triple harmonic solutions
that extends to infinity. Our model allows rigorous
computation of stability in the small μ and γ limit. The
regime of triple harmonic solutions exhibits bistability -
those solutions with largest and smallest response of the
oscillator are linearly stable and the solution with
intermediate response is unstable. © 2012 IOP Publishing
Ltd & London Mathematical Society.},
Doi = {10.1088/0951-7715/25/9/2473},
Key = {fds244166}
}

@article{fds244167,
Author = {Tovbis, A and Venakides, S},
Title = {Semiclassical limit of the scattering transform for the
focusing nonlinear Schrödinger equation},
Journal = {International Mathematics Research Notices},
Volume = {2012},
Number = {10},
Pages = {2212-2271},
Publisher = {Oxford University Press (OUP)},
Year = {2012},
Month = {May},
ISSN = {1073-7928},
url = {http://dx.doi.org/10.1093/imrn/rnr092},
Abstract = {The semiclassical limit of the focusing Nonlinear (cubic)
Schr ̈ odinger Equation corresponds to the singularly
perturbed Zakharov-Shabat (ZS) system that defines the
direct and inverse scattering transforms (IST). In this
paper, we derive explicit expressions for the leading-order
terms of these transforms, which we call semiclassical
limits of the direct and IST. Thus, we establish an explicit
connection between the decaying initial data of the form
q(x, 0) = A(x)e iS(x) and the leading order term of its
scattering data. This connection is expressed in terms of an
integral transform that can be viewed as a complexified
version of the Abel transform. Our technique is not based on
the Wentzel-Kramers-Brillouin (WKB) analysis of the ZS
system, but on the inversion of the modulation equations
that solve the inverse scattering problem in the leading
order. The results are illustrated by a number of examples.
Doi = {10.1093/imrn/rnr092},
Key = {fds244167}
}

@article{fds244165,
Author = {Tovbis, A and Venakides, S},
Title = {Nonlinear steepest descent asymptotics for semiclassical
limit of Integrable systems: Continuation in the parameter
space},
Journal = {Communications in Mathematical Physics},
Volume = {295},
Number = {1},
Pages = {139-160},
Publisher = {Springer Nature},
Year = {2010},
Month = {February},
ISSN = {0010-3616},
url = {http://dx.doi.org/10.1007/s00220-009-0984-0},
Abstract = {The initial value problem for an integrable system, such as
the Nonlinear Schrödinger equation, is solved by subjecting
the linear eigenvalue problem arising from its Lax pair to
inverse scattering, and, thus, transforming it to a matrix
Riemann-Hilbert problem (RHP) in the spectral variable. In
the semiclassical limit, the method of nonlinear steepest
descent ([4,5]), supplemented by the g-function mechanism
([3]), is applied to this RHP to produce explicit asymptotic
solution formulae for the integrable system. These formule
are based on a hyperelliptic Riemann surface R = R(x, t) in
the spectral variable, where the space-time variables (x, t)
play the role of external parameters. The curves in the x, t
plane, separating regions of different genuses of R(x, t),
are called breaking curves or nonlinear caustics. The genus
of R(x, t) is related to the number of oscillatory phases in
the asymptotic solution of the integrable system at the
point x, t. The evolution theorem ([10]) guarantees
continuous evolution of the asymptotic solution in the
space-time away from the breaking curves. In the case of the
analytic scattering data f(z; x, t) (in the NLS case, f is a
normalized logarithm of the reflection coefficient with time
evolution included), the primary role in the breaking
mechanism is played by a phase function h(z; x, t), which is
closely related to the g function. Namely, a break can be
caused ([10]) either through the change of topology of zero
level curves of h(z; x, t) (regular break), or through the
interaction of zero level curves of h(z; x, t) with
singularities of f (singular break). Every time a breaking
curve in the x, t plane is reached, one has to prove the
validity of the nonlinear steepest descent asymptotics in
the region across the curve. In this paper we prove that in
the case of a regular break, the nonlinear steepest descent
asymptotics can be "automatically" continued through the
breaking curve (however, the expressions for the asymptotic
solution will be different on the different sides of the
curve). Our proof is based on the determinantal formula for
h(z; x, t) and its space and time derivatives, obtained in
[8,9]. Although the results are stated and proven for the
focusing NLS equation, it is clear ([9]) that they can be
reformulated for AKNS systems, as well as for the nonlinear
steepest descend method in a more general setting. ©
Springer-Verlag 2010.},
Doi = {10.1007/s00220-009-0984-0},
Key = {fds244165}
}

@article{fds244164,
Author = {Layton, AT and Toyama, Y and Yang, G-Q and Edwards, GS and Kiehart, DP and Venakides, S},
Title = {Drosophila morphogenesis: tissue force laws and the modeling
of dorsal closure.},
Journal = {Hfsp Journal},
Volume = {3},
Number = {6},
Pages = {441-460},
Publisher = {HFSP},
Year = {2009},
Month = {December},
url = {http://www.ncbi.nlm.nih.gov/pubmed/20514134},
Abstract = {Dorsal closure, a stage of Drosophila development, is a
model system for cell sheet morphogenesis and wound healing.
During closure, two flanks of epidermal tissue progressively
advance to reduce the area of the eye-shaped opening in the
dorsal surface, which contains amnioserosa tissue. To
simulate the time evolution of the overall shape of the
dorsal opening, we developed a mathematical model, in which
contractility and elasticity are manifest in model
force-producing elements that satisfy force-velocity
relationships similar to muscle. The action of the elements
is consistent with the force-producing behavior of actin and
myosin in cells. The parameters that characterize the
simulated embryos were optimized by reference to
experimental observations on wild-type embryos and, to a
lesser extent, on embryos whose amnioserosa was removed by
laser surgery and on myospheroid mutant embryos. Simulations
failed to reproduce the amnioserosa-removal protocol in
either the elastic or the contractile limit, indicating that
both elastic and contractile dynamics are essential
components of the biological force-producing elements. We
found it was necessary to actively upregulate forces to
recapitulate both the double and single-canthus nick
protocols, which did not participate in the optimization of
parameters, suggesting the existence of additional key
feedback mechanisms.},
Doi = {10.2976/1.3266062},
Key = {fds244164}
}

@article{fds244169,
Author = {Lefew, WR and Venakides, S and Gauthier, DJ},
Title = {Accurate description of optical precursors and their
relation to weak-field coherent optical transients},
Journal = {Physical Review A},
Volume = {79},
Number = {6},
Pages = {063842},
Publisher = {American Physical Society (APS)},
Year = {2009},
Month = {June},
ISSN = {1050-2947},
url = {http://dx.doi.org/10.1103/PhysRevA.79.063842},
Abstract = {We study theoretically the propagation of a step-modulated
optical field as it passes through a dispersive dielectric
made up of a dilute collection of oscillators characterized
by a single narrow-band resonance. The propagated field is
given in terms of an integral of a Fourier type, which
cannot be evaluated even for simple models of the dispersive
dielectric. The fact that the oscillators have a low number
density (dilute medium) and have a narrow-band resonance
allows us to simplify the integrand. In this case, the
integral can be evaluated exactly, although it is not
possible using this method to separate out the transient
part of the propagated field known as optical precursors. We
also use an asymptotic method (saddle-point method) to
evaluate the integral. The contributions to the integral
related to the saddle points of the integrand give rise to
the optical precursors. We obtain analytic expressions for
the precursor fields and the domain over which the
asymptotic method is valid. When combined to obtain the
total transient field, we find that the agreement between
the solutions obtained by the asymptotic and the exact
methods is excellent. Our results demonstrate that
precursors can persist for many nanoseconds and the chirp in
the instantaneous frequency of the precursors can manifest
itself in beats in the transmitted intensity. Our work
strongly suggests that precursors have been observed in many
previous experiments. © 2009 The American Physical
Society.},
Doi = {10.1103/PhysRevA.79.063842},
Key = {fds244169}
}

@article{fds244168,
Author = {Tovbis, A and Venakides, S},
Title = {Determinant form of the complex phase function of the
steepest descent analysis of Riemann-Hilbert problems and
its application to the focusing nonlinear schrödinger
equation},
Journal = {International Mathematics Research Notices},
Volume = {2009},
Number = {11},
Pages = {2056-2080},
Publisher = {Oxford University Press (OUP)},
Year = {2009},
Month = {February},
ISSN = {1073-7928},
url = {http://dx.doi.org/10.1093/imrn/rnp011},
Abstract = {We derive a determinant formula for the g-function that
plays a key role in the steepest descent asymptotic analysis
of the solution of 2 × 2 matrix Riemann-Hilbert problems
(RHPs) and is closely related to a hyperelliptic Riemann
surface. We formulate a system of transcendental equations
in determinant form (modulation equations), that govern the
dependence of the branchpoints αj of the Riemann surface on
a set of external parameters. We prove that, subject to the
modulation equations, ∂g/∂αj is identically zero for
all the branchpoints. Modulation equations are also obtained
in the form of ordinary differential equations with respect
to external parameters; some applications of these equations
to the semiclassical limit of the focusing nonlinear
Schrödinger equation (NLS) are discussed. © The Author
2009.},
Doi = {10.1093/imrn/rnp011},
Key = {fds244168}
}

