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We study the semiclassical limit of the sine-Gordon (sG) equation with below threshold pure impulse initial data of Klaus-Shaw type. The Whitham averaged approximation of this system exhibits a gradient catastrophe in finite time. In accordance with a conjecture of Dubrovin, Grava and Klein, we found that in a $\mathcal{O}(\epsilon^{4/5})$ neighborhood near the gradient catastrophe point, the asymptotics of the sG solution are universally described by the Painleve I tritronquee solution. A linear map can be explicitly made from the tritronquee solution to this neighborhood. Under this map: away from the tritronquee poles, the first correction of sG is universally given by the real part of the Hamiltonian of the tritronquee solution; localized defects appear at locations mapped from the poles of tritronquee solution; the defects are proved universally to be a two parameter family of special localized solutions on a periodic background for the sG equation. We are able to characterize the solution in detail. Our approach is the rigorous steepest descent method for matrix Riemann--Hilbert problems, substantially generalizing Bertola and Tovbis's results on the nonlinear Schrodinger equation to establish universality beyond the context of solutions of a single equation.

Background: The treatment of COVID-19 patients with heparin is not always effective in preventing thrombotic complications, but can also be associated with bleeding complications, suggesting a balanced approach to anticoagulation is needed. A prior pilot study supported that thromboelastography and conventional coagulation tests could predict hemorrhage in COVID-19 in patients treated with unfractionated heparin or enoxaparin, but did not evaluate the risk of thrombosis. Methods: This single-center, retrospective study included 79 severely ill COVID-19 patients anticoagulated with intermediate or therapeutic dose unfractionated heparin. Two stepwise logistic regression models were performed with bleeding or thrombosis as the dependent variable, and thromboelastography parameters and conventional coagulation tests as the independent variables. Results: Among all 79 patients, 12 (15.2%) had bleeding events, and 20 (25.3%) had thrombosis. Multivariate logistic regression analysis identified a prediction model for bleeding (adjusted R2 = 0.787, p &lt, 0.001) comprised of increased reaction time (p = 0.016), decreased fibrinogen (p = 0.006), decreased D-dimer (p = 0.063), and increased activated partial thromboplastin time (p = 0.084). Multivariate analysis of thrombosis identified a weak prediction model (adjusted R2 = 0.348, p &lt, 0.001) comprised of increased D-dimer (p &lt, 0.001), decreased reaction time (p = 0.002), increased maximum amplitude (p &lt, 0.001), and decreased alpha angle (p = 0.014). Adjunctive thromboelastography decreased the use of packed red cells (p = 0.031) and fresh frozen plasma (p &lt, 0.001). Conclusions: Significantly, this study demonstrates the need for a precision-based titration strategy of anticoagulation for hospitalized COVID-19 patients. Since severely ill COVID-19 patients may switch between thrombotic or hemorrhagic phenotypes or express both simultaneously, institutions may reduce these complications by developing their own titration strategy using daily conventional coagulation tests with adjunctive thromboelastography.

All of the six Painlev´e equations except the first have families of rationalsolutions, which are frequently important in applications. The third Painlev´eequation in generic form depends on two parameters m and n, and it has rational solutions if and only if at least one of the parameters is an integer. We use known algebraic representations of the solutions to study numerically how the distributions of poles and zeros behave as n ∈ Z increases and how the patterns vary with m ∈ C. This study suggests that it is reasonable to consider the rational solutions in the limit of large n ∈ Z with m ∈ C being an auxiliary parameter. To analyze the rational solutions in this limit, algebraic techniques need to be supplemented by analytical ones, and the main new contribution of this paper is to develop a Riemann–Hilbert representation of the rational solutions of Painlevé-III that is amenable to asymptotic analysis. Assuming further that m is a half-integer, we derive from the Riemann–Hilbert representation a finite dimensional Hankel system for the rational solution in which n ∈ Z appears as an explicit parameter.

We study the fundamental rogue wave solutions of the focusing nonlinear Schr\"odinger equation in the limit of large order. Using a recently-proposed Riemann-Hilbert representation of the rogue wave solution of arbitrary order $k$, we establish the existence of a limiting profile of the rogue wave in the large-$k$ limit when the solution is viewed in appropriate rescaled variables capturing the near-field region where the solution has the largest amplitude. The limiting profile is a new particular solution of the focusing nonlinear Schr\"odinger equation in the rescaled variables --- the rogue wave of infinite order --- which also satisfies ordinary differential equations with respect to space and time. The spatial differential equations are identified with certain members of the Painlev\'e-III hierarchy. We compute the far-field asymptotic behavior of the near-field limit solution and compare the asymptotic formul\ae\ with the exact solution with the help of numerical methods for solving Riemann-Hilbert problems. In a certain transitional region for the asymptotics the near field limit function is described by a specific globally-defined tritronqu\'ee solution of the Painlev\'e-II equation. These properties lead us to regard the rogue wave of infinite order as a new special function., Comment: 55 pages, 27 figures

The Painlevé-III equation with parameters Θ = n+ m and Θ ∞= m- n+ 1 has a unique rational solution u(x) = u n(x; m) with u n(∞; m) = 1 whenever n∈ Z. Using a Riemann–Hilbert representation proposed in Bothner et al. (Stud Appl Math 141:626–679, 2018), we study the asymptotic behavior of u n(x; m) in the limit n→ + ∞ with m∈ C held fixed. We isolate an eye-shaped domain E in the y= n - 1x plane that asymptotically confines the poles and zeros of u n(x; m) for all values of the second parameter m. We then show that unless m is a half-integer, the interior of E is filled with a locally uniform lattice of poles and zeros, and the density of the poles and zeros is small near the boundary of E but blows up near the origin, which is the only fixed singularity of the Painlevé-III equation. In both the interior and exterior domains we provide accurate asymptotic formulæ for u n(x; m) that we compare with u n(x; m) itself for finite values of n to illustrate their accuracy. We also consider the exceptional cases where m is a half-integer, showing that the poles and zeros of u n(x; m) now accumulate along only one or the other of two “eyebrows,” i.e., exterior boundary arcs of E.

