Scholars@Duke
Stephanos Venakides

Stephanos Venakides

Professor of Mathematics
Mathematics
ven@math.duke.edu
Box 90320, Durham, NC 27708-0320
120 Science Drive, Durham, NC 27708, Durham, NC 27708
Office hours Tuesday and Thursday 3:00-4:00pm  

Selected Publications

Chiral skyrmions of large radius

Journal Article Physica D: Nonlinear Phenomena · April 1, 2021 Featured Publication We study the structure of an axially symmetric magnetic skyrmion in a ferromagnet with the Dzyaloshinskii–Moriya interaction. We examine the regime of large skyrmions and we identify rigorously the critical value of the dimensionless parameter at which the ... Full text Open Access Cite

The profile of chiral skyrmions of small radius

Journal Article Nonlinearity · July 1, 2020 Chiral skyrmions are stable particle-like solutions of the Landau-Lifshitz equation for ferromagnets with the Dzyaloshinskii-Moriya (DM) interaction, characterized by a topological number. We study the profile of an axially symmetric skyrmion and give exac ... Full text Cite

Traveling domain walls in chiral ferromagnets

Journal Article Nonlinearity · May 30, 2019 We show that chiral symmetry breaking enables traveling domain wall solution for the conservative Landau-Lifshitz equation of a uniaxial ferromagnet with Dzyaloshinskii-Moriya interaction. In contrast to related domain wall models including stray-field bas ... Full text Cite

Domain decomposition for quasi-periodic scattering by layered media via robust boundary-integral equations at all frequencies

Journal Article Communications in Computational Physics · January 1, 2019 We develop a non-overlapping domain decomposition method (DDM) for scalar wave scattering by periodic layered media. Our approach relies on robust boundary-integral equation formulations of Robin-to-Robin (RtR) maps throughout the frequency spectrum, inclu ... Full text Cite

Mathematical models of dorsal closure.

Journal Article Progress in biophysics and molecular biology · September 2018 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 dist ... Full text Cite

Cell Sheet Morphogenesis: Dorsal Closure in Drosophila melanogaster as a Model System.

Journal Article Annual review of cell and developmental biology · October 2017 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 elong ... Full text Cite

Superalgebraically convergent smoothly windowed lattice sums for doubly periodic Green functions in three-dimensional space

Journal Article Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences · July 1, 2016 This 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 scatte ... Full text Cite

Lossless polariton solitons

Journal Article Physica D: Nonlinear Phenomena · February 15, 2016 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 solu ... Full text Cite

Continuous and discontinuous dark solitons in polariton condensates

Journal Article Physical Review B - Condensed Matter and Materials Physics · April 9, 2015 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-ve ... Full text Cite

Long-time limit studies of an obstruction in the g-function mechanism for semiclassical focusing NLS

Journal Article · 2015 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 p ... Link to item Cite

Smooth parametric dependence of asymptotics of the semiclassical focusing NLS

Journal Article Analysis and PDE · January 1, 2015 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 asymp ... Full text Cite

Destructive impact of imperfect beam collimation in extraordinary optical transmission.

Journal Article Journal of the Optical Society of America. A, Optics, image science, and vision · June 2013 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 transmi ... Full text Cite

An exactly solvable model for nonlinear resonant scattering

Journal Article Nonlinearity · September 1, 2012 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 ... Full text Cite

Semiclassical limit of the scattering transform for the focusing nonlinear Schrödinger equation

Journal Article International Mathematics Research Notices · May 21, 2012 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 expressi ... Full text Cite

Perturbation of Riemann-Hilbert jump contours: smooth parametric dependence with application to semiclassical focusing NLS

Journal Article · August 25, 2011 A perturbation of a class of scalar Riemann-Hilbert problems (RHPs) with the jump contour as a finite union of oriented simple arcs in the complex plane and the jump function with a $z\log z$ type singularity on the jump contour is considered. The jump fun ... Link to item Cite

Nonlinear steepest descent asymptotics for semiclassical limit of Integrable systems: Continuation in the parameter space

Journal Article Communications in Mathematical Physics · February 1, 2010 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 pr ... Full text Cite

Drosophila morphogenesis: tissue force laws and the modeling of dorsal closure.