@article{fds320541,
Author = {Tovbis, A and Venakides, S},
Title = {Determinant form of modulation equations for the
semiclassical focusing Nonlinear Schr\" odinger
equation},
Year = {2009},
Abstract = {We derive a determinant formula for the WKB exponential of
singularly perturbed Zakharov-Shabat system that corresponds
to the semiclassical (zero dispersion) limit of the focusing
Nonlinear Schr\" odinger equation. The derivation is based
on the Riemann-Hilbert Problem (RHP) representation of the
WKB exponential. We also prove its independence of the
branchpoints of the corresponding hyperelliptic surface
assuming that the modulation equations are
satisfied.},
Key = {fds320541}
}

@article{fds341090,
Author = {Ptitsyna, N and Shipman, SP and Venakides, S},
Title = {Fano resonance of waves in periodic slabs},
Journal = {Mathematical Methods in Electromagnetic Theory, Mmet,
Conference Proceedings},
Pages = {73-78},
Year = {2008},
Month = {September},
url = {http://dx.doi.org/10.1109/MMET.2008.4580900},
Abstract = {We investigate Fano-type anomalous transmission of energy of
plane waves across lossless slab scatterers with periodic
structure in the presence of non-robust guided modes. Our
approach is based on rigorous analytic perturbation of the
scattering problem near a guided mode and applies to very
general structures, continuous and discrete. © 2008
IEEE.},
Doi = {10.1109/MMET.2008.4580900},
Key = {fds341090}
}

@article{fds320429,
Author = {Tovbis, A and Venakides, S and Zhou, X},
Title = {Semiclassical Focusing Nonlinear Schrodinger equation in the
approach},
Journal = {Surveys on Discrete and Computational Geometry: Twenty Years
Later},
Volume = {458},
Pages = {117-144},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Baik, J and Kriecherbauer, T and Li, LC and McLaughlin, KDT and Tomei,
C},
Year = {2008},
ISBN = {978-0-8218-4240-9},
Key = {fds320429}
}

@article{fds244172,
Author = {Tovbis, A and Venakides, S and Zhou, X},
Title = {Semiclassical focusing nonlinear schrödinger equation i:
Inverse scattering map and its evolution for radiative
initial data},
Journal = {International Mathematics Research Notices},
Volume = {2007},
Number = {Article ID rnm094, 54 pages. doi:10.},
Publisher = {Oxford University Press (OUP)},
Year = {2007},
Month = {December},
ISSN = {1073-7928},
url = {http://dx.doi.org/10.1093/imrn/rnm094},
Abstract = {We consider the semiclassical limit for the focusing
nonlinear (cubic) Schrödinger Equation (NLS) in the pure
radiational case. We present a method of reconstructing the
leading order terms of the solitonless initial data and of
its evolution for a wide class of the corresponding
reflection coefficients. © The Author 2007.},
Doi = {10.1093/imrn/rnm094},
Key = {fds244172}
}

@article{fds244171,
Author = {Buckingham, R and Venakides, S},
Title = {Long-time asymptotics of the nonlinear Schrödinger equation
shock problem},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {60},
Number = {9},
Pages = {1349-1414},
Publisher = {WILEY},
Year = {2007},
Month = {September},
ISSN = {0010-3640},
MRCLASS = {35Q55 (35B40 37K15)},
MRNUMBER = {MR2337507},
url = {http://dx.doi.org/10.1002/cpa.20179},
Abstract = {The long-time asymptotics of two colliding plane waves
governed by the focusing nonlinear Schrödinger equation are
analyzed via the inverse scattering method. We find three
asymptotic regions in space-time: a region with the original
wave modified by a phase perturbation, a residual region
with a one-phase wave, and an intermediate transition region
with a modulated two-phase wave. The leading-order terms for
the three regions are computed with error estimates using
the steepest-descent method for Riemann-Hilbert problems.
The nondecaying initial data requires a new adaptation of
this method. A new breaking mechanism involving a complex
conjugate pair of branch points emerging from the real axis
is observed between the residual and transition regions.
Also, the effect of the collision is felt in the plane-wave
state well beyond the shock front at large times. © 2007
Wiley Periodicals, Inc.},
Doi = {10.1002/cpa.20179},
Key = {fds244171}
}

@article{fds244170,
Author = {Peralta, XG and Toyama, Y and Hutson, MS and Montague, R and Venakides,
S and Kiehart and, DP and Edwards, GS},
Title = {Resiliency, coordination, and synchronization of dorsal
closure during Drosophila morphogenesis},
Journal = {Biophysical Journal},
Volume = {92},
Number = {7},
Pages = {2583-2596},
Year = {2007},
Month = {April},
ISSN = {0006-3495},
url = {http://www.ncbi.nlm.nih.gov/pubmed/17218455},
Abstract = {Tissue dynamics during dorsal closure, a stage of Drosophila
development, provide a model system for cell sheet
morphogenesis and wound healing. Dorsal closure is
characterized by complex cell sheet movements, driven by
multiple tissue specific forces, which are coordinated in
space, synchronized in time, and resilient to UV-laser
perturbations. The mechanisms responsible for these
attributes are not fully understood. We measured spatial,
kinematic, and dynamic antero-posterior asymmetries to
biophysically characterize both resiliency to laser
perturbations and failure of closure in mutant embryos and
compared them to natural asymmetries in unperturbed,
wild-type closure. We quantified and mathematically modeled
two processes that are upregulated to provide
resiliency.contractility of the amnioserosa and formation of
a seam between advancing epidermal sheets, i.e., zipping.
Both processes are spatially removed from the laser-targeted
site, indicating they are not a local response to
laser-induced wounding and suggesting mechanosensitive
and/or chemosensitive mechanisms for upregulation. In mutant
embryos, tissue junctions initially fail at the anterior end
indicating inhomogeneous mechanical stresses attributable to
head involution, another developmental process that occurs
concomitant with the end stages of closure. Asymmetries in
these mutants are reversed compared to wild-type, and
inhomogeneous stresses may cause asymmetries in wild-type
closure.},
Doi = {10.1529/biophysj.106.094110},
Key = {fds244170}
}

@article{fds304498,
Author = {Peralta, XG and Toyama, Y and Hutson, MS and Montague, R and Venakides,
S and Kiehart, DP and Edwards, GS},
Title = {Upregulation of forces and morphogenic asymmetries in dorsal
closure during Drosophila development.},
Journal = {Biophysical Journal},
Volume = {92},
Number = {7},
Pages = {2583-2596},
Year = {2007},
Month = {April},
ISSN = {0006-3495},
url = {http://www.ncbi.nlm.nih.gov/pubmed/17218455},
Abstract = {Tissue dynamics during dorsal closure, a stage of Drosophila
development, provide a model system for cell sheet
morphogenesis and wound healing. Dorsal closure is
characterized by complex cell sheet movements, driven by
multiple tissue specific forces, which are coordinated in
space, synchronized in time, and resilient to UV-laser
perturbations. The mechanisms responsible for these
attributes are not fully understood. We measured spatial,
kinematic, and dynamic antero-posterior asymmetries to
biophysically characterize both resiliency to laser
perturbations and failure of closure in mutant embryos and
compared them to natural asymmetries in unperturbed,
wild-type closure. We quantified and mathematically modeled
two processes that are upregulated to provide
resiliency--contractility of the amnioserosa and formation
of a seam between advancing epidermal sheets, i.e., zipping.
Both processes are spatially removed from the laser-targeted
site, indicating they are not a local response to
laser-induced wounding and suggesting mechanosensitive
and/or chemosensitive mechanisms for upregulation. In mutant
embryos, tissue junctions initially fail at the anterior end
indicating inhomogeneous mechanical stresses attributable to
head involution, another developmental process that occurs
concomitant with the end stages of closure. Asymmetries in
these mutants are reversed compared to wild-type, and
inhomogeneous stresses may cause asymmetries in wild-type
closure.},
Doi = {10.1529/biophysj.106.094110},
Key = {fds304498}
}

@article{fds320430,
Author = {Buckingham, R and Tovbis, A and Venakides, S and Zhou,
X},
Title = {The semiclassical focusing nonlinear Schrodinger
equation},
Journal = {Recent Advances in Nonlinear Partial Differential Equations
and Applications},
Volume = {65},
Series = {Proceedings of Symposia in Applied Mathematics},
Pages = {47-80},
Booktitle = {"Recent Advances in Nonlinear Partial Differentila Equations
and Applications''},
Publisher = {AMER MATHEMATICAL SOC},
Editor = {Bonilla, LL and Carpio, A and Vega, JM and Venakides,
S},
Year = {2007},
ISBN = {978-0-8218-4211-9},
Key = {fds320430}
}