Journal Article HFSP journal · December 2009 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, wh ... Full text Cite

Accurate description of optical precursors and their relation to weak-field coherent optical transients

Journal Article Physical Review A - Atomic, Molecular, and Optical Physics · June 29, 2009 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 a ... Full text Cite

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 Article International Mathematics Research Notices · February 1, 2009 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 s ... Full text Cite

Determinant form of modulation equations for the semiclassical focusing Nonlinear Schr\" odinger equation

Journal Article · 2009 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-H ... Link to item Cite

Fano resonance of waves in periodic slabs

Journal Article Mathematical Methods in Electromagnetic Theory, MMET, Conference Proceedings · September 19, 2008 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 n ... Full text Cite

Semiclassical focusing nonlinear schrödinger equation i: Inverse scattering map and its evolution for radiative initial data

Journal Article International Mathematics Research Notices · December 1, 2007 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 cl ... Full text Cite

Long-time asymptotics of the nonlinear Schrödinger equation shock problem

Journal Article Communications on Pure and Applied Mathematics · September 1, 2007 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 pha ... Full text Cite

Resiliency, coordination, and synchronization of dorsal closure during Drosophila morphogenesis

Journal Article Biophysical Journal · April 2007 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 mult ... Cite

Upregulation of forces and morphogenic asymmetries in dorsal closure during Drosophila development.

Journal Article Biophysical journal · April 2007 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, whi ... Full text Cite

The semiclassical focusing nonlinear Schrodinger equation

Conference Recent Advances in Nonlinear Partial Differential Equations and Applications · 2007 Link to item Cite

On the long-time limit of semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation: Pure radiation case

Journal Article Communications on Pure and Applied Mathematics · January 1, 2006 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. ... Full text Cite

Resonant transmission near nonrobust periodic slab modes.

Journal Article Physical review. E, Statistical, nonlinear, and soft matter physics · February 2005 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 asymp ... Full text Open Access Cite

Force regulation during dorsal closure in Drosophila

Conference Molecular Biology of the Cell · November 2004 Link to item Cite

On semiclassical (zero dispersion limit) solutions of the focusing nonlinear Schrödinger equation

Journal Article Communications on Pure and Applied Mathematics · July 1, 2004 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 and pure radiation. In ... Full text Cite

Resonance and bound states in photonic crystal slabs

Journal Article SIAM Journal on Applied Mathematics · October 1, 2003 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 Fl ... Full text Cite

Forces for morphogenesis investigated with laser microsurgery and quantitative modeling.

Journal Article Science (New York, N.Y.) · April 2003 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 ... Full text Cite

Optimization of Resonances in Photonic Crystal Slabs

Journal Article Proceedings of SPIE - The International Society for Optical Engineering · January 1, 2003 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. ... Full text Cite

Boundary-integral calculations of two-dimensional electromagnetic scattering in infinite photonic crystal slabs: Channel defects and resonances

Journal Article SIAM Journal on Applied Mathematics · July 1, 2002 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 di ... Full text Cite

Unified approach to KdV modulations

Journal Article Communications on Pure and Applied Mathematics · October 1, 2001 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 m ... Full text Cite

A riemann-Hilbert approach to asymptotic questions for orthogonal polynomials

Journal Article Journal of Computational and Applied Mathematics · August 1, 2001 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 (1 ... Full text Cite

Soliton turbulence as a thermodynamic limit of stochastic soliton lattices

Journal Article Physica D: Nonlinear Phenomena · May 15, 2001 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 densi ... Full text Cite

1:2 resonance mediated second harmonic generation in a 1-D nonlinear discrete periodic medium

Journal Article SIAM Journal on Applied Mathematics · January 1, 2001 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) ... Full text Cite

The eigenvalue problem for the focusing nonlinear Schrödinger equation: New solvable cases

Journal Article Physica D: Nonlinear Phenomena · November 15, 2000 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 ... Full text Cite

Boundary integral calculations of two-dimensional electromagnetic scattering by photonic crystal Fabry-Perot structures

Journal Article SIAM Journal on Applied Mathematics · January 1, 2000 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 q ... Full text Cite

Studying the asymptotics of Selberg-type integrals

Conference Applied and Industrial Mathematics, Venice-2, 1998 · 2000 Link to item Cite

Two-dimensional photonic crystal fabry-perot resonators with lossy dielectrics

Journal Article IEEE Transactions on Microwave Theory and Techniques · December 1, 1999 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' ... Full text Cite

Strong asymptotics of orthogonal polynomials with respect to exponential weights

Journal Article Communications on Pure and Applied Mathematics · January 1, 1999 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 formu ... Full text Cite

Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

Journal Article Communications on Pure and Applied Mathematics · January 1, 1999 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 Pla ... Full text Cite

Wave propagation and resonance in a one-dimensional nonlinear discrete periodic medium

Journal Article SIAM Journal on Applied Mathematics · January 1, 1999 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 gen ... Full text Cite

Long-time asymptotics for the pure radiation solution of the sine-Gordon equation

Journal Article Communications in Partial Differential Equations · January 1, 1999 Full text Cite

Existence and modulation of traveling waves in particle chains

Journal Article Communications on Pure and Applied Mathematics · January 1, 1999 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 nth particle at time t along the chain axis ... Full text Cite

Renormalization of the τ-functions for integrable systems: A model problem

Journal Article Communications on Pure and Applied Mathematics · January 1, 1998 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 p ... Full text Cite

An extension of the steepest descent method for Riemann-Hilbert problems: the small dispersion limit of the Korteweg-de Vries (KdV) equation.