@article{fds244160,
Author = {Tovbis, A and Venakides, S and Zhou, X},
Title = {On the long-time limit of semiclassical (zero dispersion
limit) solutions of the focusing nonlinear Schrödinger
Journal = {Communications on Pure and Applied Mathematics},
Volume = {59},
Number = {10},
Pages = {1379-1432},
Publisher = {WILEY},
Year = {2006},
Month = {October},
ISSN = {0010-3640},
MRNUMBER = {MR2248894},
url = {http://dx.doi.org/10.1002/cpa.20142},
Abstract = {In a previous paper [13] we calculated the leading-order
term q 0(x, t, ε) of the solution of q(x, t, ε), the
focusing nonlinear (cubic) Schrödinger (NLS) equation in
the semiclassical limit (ε → 0) for a certain
one-parameter family of initial conditions. This family
contains both solitons and pure radiation. In the pure
radiation case, our result is valid for all times t ≥ 0.
The aim of the present paper is to calculate the long-term
behavior of the semiclassical solution q(x, t, ε) in the
pure radiation case. As before, our main tool is the
Riemann-Hilbert problem (RHP) formulation of the inverse
scattering problem and the corresponding system of "moment
and integral conditions," known also as a system of
"modulation equations." © 2006 Wiley Periodicals,
Inc.},
Doi = {10.1002/cpa.20142},
Key = {fds244160}
}

@article{fds244174,
Author = {Shipman, SP and Venakides, S},
Title = {Resonant transmission near nonrobust periodic slab
modes.},
Journal = {Physical Review. E, Statistical, Nonlinear, and Soft Matter
Physics},
Volume = {71},
Number = {2 Pt 2},
Pages = {026611},
Year = {2005},
Month = {February},
ISSN = {1539-3755},
url = {http://www.ncbi.nlm.nih.gov/pubmed/15783445},
Abstract = {We present a precise theoretical explanation and prediction
of certain resonant peaks and dips in the electromagnetic
transmission coefficient of periodically structured slabs in
the presence of nonrobust guided slab modes. We also derive
the leading asymptotic behavior of the related phenomenon of
resonant enhancement near the guided mode. The theory
applies to structures in which losses are negligible and to
very general geometries of the unit cell. It is based on
boundary-integral representations of the electromagnetic
fields. These depend on the frequency and on the Bloch wave
vector and provide a complex-analytic connection in these
parameters between generalized scattering states and guided
slab modes. The perturbation of three coincident zeros-those
of the dispersion relation for slab modes, the reflection
constant, and the transmission constant-is central to
calculating transmission anomalies both for lossless
dielectric materials and for perfect metals.},
Doi = {10.1103/physreve.71.026611},
Key = {fds244174}
}

@article{fds320431,
Author = {Peralta, XG and Toyama, Y and Wells, A and Tokutake, Y and Hutson, MS and Venakides, S and Kiehart, DP and Edwards, GS},
Title = {Force regulation during dorsal closure in
Drosophila},
Journal = {Molecular Biology of the Cell},
Volume = {15},
Pages = {403A-403A},
Publisher = {AMER SOC CELL BIOLOGY},
Year = {2004},
Month = {November},
Key = {fds320431}
}

@article{fds244173,
Author = {Tovbis, A and Venakides, S and Zhou, X},
Title = {On semiclassical (zero dispersion limit) solutions of the
focusing nonlinear Schrödinger equation},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {57},
Number = {7},
Pages = {877-985},
Publisher = {WILEY},
Year = {2004},
Month = {July},
ISSN = {0010-3640},
MRCLASS = {35Q55 (35C20 37K15 37K40)},
MRNUMBER = {MR2044068 (2005c:35269)},
url = {http://dx.doi.org/10.1002/cpa.20024},
Abstract = {We calculate the leading-order term of the solution of the
focusing nonlinear (cubic) Schrödinger equation (NLS) in
the semiclassical limit for a certain one-parameter family
of initial conditions. This family contains both solitons
is valid for all times t ≥ 0. We utilize the
Riemann-Hilbert problem formulation of the inverse
scattering problem to obtain the leading-order term of the
solution. Error estimates are provided. © 2004 Wiley
Periodicals, Inc.},
Doi = {10.1002/cpa.20024},
Key = {fds244173}
}

@article{fds244159,
Author = {Lipton, RP and Shipman, SP and Venakides, S},
Title = {Optimization of Resonances in Photonic Crystal
Slabs},
Journal = {Smart Structures and Materials 2005: Active Materials:
Behavior and Mechanics},
Volume = {5184},
Pages = {168-177},
Publisher = {SPIE},
Year = {2003},
Month = {December},
url = {http://dx.doi.org/10.1117/12.505091},
Abstract = {Variational methods are applied to the design of a
two-dimensional lossless photonic crystal slab to optimize
resonant scattering phenomena. The method is based on
varying properties of the transmission coefficient that are
connected to resonant behavior. Numerical studies are based
on boundary-integral methods for crystals consisting of
multiple scatterers. We present an example in which we
modify a photonic crystal consisting of an array of
dielectric rods in air so that a weak transmission anomaly
is transformed into a sharp resonance.},
Doi = {10.1117/12.505091},
Key = {fds244159}
}

@article{fds244175,
Author = {Shipman, SP and Venakides, S},
Title = {Resonance and bound states in photonic crystal
slabs},
Journal = {Siam Journal on Applied Mathematics},
Volume = {64},
Number = {1},
Pages = {322-342},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2003},
Month = {October},
ISSN = {0036-1399},
url = {http://dx.doi.org/10.1137/S0036139902411120},
Abstract = {Using boundary-integral projections for time-harmonic
electromagnetic (EM) fields, and their numerical
implementation, we analyze EM resonance in slabs of
two-phase dielectric photonic crystal materials. We
characterize resonant frequencies by a complex Floquet-Bloch
dispersion relation ω = W (β) defined by the existence of
a nontrivial nullspace of a pair of boundary-integral
projections parameterized by the wave number β and the
time-frequency ω. At resonant frequencies, the crystal slab
supports a source-free EM field. We link complex resonant
frequencies, where the imaginary part is small, to resonant
scattering behavior of incident source fields at nearby real
frequencies and anomalous transmission of energy through the
slab. At a real resonant frequency, the source-free field
supported by the slab is a bound state. We present numerical
examples which demonstrate the effects of structural defects
on the resonant properties of a crystal slab and surface
waves supported by a dielectric defect.},
Doi = {10.1137/S0036139902411120},
Key = {fds244175}
}

@article{fds244176,
Author = {Hutson, MS and Tokutake, Y and Chang, M-S and Bloor, JW and Venakides,
S and Kiehart, DP and Edwards, GS},
Title = {Forces for morphogenesis investigated with laser
microsurgery and quantitative modeling.},
Journal = {Science (New York, N.Y.)},
Volume = {300},
Number = {5616},
Pages = {145-149},
Year = {2003},
Month = {April},
url = {http://www.ncbi.nlm.nih.gov/pubmed/12574496},
Abstract = {We investigated the forces that connect the genetic program
of development to morphogenesis in Drosophila. We focused on
dorsal closure, a powerful model system for development and
wound healing. We found that the bulk of progress toward
closure is driven by contractility in supracellular "purse
strings" and in the amnioserosa, whereas adhesion-mediated
zipping coordinates the forces produced by the purse strings
and is essential only for the end stages. We applied
quantitative modeling to show that these forces, generated
in distinct cells, are coordinated in space and synchronized
in time. Modeling of wild-type and mutant phenotypes is
predictive; although closure in myospheroid mutants
ultimately fails when the cell sheets rip themselves apart,
our analysis indicates that beta(PS) integrin has an
earlier, important role in zipping.},
Doi = {10.1126/science.1079552},
Key = {fds244176}
}

@article{fds320432,
Author = {Hutson, S and Tokutake, Y and Chang, M and Bloor, JW and Venakides, S and Kiehart, DP and Edwards, GS},
Title = {Measuring the forces that drive morphogenesis:
Laser-microsurgery and quantitative modeling applied to
dorsal closure in Drosophila},
Journal = {Molecular Biology of the Cell},
Volume = {13},
Pages = {476A-476A},
Publisher = {American Society for Cell Biology},
Year = {2002},
Month = {November},
Key = {fds320432}
}

@article{fds244158,
Author = {Haider, MA and Shipman, SP and Venakides, S},
Title = {Boundary-integral calculations of two-dimensional
electromagnetic scattering in infinite photonic crystal
slabs: Channel defects and resonances},
Journal = {Siam Journal on Applied Mathematics},
Volume = {62},
Number = {6},
Pages = {2129-2148},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2002},
Month = {July},
url = {http://dx.doi.org/10.1137/S003613990138531X},
Abstract = {We compute the transmission of two-dimensional (2D)
electromagnetic waves through a square lattice of lossless
dielectric rods with a channel defect. The lattice is finite
in the direction of propagation of the incident wave and
periodic in a transverse direction. We revisit a
boundary-integral formulation of 2D electromagnetic
scattering [Venakides, Haider, and Papanicolaou, SIAM J.
Appl. Math., 60 (2000), pp. 1686-1706] that is Fredholm of
the first kind and develop a second-kind formulation. We
refine the numerical implementation in the above paper by
exploiting separability in the Green's function to evaluate
the far-field influence more efficiently. The resulting cost
savings in computing and solving the discretized linear
system leads to an accelerated method. We use it to analyze
E-polarized electromagnetic scattering of normally incident
waves on a structure with a periodic channel defect. We find
three categories of resonances: waveguide modes in the
channel, high-amplitude fields in the crystal at frequencies
near the edge of the frequency bandgap, and very
high-amplitude standing fields at frequencies in a
transmission band that are normal to the direction of the
incident wave. These features are captured essentially
identically with the first-kind as with the second-kind
formulation.},
Doi = {10.1137/S003613990138531X},
Key = {fds244158}
}

@article{fds244155,
Author = {El, GA and Krylov, AL and Venakides, S},
Title = {Unified approach to KdV modulations},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {54},
Number = {10},
Pages = {1243-1270},
Publisher = {WILEY},
Year = {2001},
Month = {October},
url = {http://dx.doi.org/10.1002/cpa.10002},
Abstract = {We develop a unified approach to integrating the Whitham
modulation equations. Our approach is based on the
formulation of the initial-value problem for the
zero-dispersion KdV as the steepest descent for the scalar
Riemann-Hilbert problem [6] and on the method of generating
differentials for the KdV-Whitham hierarchy [9]. By assuming
the hyperbolicity of the zero-dispersion limit for the KdV
with general initial data, we bypass the inverse scattering
transform and produce the symmetric system of algebraic
equations describing motion of the modulation parameters
plus the system of inequalities determining the number the
oscillating phases at any fixed point on the (x, t)-plane.
The resulting system effectively solves the zero-dispersion
KdV with an arbitrary initial datum. © 2001 John Wiley &
Sons, Inc.},
Doi = {10.1002/cpa.10002},
Key = {fds244155}
}

@article{fds244157,
Author = {Deift, P and Kriecherbauer, T and McLaughlin, KR and Venakides, S and Zhou, X},
Title = {A riemann-Hilbert approach to asymptotic questions for
orthogonal polynomials},
Journal = {Journal of Computational and Applied Mathematics},
Volume = {133},
Number = {1-2},
Pages = {47-63},
Publisher = {Elsevier BV},
Year = {2001},
Month = {August},
ISSN = {0377-0427},
url = {http://dx.doi.org/10.1016/S0377-0427(00)00634-8},
Abstract = {A few years ago the authors introduced a new approach to
study asymptotic questions for orthogonal polynomials. In
this paper we give an overview of our method and review the
results which have been obtained in Deift et al. (Internat.
Math. Res. Notices (1997) 759, Comm. Pure Appl. Math. 52
(1999) 1491, 1335), Deift (Orthogonal Polynomials and Random
Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes,
Vol. 3, New York University, 1999), Kriecherbauer and
McLaughlin (Internat. Math. Res. Notices (1999) 299) and
Baik et al. (J. Amer. Math. Soc. 12 (1999) 1119). We mainly
consider orthogonal polynomials with respect to weights on
the real line which are either (1) Freud-type weights dα(x)
= e-Q(x) dx (Q polynomial or Q(x) = xβ, β>0), or (2)
varying weights dαn(x) = e-nV(x) dx (V analytic,
limx→∞V(x)/logx = ∞). We obtain Plancherel-Rotach-type
asymptotics in the entire complex plane as well as
asymptotic formulae with error estimates for the leading
coefficients, for the recurrence coefficients, and for the
zeros of the orthogonal polynomials. Our proof starts from
an observation of Fokas et al. (Comm. Math. Phys. 142 (1991)
313) that the orthogonal polynomials can be determined as
solutions of certain matrix valued Riemann-Hilbert problems.
We analyze the Riemann-Hilbert problems by a steepest
descent type method introduced by Deift and Zhou (Ann. Math.
137 (1993) 295) and further developed in Deift and Zhou
(Comm. Pure Appl. Math. 48 (1995) 277) and Deift et al.
(Proc. Nat. Acad. Sci. USA 95 (1998) 450). A crucial step in
our analysis is the use of the well-known equilibrium
measure which describes the asymptotic distribution of the
zeros of the orthogonal polynomials. © 2001 Elsevier
Doi = {10.1016/S0377-0427(00)00634-8},
Key = {fds244157}
}

@article{fds244156,
Author = {El, GA and Krylov, AL and Molchanov, SA and Venakides,
S},
Title = {Soliton turbulence as a thermodynamic limit of stochastic
soliton lattices},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {152-153},
Pages = {653-664},
Publisher = {Elsevier BV},
Year = {2001},
Month = {May},
url = {http://dx.doi.org/10.1016/S0167-2789(01)00198-1},
Abstract = {We use the recently introduced notion of stochastic soliton
lattice for quantitative description of soliton turbulence.
We consider the stochastic soliton lattice on a special
band-gap scaling of the spectral surface of genus N so that
the integrated density of states remains finite as N → ∞
(thermodynamic type limit). We prove existence of the
limiting stationary ergodic process and associate it with
the homogeneous soliton turbulence. The phase space of the
soliton turbulence is a one-dimensional space with the
random Poisson measure. The zero-density limit of the
soliton turbulence coincides with the Frish-Lloyd potential
of the quantum theory of disordered systems. © 2001
Doi = {10.1016/S0167-2789(01)00198-1},
Key = {fds244156}
}

@article{fds244154,
Author = {Georgieva, A and Kriecherbauer, T and Venakides,
S},
Title = {1:2 resonance mediated second harmonic generation in a 1-D
nonlinear discrete periodic medium},
Journal = {Siam Journal on Applied Mathematics},
Volume = {61},
Number = {5},
Pages = {1802-1815},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2001},
Month = {January},
url = {http://dx.doi.org/10.1137/S0036139999365341},
Abstract = {We derive traveling wave solutions in a nonlinear diatomic
particle chain near the 1:2 resonance (κ*, ω*), where ω*
= D(κ*), 2ω* = D(2κ*) and ω = D(κ) is the linear
dispersion relation. To leading order, the waves have form
±εsin(κn - ωt) + δsin(2κn - 2ωt), where the
near-resonant acoustic frequency ω and the amplitude ε of
the first harmonic are given to first order in terms of the
wavenumber difference κ - κ* and the amplitude δ of the
second harmonic. These traveling wave solutions are unique
within a certain set of symmetries. We find that there is a
continuous line in parameter space that transfers energy
from the first to the second harmonic, even in cases where
initially almost all energy is in the first harmonic,
connecting these waves to pure optical waves that have no
first harmonic content.},
Doi = {10.1137/S0036139999365341},
Key = {fds244154}
}

@article{fds244153,
Author = {Tovbis, A and Venakides, S},
Title = {The eigenvalue problem for the focusing nonlinear
Schrödinger equation: New solvable cases},
Journal = {Physica D: Nonlinear Phenomena},
Volume = {146},
Number = {1-4},
Pages = {150-164},
Publisher = {Elsevier BV},
Year = {2000},
Month = {November},
url = {http://dx.doi.org/10.1016/S0167-2789(00)00126-3},
Abstract = {In this paper, we study the semi-classical limit of the
Zakharov-Shabat eigenvalue problem for the focusing of NLS
with some specific initial data. In all these cases, the
eigenvalue problem is reduced to connection problems for the
hypergeometric equation and for other classical equations.
The special initial data [Suppl. Prog. Theor. Phys. 55
(1974) 284] is contained in our family of initial data,
parameterized by a real parameter μ, as a particular case
μ=0. We find that beyond a certain value of the parameter
μ, the pure-point spectrum becomes empty and all the
scattering information is contained in the reflection
coefficient.},
Doi = {10.1016/S0167-2789(00)00126-3},
Key = {fds244153}
}

@article{fds244152,
Author = {Venakides, S and Haider, MA and Papanicolaou, V},
Title = {Boundary integral calculations of two-dimensional
electromagnetic scattering by photonic crystal Fabry-Perot
structures},
Journal = {Siam Journal on Applied Mathematics},
Volume = {60},
Number = {5},
Pages = {1686-1706},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {2000},
Month = {May},
url = {http://dx.doi.org/10.1137/s0036139999350779},
Abstract = {We compute the transmission properties of two-dimensional
(2-D) electromagnetic transverse magnetic (TM) waves that
are normally incident on a Fabry-Perot structure with
mirrors consisting of photonic crystals. We use a boundary
integral formulation with quadratic boundary elements and
utilize the Ewald representation for Green's functions. We
trace the frequencies of the Fabry-Perot cavity modes
traversing the photonic bandgap as the cavity length
increases and calculate corresponding Q-values.},
Doi = {10.1137/s0036139999350779},
Key = {fds244152}
}