Journal Article Proceedings of the National Academy of Sciences of the United States of America · January 1998 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 ... Full text Cite

New Results in Small Dispersion KdV by an Extension of the Steepest Descent Method for Riemann-Hilbert Problems

Journal Article International Mathematics Research Notices · December 1, 1997 Cite

Asymptotics for Polynomials Orthogonal with Respect to Varying Exponential Weights

Journal Article International Mathematics Research Notices · December 1, 1997 Cite

Periodic generation and propagation of traveling fronts in dc voltage biased semiconductor superlattices

Journal Article SIAM Journal on Applied Mathematics · January 1, 1997 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, suppleme ... Full text Cite

Asymptotics for polynomials orthogonal with respect to varying exponential weights

Journal Article International Mathematics Research Notices · 1997 Link to item Cite

Forced lattice vibrations: Part I

Journal Article Communications on Pure and Applied Mathematics · January 1, 1995 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 pertu ... Full text Cite

Forced lattice vibrations: Part II

Journal Article Communications on Pure and Applied Mathematics · January 1, 1995 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 per ... Full text Cite

Forced Lattice Vibrations -- A Videotext

Journal Article · September 26, 1994 We begin with a description of recent numerical and analytical results that are closely related to the results of this paper. ... Link to item Cite

Gunn effect: Instability of the steady state and stability of the solitary wave in long extrinsic semiconductors

Journal Article SIAM Journal on Applied Mathematics · January 1, 1994 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 ... Full text Cite

The collisionless shock region for the long‐time behavior of solutions of the KdV equation

Journal Article Communications on Pure and Applied Mathematics · January 1, 1994 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. Copyright © 1994 Wile ... Full text Cite

Periodic limit of inverse scattering

Journal Article Communications on Pure and Applied Mathematics · January 1, 1993 I t is well known that a p‐periodic potential Q(x) can be reconstructed from spectral data of the corresponding Hill operator −(d2/dx2) + Q(x) in terms of a Riemann θ‐function. We regard the periodic potential Q(x) as the pointwise limit of a scattering po ... Full text Cite

The toda shock problem

Journal Article Communications on Pure and Applied Mathematics · January 1, 1991 Full text Cite

Approximate traveling waves in linear reaction-hyperbolic equations

Journal Article SIAM Journal on Applied Mathematics · January 1, 1990 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, ... Full text Cite

The korteweg‐de vries equation with small dispersion: Higher order lax‐levermore theory

Journal Article Communications on Pure and Applied Mathematics · January 1, 1990 Full text Cite

The continuum limit of theta functions

Journal Article Communications on Pure and Applied Mathematics · January 1, 1989 Full text Cite

The infinite period limit of the inverse formalism for periodic potentials

Journal Article Communications on Pure and Applied Mathematics · January 1, 1988 Full text Cite

The Zero Dispersion Limit of the Korteweg-Devries Equation with Periodic Initial Data

Journal Article Transactions of the American Mathematical Society · May 1987 Full text Link to item Cite

The zero dispersion limit of the korteweg-de vries equation with periodic initial data

Journal Article Transactions of the American Mathematical Society · January 1, 1987 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 perturb ... Full text Cite

Long time asymptotics of the korteweg-de vries equation

Journal Article Transactions of the American Mathematical Society · January 1, 1986 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 ... Full text Cite

Long-Time Asymptotics of the Korteweg-Devries Equation

Journal Article Transcations of the American Mathematical Society · January 1986 Full text Link to item Cite

The generation of modulated wavetrains in the solution of the Korteweg—de vries equation

Journal Article Communications on Pure and Applied Mathematics · January 1, 1985 Full text Cite

The zero dispersion limit of the korteweg‐de vries equation for initial potentials with non‐trivial reflection coefficient

Journal Article Communications on Pure and Applied Mathematics · January 1, 1985 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 ... Full text Cite

Traveling domain walls in chiral ferromagnets

Journal Article Discontinuity, Nonlinearity, and Complexity Full text Cite