@article{fds320434,
Author = {Reed, D and Venakides, S},
Title = {Studying the asymptotics of Selberg-type
integrals},
Journal = {Applied and Industrial Mathematics, Venice 2,
1998},
Pages = {187-198},
Publisher = {SPRINGER},
Editor = {Spigler, R},
Year = {2000},
Month = {January},
ISBN = {0-7923-6152-0},
Key = {fds320434}
}

@article{fds320433,
Author = {VENAKIDES, S},
Title = {Boundary integral calculations of two-dimensional
electromagnetic scattering by photonic crystal Fabri-Peror
structures},
Journal = {Siam J. Appl. Math.},
Volume = {60},
Pages = {1686-1706},
Year = {2000},
url = {http://dx.doi.org/10.1137/S0036139999350779},
Doi = {10.1137/S0036139999350779},
Key = {fds320433}
}

@article{fds244145,
Author = {Beaky, MM and Burk, JB and Everitt, HO and Haider, MA and Venakides,
S},
Title = {Two-dimensional photonic crystal fabry-perot resonators with
lossy dielectrics},
Journal = {Ieee Transactions on Microwave Theory and
Techniques},
Volume = {47},
Number = {11},
Pages = {2085-2091},
Publisher = {Institute of Electrical and Electronics Engineers
(IEEE)},
Year = {1999},
Month = {December},
ISSN = {0018-9480},
url = {http://dx.doi.org/10.1109/22.798003},
Abstract = {Square and triangular lattice two-dimensional (2-D) photonic
crystals (PC's) composed of lossy dielectric rods in air
were constructed with a microwave bandgap between 4-8 GHz.
Fabry-Perot resonators of varying length were constructed
from two of these PC's of adjustable thickness and
reflectivity. The quality factor of cavity modes supported
in the resonators was found to increase with increasing PC
mirror thickness, but only to a point dictated by the
lossiness of the dielectric rods. A 2-D periodic Green's
function simulation was found to model the data accurately
and quickly using physical parameters obtained in separate
measurements. Simple rules are developed for designing
optimal resonators in the presence of dielectric loss. ©
1999 IEEE.},
Doi = {10.1109/22.798003},
Key = {fds244145}
}

@article{fds244147,
Author = {Filip, AM and Venakides, S},
Title = {Existence and modulation of traveling waves in particle
chains},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {52},
Number = {6},
Pages = {693-735},
Year = {1999},
Month = {June},
Abstract = {We consider an infinite particle chain whose dynamics are
governed by the following system of differential equations:
q̈n= V′ (qn-1- qn) - V′ (qn- qn+1), n = 1,2, . . . ,
where qn(t) is the displacement of the nthparticle at time t
along the chain axis and · denotes differentiation with
respect to time. We assume that all particles have unit mass
and that the interaction potential V between adjacent
particles is a convex C∞function. For this system, we
prove the existence of C∞, time-periodic, traveling-wave
solutions of the form qn(t) = q(wt - kn) + βt - αn, where
q is a periodic function q(z) = q(z + 1) (the period is
normalized to equal 1), w and k are, respectively, the
frequency and the wave number, α is the mean particle
spacing, and β can be chosen to be an arbitrary parameter.
We present two proofs, one based on a variational principle
and the other on topological methods, in particular degree
theory. For small-amplitude waves, based on perturbation
techniques, we describe the form of the traveling waves, and
we derive the weakly nonlinear dispersion relation. For the
fully nonlinear case, when the amplitude of the waves is
high, we use numerical methods to compute the traveling-wave
solution and the nonlinear dispersion relation. We finally
apply Whitham's method of averaged Lagrangian to derive the
modulation equations for the wave parameters α, β, k, and
w. © 1999 John Wiley & Sons, Inc.},
Key = {fds244147}
}

@article{fds244148,
Author = {Cheng, PJ and Venakides, S and Zhou, X},
Title = {Long-time asymptotics for the pure radiation solution of the
sine-Gordon equation},
Journal = {Communications in Partial Differential Equations},
Volume = {24},
Number = {7-8},
Pages = {1195-1262},
Publisher = {Informa UK Limited},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1080/03605309908821464},
Doi = {10.1080/03605309908821464},
Key = {fds244148}
}

@article{fds244149,
Author = {Georgieva, A and Kriecherbauer, T and Venakides,
S},
Title = {Wave Propagation and Resonance in a One-Dimensional
Nonlinear Discrete Periodic Medium},
Journal = {Siam Journal on Applied Mathematics},
Volume = {60},
Number = {1},
Pages = {272-294},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1137/s0036139998340315},
Abstract = {We consider wave propagation in a nonlinear infinite
diatomic chain of particles as a discrete model of
propagation in a medium whose properties vary periodically
in space. The particles have alternating masses M1 and M2
and interact in accordance to a general nonlinear force F
acting between the nearest neighbors. Their motion is
described by the system of equations qqn =
1/M1(F(yn-1-yn)-F(yn-yn+1)), qqn+1 = 1/M2(F(yn-yn+1)-F(yn+1-yn+2)),
where {yn}n = -∞∞ is the position of the nth particle.
Using Fourier series methods and tools from bifurcation
theory, we show that, for nonresonant wave-numbers k, this
system admits nontrivial small-amplitude traveling wave
solutions g and h, depending only on the linear combination
z = kn-ωt. We determine the nonlinear dispersion relation.
We also show that the system sustains binary oscillations
with arbitrarily large amplitude.},
Doi = {10.1137/s0036139998340315},
Key = {fds244149}
}

@article{fds244150,
Author = {Deift, P and Kriecherbauer, T and McLaughlin, KTR and Venakides, S and Zhou, X},
Title = {Uniform asymptotics for polynomials orthogonal with respect
to varying exponential weights and applications to
universality questions in random matrix theory},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {52},
Number = {11},
Pages = {1335-1425},
Publisher = {WILEY},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2},
Abstract = {We consider asymptotics for orthogonal polynomials with
respect to varying exponential weights wn(x)dx = e-nV(x)dx
on the line as n → ∞. The potentials V are assumed to be
real analytic, with sufficient growth at infinity. The
principle results concern Plancherel-Rotach-type asymptotics
for the orthogonal polynomials down to the axis. Using these
asymptotics, we then prove universality for a variety of
statistical quantities arising in the theory of random
matrix models, some of which have been considered recently
in [31] and also in [4]. An additional application concerns
the asymptotics of the recurrence coefficients and leading
The orthogonal polynomial problem is formulated as a
Riemann-Hilbert problem following [19, 20]. The
Riemann-Hilbert problem is analyzed in turn using the
steepest-descent method introduced in [12] and further
developed in [11, 13]. A critical role in our method is
played by the equilibrium measure dμv for V as analyzed in
[8]. © 1999 John Wiley & Sons, Inc.},
Doi = {10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2},
Key = {fds244150}
}

@article{fds244151,
Author = {Deift, P and Kriecherbauer, T and Mclaughlin, KTR and Venakides, S and Zhou, X},
Title = {Strong asymptotics of orthogonal polynomials with respect to
exponential weights},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {52},
Number = {12},
Pages = {1491-1552},
Year = {1999},
Month = {January},
url = {http://dx.doi.org/10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2},
Abstract = {We consider asymptotics of orthogonal polynomials with
respect to weights w(x)dx = e-Q(x)dx on the real line, where
Q(x) = Σ2mk=0qkxk, q2m > 0, denotes a polynomial of even
order with positive leading coefficient. The orthogonal
polynomial problem is formulated as a Riemann-Hilbert
problem following [22, 23]. We employ the
steepest-descent-type method introduced in [18] and further
developed in [17, 19] in order to obtain uniform
Plancherel-Rotach-type asymptotics in the entire complex
plane, as well as asymptotic formulae for the zeros, the
leading coefficients, and the recurrence coefficients of the
orthogonal polynomials. © 1999 John Wiley & Sons,
Inc.},
Doi = {10.1002/(SICI)1097-0312(199912)52:12<1491::AID-CPA2>3.0.CO;2},
Key = {fds244151}
}

@article{fds244144,
Author = {Deift, P and Venakides, S and Zhou, X},
Title = {An extension of the steepest descent method for
Riemann-Hilbert problems: the small dispersion limit of the
Korteweg-de Vries (KdV) equation.},
Journal = {Proceedings of the National Academy of Sciences of the
United States of America},
Volume = {95},
Number = {2},
Pages = {450-454},
Year = {1998},
Month = {January},
ISSN = {0027-8424},
url = {http://www.ncbi.nlm.nih.gov/pubmed/11038618},
Abstract = {This paper extends the steepest descent method for
Riemann-Hilbert problems introduced by Deift and Zhou in a
critical new way. We present, in particular, an algorithm,
to obtain the support of the Riemann-Hilbert problem for
leading asymptotics. Applying this extended method to small
dispersion KdV (Korteweg-de Vries) equation, we (i) recover
the variational formulation of P. D. Lax and C. D. Levermore
[(1979) Proc. Natl. Acad. Sci. USA76, 3602-3606] for the
weak limit of the solution, (ii) derive, without using an
ansatz, the hyperelliptic asymptotic solution of S.
Venakides that describes the oscillations; and (iii) are now
able to compute the phase shifts, integrating the modulation
equations exactly. The procedure of this paper is a version
of fully nonlinear geometrical optics for integrable
systems. With some additional analysis the theory can
provide rigorous error estimates between the solution and
its computed asymptotic expression.},
Doi = {10.1073/pnas.95.2.450},
Key = {fds244144}
}

@article{fds244146,
Author = {McDonald, MA and Venakides, S},
Title = {Renormalization of the τ-functions for integrable systems:
A model problem},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {51},
Number = {8},
Pages = {937-966},
Publisher = {WILEY},
Year = {1998},
Month = {January},
url = {http://dx.doi.org/10.1002/(SICI)1097-0312(199808)51:8<937::AID-CPA3>3.0.CO;2-6},
Abstract = {We introduce a renormalization procedure for the τ-function
of integrable systems. We illustrate the procedure using the
supercritical Toda shock problem as a model problem. We
start with a finite chain and take the limit of the solution
as the number of particles N → ∞. This results in a new
formula for the τ-function for the problem with an infinite
chain. We apply the renormalized formula to rederive
leading-order effects of the supercritical Toda shock
problem. © 1998 John Wiley & Sons, Inc.},
Doi = {10.1002/(SICI)1097-0312(199808)51:8<937::AID-CPA3>3.0.CO;2-6},
Key = {fds244146}
}

@article{fds244141,
Author = {Deift, P and Kriecherbauer, T and McLaughlin, KTR and Venakides, S and Zhou, X},
Title = {Asymptotics for Polynomials Orthogonal with Respect to
Varying Exponential Weights},
Journal = {International Mathematics Research Notices},
Number = {16},
Pages = {X-782},
Year = {1997},
Month = {December},
Key = {fds244141}
}

@article{fds244142,
Author = {Deift, P and Venakides, S and Zhou, X},
Title = {New Results in Small Dispersion KdV by an Extension of the
Steepest Descent Method for Riemann-Hilbert
Problems},
Journal = {International Mathematics Research Notices},
Number = {6},
Pages = {284-299},
Year = {1997},
Month = {December},
Key = {fds244142}
}

@article{fds244143,
Author = {Bonilla, LL and Kindelan, M and Moscoso, M and Venakides,
S},
Title = {Periodic generation and propagation of traveling fronts in
dc voltage biased semiconductor superlattices},
Journal = {Siam Journal on Applied Mathematics},
Volume = {57},
Number = {6},
Pages = {1588-1614},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1997},
Month = {January},
url = {http://dx.doi.org/10.1137/S0036139995288885},
Abstract = {The continuum limit of a recently proposed model for charge
transport in resonant-tunneling semiconductor superlattices
(SLs) is analyzed. It is described by a nonlinear hyperbolic
integrodifferential equation on a one-dimensional spatial
support, supplemented by shock and entropy conditions. For
appropriate parameter values, a time-periodic solution is
found in numerical simulations of the model. An asymptotic
theory shows that the time-periodic solution is due to
recycling and motion of shock waves representing domain
walls connecting regions of the SL where the electric field
is almost uniform.},
Doi = {10.1137/S0036139995288885},
Key = {fds244143}
}

@article{fds320436,
Author = {Deift, P and Venakides, S and Zhou, X},
Title = {New results in small dispersion kdV by an extension of the
steepest descent method for Riemann-Hilbert
problems},
Journal = {International Mathematics Research Notices},
Number = {6},
Pages = {285-299},
Publisher = {DUKE UNIV PRESS},
Year = {1997},
Month = {January},
Key = {fds320436}
}

@article{fds320435,
Author = {Deift, P and Kriecherbauer, T and McLaughlin, KTR and Venakides, S and Zhou, X},
Title = {Asymptotics for polynomials orthogonal with respect to
varying exponential weights},
Journal = {International Mathematics Research Notices},
Number = {16},
Pages = {759-782},
Publisher = {Oxford University Press (OUP): Policy B - Oxford Open Option
A},
Year = {1997},
Key = {fds320435}
}

@article{fds320437,
Author = {Deift, P and Kriecherbauer, T and Venakides, S},
Title = {Forced lattice vibrations: Part I},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {48},
Number = {11},
Pages = {1187-1249},
Publisher = {WILEY},
Year = {1995},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160481102},
Abstract = {This is the First part of a two‐part series on forced
lattice vibrations in which a semi‐infinite lattice of
one‐dimensional particles {xn}n≧1 (Formula Presented.)
is driven from one end by a particle x0. This particle
undergoes a given, periodically perturbed, uniform motion,
x0(t) = at + h(yt), where a and γ are constants and h(·)
has period 2π. For a wide variety of restoring forces F
(i.e., F′ > 0), numerical calculations indicate the
existence of a sequence of thresholds γ1 = γ1(a, h, F) >
γ2 = γ2(a,h,F) > … > γk = γk(a,h,F) > …, γk → 0,
as k → ∞. If γk > γ > γk+1, a k‐phase wave that is
well described by the wave form, (Formula Presented.)
emerges and travels through the lattice. The goal of this
series is to describe the emergence and calculate some
properties of these wave forms. In Part I the authors first
consider the case where F(x) = ex (i.e., Toda forces) but h
is arbitrary, and show how to compute a basic diagnostic
(see J(λ), formula (1.26)) for the system in terms of the
solution of an associated scalar Riemann‐Hilbert problem,
once a certain finite set of numbers is known. In another
direction, the authors consider the case where F(x) is
restoring but arbitrary, and h is small. Here the authors
prove a general result, asserting that if there exists a
sufficiently ample family of traveling‐wave solutions of
the doubly infinite lattice, (Formula Presented.) then it is
possible to construct time‐periodic k‐phase wave
solutions with asymptotics in n of type (iii) for the driven
system (i). In Part II, the authors prove that sufficiently
ample families of traveling‐wave solutions of the system
(iv) exist in the cases γ > γ1 and γ1 > γ > γ2 for
general restoring forces F. In the case with Toda forces,
F(x) = ex, the authors prove that sufficiently ample
Wiley Periodicals, Inc., A Wiley Company},
Doi = {10.1002/cpa.3160481102},
Key = {fds320437}
}

@article{fds320438,
Author = {Deift, P and Kriecherbauer, T and Venakides, S},
Title = {Forced lattice vibrations: Part II},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {48},
Number = {11},
Pages = {1251-1298},
Publisher = {WILEY},
Year = {1995},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160481103},
Abstract = {This is the second part of a two‐part series on forced
lattice vibrations in which a semi‐infinite lattice of
one‐dimensional particles {xn}n≧1, (Formula Presented.)
is driven from one end by a particle x0. This particle
undergoes a given, periodically perturbed, uniform motion
x0(t) = 2at + h(yt) where a and γ are constants and h(·)
has period 2π. Results and notation from Part I are used
freely and without further comment. Here the authors prove
that sufficiently ample families of traveling‐wave
solutions of the doubly infinite system (Formula Presented.)
exist in the cases γ > γ1 and γ1 > γ > γ2 for general
restoring forces F. In the case with Toda forces, F(x) = ex,
the authors prove that sufficiently ample families of
traveling‐wave solutions exist for all k, γk > γ >
γk+1. By a general result proved in Part I, this implies
that there exist time‐periodic solutions of the driven
system (i) with k‐phase wave asymptotics in n of the type
(Formula Presented.) with k = 0 or 1 for general F and k
arbitrary for F(x) = ex (when k = 0, take γ0 = ∞ and X0
Company},
Doi = {10.1002/cpa.3160481103},
Key = {fds320438}
}

@article{fds244140,
Author = {Bonilla, LL and Higuera, FJ and Venakides, S},
Title = {The Gunn Effect: Instability of the Steady State and
Stability of the Solitary Wave in Long Extrinsic
Semiconductors},
Journal = {Siam Journal on Applied Mathematics},
Volume = {54},
Number = {6},
Pages = {1521-1541},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1994},
Month = {December},
url = {http://dx.doi.org/10.1137/s0036139992236554},
Abstract = {A linear stability analysis of the stationary solution of a
one-dimensional drift-diffusion model used to describe the
Gunn effect in GaAs is performed. It is shown that for long
semiconductor samples under dc voltage bias conditions, and
small diffusivity, the steady state may lose stability via a
Hopf bifurcation. In the limit of infinitely long samples,
there is a quasicontinuum of oscillatory modes of the
positive real part for voltages larger than a certain
critical value. The linear stability of the solitary wave
characteristic of the Gunn effect is proved for an idealized
electron velocity curve in the zero diffusion
limit.},
Doi = {10.1137/s0036139992236554},
Key = {fds244140}
}

@article{fds320439,
Author = {Deift, P and Venakides, S and Zhou, X},
Title = {The collisionless shock region for the long‐time behavior
of solutions of the KdV equation},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {47},
Number = {2},
Pages = {199-206},
Publisher = {WILEY},
Year = {1994},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160470204},
Abstract = {The authors further develop the nonlinear steepest descent
method of [5] and [6] to give a full description of the
collisionless shock region for solutions of the KdV equation
with decaying initial data. © 1994 John Wiley & Sons, Inc.
Company},
Doi = {10.1002/cpa.3160470204},
Key = {fds320439}
}

@article{fds320440,
Author = {Zhang, T and Venakides, S},
Title = {Periodic limit of inverse scattering},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {46},
Number = {6},
Pages = {819-865},
Publisher = {WILEY},
Year = {1993},
Month = {July},
url = {http://dx.doi.org/10.1002/cpa.3160460603},
Abstract = {I t is well known that a p‐periodic potential Q(x) can be
reconstructed from spectral data of the corresponding Hill
operator −(d 2 /dx 2 ) + Q(x) in terms of a Riemann
θ‐function. We regard the periodic potential Q(x) as the
pointwise limit of a scattering potential Q N , c (x)
(defined to equal Q(x) when −Np ≦ x ≦ Np, to equal
zero when x < Np) and to equal c 2 when x > (Np) as N →
∞ and c 2 → ∞. The scattering potential Q N , c (x)
can be recovered from the scattering data of the
corresponding Schrödinger operator in terms of a Dyson
determinant according to a well known‐theory. We derive
the Riemann θ‐function corresponding to the periodic
potential Q(x) by taking the above limit of the Dyson
determinant for the scattering potential. We first calculate
the scattering data of the potential Q N , c (x) through
recursive formulas in terms of the left transmission and
reflection coefficients T and R of the potential which is
equal to Q(x) when 0 ≦ x ≦ p and equal to zero
otherwise. We use these data to express the Dyson
determinant of Q N , c (x). We then expand the Dyson
determinant into a Fredholm series and compute the main
contributions to the expansion in the asymptotic limit N →
∞ and c 2 → ∞ using a method developed by Lax,
Levermore, and Venakides in their study of the small
dispersion limit of the initial value problem of
Korteweg‐de Vries equation. The computation of the leading
order contributions reduces to a quadratic functional
maximization problem constrained by a positivity condition
and by a mass quantization condition. The solutions to this
maximization problem constitute the differentials on a
Riemann surface, the main ingredients for the Riemann
θ‐function corresponding to the periodic potential. The
limit of the Dyson determinant for Q N , c (x) as N → ∞
and c 2 → ∞ is shown to equal the exact Riemann
θ‐function corresponding to the periodic potential Q(x)
times an exponential function with exponent being a
quadratic polynomial in x. Our calculation includes the
correct phase shifts of the θ‐function. © 1993 John
Inc., A Wiley Company},
Doi = {10.1002/cpa.3160460603},
Key = {fds320440}
}

@article{fds320441,
Author = {Venakides, S and Deift, P and Oba, R},
Title = {The toda shock problem},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {44},
Number = {8-9},
Pages = {1171-1242},
Publisher = {WILEY},
Year = {1991},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160440823},
Doi = {10.1002/cpa.3160440823},
Key = {fds320441}
}

@article{fds320442,
Author = {VENAKIDES, S},
Title = {THE KORTEWEG-DEVRIES EQUATION WITH SMALL DISPERSION -
HIGHER-ORDER LAX LEVERMORE THEORY},
Journal = {Applied and Industrial Mathematics},
Volume = {56},
Pages = {255-262},
Editor = {SPIGLER, R},
Year = {1991},
Month = {January},
ISBN = {0-7923-0521-3},
Key = {fds320442}
}

@article{fds244162,
Author = {Reed, MC and Venakides, S and Blum, JJ},
Title = {Approximate traveling waves in linear reaction-hyperbolic
equations},
Journal = {Siam Journal on Applied Mathematics},
Volume = {50},
Number = {1},
Pages = {167-180},
Publisher = {Society for Industrial & Applied Mathematics
(SIAM)},
Year = {1990},
Month = {January},
url = {http://dx.doi.org/10.1137/0150011},
Abstract = {Linear reaction-hyperbolic equations of a general type
arising in certain physiological problems do not have
traveling wave solutions, but numerical computations have
shown that they possess approximate traveling waves. Using
singular perturbation theory, it is shown that as the rates
of the chemical reactions approach ∞, solutions approach
traveling waves. The speed of the limiting wave and the
first term in the asymptotic expansion are computed in terms
of the underlying chemical mechanisms.},
Doi = {10.1137/0150011},
Key = {fds244162}
}

@article{fds320443,
Author = {Venakides, S},
Title = {The korteweg‐de vries equation with small dispersion:
Higher order lax‐levermore theory},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {43},
Number = {3},
Pages = {335-361},
Publisher = {WILEY},
Year = {1990},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160430303},
Doi = {10.1002/cpa.3160430303},
Key = {fds320443}
}

@article{fds320444,
Author = {Venakides, S},
Title = {The continuum limit of theta functions},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {42},
Number = {6},
Pages = {711-728},
Publisher = {WILEY},
Year = {1989},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160420602},
Doi = {10.1002/cpa.3160420602},
Key = {fds320444}
}

@article{fds320445,
Author = {VENAKIDES, S},
Title = {THE SMALL DISPERSION LIMIT OF THE KORTEWEG-DEVRIES
EQUATION},
Journal = {Differential Equations //},
Volume = {118},
Pages = {725-737},
Publisher = {Marcel Dekker},
Editor = {DAFERMOS, CM and LADAS, G and PAPANICOLAOU, G},
Year = {1989},
Month = {January},
ISBN = {0-8247-8077-9},
Key = {fds320445}
}

@article{fds320446,
Author = {Venakides, S},
Title = {The infinite period limit of the inverse formalism for
periodic potentials},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {41},
Number = {1},
Pages = {3-17},
Publisher = {WILEY},
Year = {1988},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160410103},
Doi = {10.1002/cpa.3160410103},
Key = {fds320446}
}

@article{fds320447,
Author = {Venakides, S},
Title = {The Zero Dispersion Limit of the Korteweg-Devries Equation
with Periodic Initial Data},
Journal = {Transactions of the American Mathematical
Society},
Volume = {301},
Number = {1},
Pages = {189-226},
Publisher = {American Mathematical Society},
Year = {1987},
Month = {May},
url = {http://dx.doi.org/10.2307/2000334},
Doi = {10.2307/2000334},
Key = {fds320447}
}

@article{fds320448,
Author = {Venakides, S},
Title = {The zero dispersion limit of the korteweg-de vries equation
with periodic initial data},
Journal = {Transactions of the American Mathematical
Society},
Volume = {301},
Number = {1},
Pages = {189-226},
Publisher = {American Mathematical Society (AMS)},
Year = {1987},
Month = {January},
url = {http://dx.doi.org/10.1090/S0002-9947-1987-0879569-7},
Abstract = {We study the initial value problem for the Korteweg-de Vries
equation (FORMULA PRESENTED) in the limit of small
dispersion, i.e., 0. When the unperturbed equation (FORMULA
PRESENTED) develops a shock, rapid oscillations arise in the
solution of the perturbed equation (i) In our study: a. We
compute the weak limit of the solution of (i) for periodic
initial data as 0. b. We show that in the neighborhood of a
point (x, t) the solution u(x, t,) can be approximated
either by a constant or by a periodic or by a quasiperiodic
solution of equation (i). In the latter case the associated
wavenumbers and frequencies are of order O(1/). c. We
compute the number of phases and the wave parameters
associated with each phase of the approximating solution as
functions of x and t. d. We explain the mechanism of the
generation of oscillatory phases. Our computations in a and
c are subject to the solution of the Lax-Levermore evolution
equations (7.7). Our results in b-d rest on a plausible
averaging assumption. © 1987 American Mathematical
Society.},
Doi = {10.1090/S0002-9947-1987-0879569-7},
Key = {fds320448}
}

@article{fds320449,
Author = {Venakides, S},
Title = {Long time asymptotics of the korteweg-de vries
equation},
Journal = {Transactions of the American Mathematical
Society},
Volume = {293},
Number = {1},
Pages = {411-419},
Publisher = {American Mathematical Society (AMS)},
Year = {1986},
Month = {January},
url = {http://dx.doi.org/10.1090/S0002-9947-1986-0814929-0},
Abstract = {We study the long time evolution of the solution to the
Korteweg- de Vries equation with initial data u(x) which
satisfy lim y(.x) = -1, lim U(x) = 0. (Formula presented) We
show that as t →∞the step emits a wavetrain of solitons
which asymptotically have twice the amplitude of the initial
step. We derive a lower bound of the number of solitons
separated at time t for t large. © 1986 American
Mathematical Society.},
Doi = {10.1090/S0002-9947-1986-0814929-0},
Key = {fds320449}
}

@article{fds320450,
Author = {Venakides, S},
Title = {Long Time Asymptotics of the Korteweg-de Vries
Equation},
Journal = {Transactions of the American Mathematical
Society},
Volume = {293},
Number = {1},
Pages = {411-411},
Publisher = {JSTOR},
Year = {1986},
Month = {January},
url = {http://dx.doi.org/10.2307/2000288},
Doi = {10.2307/2000288},
Key = {fds320450}
}

@article{fds320451,
Author = {Venakides, S},
Title = {The generation of modulated wavetrains in the solution of
the Korteweg—de vries equation},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {38},
Number = {6},
Pages = {883-909},
Publisher = {WILEY},
Year = {1985},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160380616},
Doi = {10.1002/cpa.3160380616},
Key = {fds320451}
}

@article{fds320452,
Author = {Venakides, S},
Title = {The zero dispersion limit of the korteweg‐de vries
equation for initial potentials with non‐trivial
reflection coefficient},
Journal = {Communications on Pure and Applied Mathematics},
Volume = {38},
Number = {2},
Pages = {125-155},
Publisher = {WILEY},
Year = {1985},
Month = {January},
url = {http://dx.doi.org/10.1002/cpa.3160380202},
Abstract = {The inverse scattering method is used to determine the
distribution limit as ϵ → 0 of the solution u(x, t, ϵ)
of the initial value problem. Ut − 6uux + ϵ2uxxx = 0,
u(x, 0) = v(x), where v(x) is a positive bump which decays
sufficiently fast as x x→±α. The case v(x) ≪ 0 has
been solved by Peter D. Lax and C. David Levermore [8], [9],
[10]. The computation of the distribution limit of u(x, t,
ϵ) as ϵ → 0 is reduced to a quadratic maximization
Periodicals, Inc., A Wiley Company},
Doi = {10.1002/cpa.3160380202},
Key = {fds320452}
}

@article{fds10163,
Author = {El, G.A. and Krylov, A.L. and Molchanov, S.A. and Venakides,
S.},
Title = {Soliton turbulence as a thermodynamic limit of stochastic
soliton lattices. In Advances in nonlinear mathematics and
science.},
Journal = {Physica D 152/153 (2001), 653--664},
Key = {fds10163}
}

@article{fds9749,
Author = {S. Venakides and M. Haider and V. Papanicolaou},
Title = {Boundary Integral Calculations of 2-d Electromagnetic
Scattering by Photonic Crystal Fabry-Perot
Structures},
Journal = {SIAM J. Appl. Math. vol. 60/5, (2000), pp.
1636-1706},
Key = {fds9749}
}

@article{fds9752,
Author = {A. Georgieva and T. Kriecherbauer and Stephanos
Venakides},
Title = {Wave Propagation and Resonance in a 1-d Nonlinear Discrete
Periodic Medium},
Journal = {SIAM J. Appl. Math., vol. 60/1, (1999), pp.
272-294},
Key = {fds9752}
}

@article{fds9506,
Author = {P. Deift and T. Kriecherbauer and K. T-R McLaughlin and S. Venakides and X. Zhou},
Title = {Strong Asymptotics of Orhtogonal Polynomials with Respect to
Exponential Weights},
Journal = {CPAM, vol.52 (1999) 1491-1552.},
Key = {fds9506}
}

@article{fds8925,
Author = {M. McDonald and S. Venakides},
Title = {Renormalization of the Tau Function for Integrable Systems:
A Model Problem},
Journal = {CPAM, Vol 51, 1998, 937-966.},
Key = {fds8925}
}

@article{fds8924,
Author = {P. Deift and S. Venakides and X. Zhou},
Title = {An Extension of the Method of Steepest Descent for
Riemann-Hilbert Problems: The Small Dispersion Limit of the
Korteweg-de Vries (KdV) Equation},
Journal = {Proc. Ntl. Acad. Sc. USA, vol. 95, Jan 1998,
450-454.},
Key = {fds8924}
}

@article{fds9390,
Author = {P. Deift and S. Venakides and X. Zhou},
Title = {New Results in the Small-Dispersion KdV by an Extension of
the Method of Steepest Descent for Riemann-Hilbert
Problems},
Journal = {IMRN, 1997, N0. 6, 285-299.},
Key = {fds9390}
}

@article{fds9389,
Author = {P. Deift and T. Kriecherbauer and K. T-R McLaughlin and S. Venakides and X. Zhou},
Title = {Asymptotics of Polynomials Orthogonal with Respect to
Varying Exponential Weights},
Journal = {IMRN, 1997 No 16, pp. 759-782},
Key = {fds9389}
}

@article{fds9395,
Author = {P. Deift and T. Kriecherbauer and S. Venakides},
Title = {Forced Lattice Vibrations Part II},
Journal = {Comm. Pure Appl. Math. 48, 1995, 1251-1298.},
Key = {fds9395}
}

@article{fds9394,
Author = {P. Deift and T. Kriecherbauer and S. Venakides},
Title = {Forced Lattice Vibrations Part I},
Journal = {Comm. Pure Appl. Math. 48,1995, 1187-1250.},
Key = {fds9394}
}

@article{fds9410,
Author = {L. L. Bonilla and F. Higuera and S. Venakides},
Title = {The Stability of the Steady State of the Gunn
Oscillator},
Journal = {SIAM J. Appl. Math. vol. 54, No 6, (1994), pp.
1521-1541.},
Key = {fds9410}
}

@article{fds9409,
Author = {P. Deift and S. Venakides and X. Zhou},
Title = {The Collisionless Shock Region for the Long Time Behavior of
the Solutions of the KdV Equation},
Journal = {CPAM. vol. 47, (1994), pp. 199-206.},
Key = {fds9409}
}

@article{fds9408,
Author = {P. D. Lax and C. D. Levermore and S. Venakides},
Title = {The Generation and Propagation of Oscillations in Dispersive
IVP's and their Limiting Behavior},
Journal = {Important Developments in Soliton Theory 1980--1990}, T.
Fokas and V.E. Zakharov eds., Springer-Verlag, Berlin
(1992).},
Key = {fds9408}
}

@article{fds9399,
Author = {S. Venakides},
Title = {The solution of completely integrable systems in the
continuum limit of the spectral data},
Journal = {IMA Proceedings, vol. 2, (1986) pp. 337-356..},
Key = {fds9399}
}

@article{fds9396,
Author = {S. Venakides},
Title = {The zero-dispersion limit of the Korteweg-de Vries equation
with non-trivial reflection coefficient},
Journal = {Comm. Pure and Appl. Math. 38, pp. 125-155,
1985.},
Key = {fds9396}
}

%% Preprints
@article{fds226041,
Author = {Stavros Komineas and Stephen P. Shipman and Stephanos
Venakides},
Title = {Lossless Polariton Solitons},
Journal = {arXiv},
Year = {2014},
url = {http://arxiv.org/abs/1409.4067},
Abstract = {Photons and excitons in a semiconductor microcavity interact
to form exciton-polariton condensates. These are governed by
a nonlinear quantum-mechanical system involving exciton and
photon wavefunctions. We calculate all non-traveling
harmonic soliton solutions for the one-dimensional lossless
system. There are two frequency bands of bright solitons
when the inter-exciton interactions produce a repulsive
nonlinearity and two frequency bands of dark solitons when
the nonlinearity is attractive. In addition, there are two
frequency bands for which the exciton wavefunction is
discontinuous at its symmetry point, where it undergoes a
phase jump of π. A band of continuous dark solitons merges
with a band of discontinuous dark solitons, forming a larger
band over which the soliton far-field amplitude varies from
0 to ∞; the discontinuity is initiated when the operating
frequency exceeds the free exciton frequency. The far fields
of the solitons in the lowest and highest frequency bands
(one discontinuous and one continuous dark) are linearly
unstable, whereas the other four bands have linearly stable
far fields, including the merged band of dark
solitons.},
Key = {fds226041}
}

dept@math.duke.edu
ph: 919.660.2800
fax: 919.660.2821

Mathematics Department
Duke University, Box 90320
Durham, NC 27708-0320