This Special Issue on Dispersive Hydrodynamics is dedicated to the memory and work of G.B. Whitham who was one of the pioneers in this field of physical applied mathematics. Some of the papers appearing here are related to work reported on at the workshop “Dispersive Hydrodynamics: The Mathematics of Dispersive Shock Waves and Applications” held in May 2015 at the Banff International Research Station. This Preface provides a broad overview of the field and summaries of the various contributions to the Special Issue, placing them in a unified context.

We review various methods for the analysis of initial-value problems for integrable dispersive equations in the weak-dispersion or semiclassical regime. Some methods are sufficiently powerful to rigorously explain the generation of modulated wavetrains, so-called dispersive shock waves, as the result of shock formation in a limiting dispersionless system. They also provide a detailed description of the solution near caustic curves that delimit dispersive shock waves, revealing fascinating universal wave patterns., Comment: 20 pages, 9 figures. Submitted to Physica D

We present a new and relatively elementary method for studying the solution of the initial-value problem for dispersive linear and integrable equations in the large-t limit, based on a generalization of steepest descent techniques for Riemann-Hilbert problems to the setting of \({\overline {\partial }}\)-problems. Expanding upon prior work (Dieng and McLaughlin, Long-time asymptotics for the NLS equation via \({\overline {\partial }}\) methods, arXiv:0805.2807, 2008) of the first two authors, we develop the method in detail for the linear and defocusing nonlinear Schrodinger equations, and show how in the case of the latter it gives sharper asymptotics than previously known under essentially minimal regularity assumptions on initial data.

The increasing tritronqu\'ee solutions of the Painlev\'e-II equation with parameter $\alpha$ exhibit square-root asymptotics in the maximally-large sector $|\arg(x)

We compute the scattering data of the Benjamin-Ono equation for arbitrary rational initial conditions with simple poles. Specifically, we obtain explicit formulas for the Jost solutions and eigenfunctions of the associated spectral problem, yielding an Evans function for the eigenvalues and formulas for the phase constants and reflection coefficient.

We study the unstable spectrum close to the imaginary axis for the linearization of the nonlinear Klein–Gordon equation about a periodic traveling wave in a co-moving frame. We define dynamical Hamiltonian–Hopf instabilities as points in the stable spectrum that are accumulation points for unstable spectrum, and show how they can be determined from the knowledge of the discriminant of Hill’s equation for an associated periodic potential. This result allows us to give simple criteria for the existence of dynamical Hamiltonian–Hopf instabilities in terms of instability indices previously shown to be useful in stability analysis of periodic traveling waves. We also discuss how these methods can be applied to more general nonlinear wave equations.

Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods in practice is that calculation of the spectral transforms of the initial and boundary data requires knowledge of too many boundary conditions, more than are required to make the problem well-posed. The elimination of the unknown boundary values is frequently addressed in the spectral domain via the so-called global relation, and types of boundary conditions for which the global relation can be solved are called linearizable. For the defocusing nonlinear Schrodinger equation, the global relation is only known to be explicitly solvable in rather restrictive situations, namely homogeneous boundary conditions of Dirichlet, Neumann, and Robin (mixed) type. General nonhomogeneous boundary conditions are not known to be linearizable. In this paper, we propose an explicit approximation for the nonlinear Dirichlet-to-Neumann map supplied by the defocusing nonlinear Schrodinger equation and use it to provide approximate solutions of general nonhomogeneous boundary value problems for this equation posed as an initial-boundary value problem on the half-line. Our method sidesteps entirely the solution of the global relation. The accuracy of our method is proven in the semiclassical limit, and we provide explicit asymptotics for the solution in the interior of the quarter-plane space-time domain.

We study the diffusion (or heat) equation on a finite 1-dimensional spatial domain, but we replace one of the boundary conditions with a "nonlocal condition", through which we specify a weighted average of the solution over the spatial interval. We provide conditions on the regularity of both the data and weight for the problem to admit a unique solution, and also provide a solution representation in terms of contour integrals. The solution and well-posedness results rely upon an extension of the Fokas (or unified) transform method to initial-nonlocal value problems for linear equations; the necessary extensions are described in detail. Despite arising naturally from the Fokas transform method, the uniqueness argument appears to be novel even for initial-boundary value problems., Comment: 21 pages, 3 figures

The rational solutions of the Painlev\'e-II equation appear in several applications and are known to have many remarkable algebraic and analytic properties. They also have several different representations, useful in different ways for establishing these properties. In particular, Riemann-Hilbert representations have proven to be useful for extracting the asymptotic behavior of the rational solutions in the limit of large degree (equivalently the large-parameter limit). We review the elementary properties of the rational Painlev\'e-II functions, and then we describe three different Riemann-Hilbert representations of them that have appeared in the literature: a representation by means of the isomonodromy theory of the Flaschka-Newell Lax pair, a second representation by means of the isomonodromy theory of the Jimbo-Miwa Lax pair, and a third representation found by Bertola and Bothner related to pseudo-orthogonal polynomials. We prove that the Flaschka-Newell and Bertola-Bothner Riemann-Hilbert representations of the rational Painlev\'e-II functions are explicitly connected to each other. Finally, we review recent results describing the asymptotic behavior of the rational Painlev\'e-II functions obtained from these Riemann-Hilbert representations by means of the steepest descent method.

The three-wave resonant interaction equations are a non-dispersive system of partial differential equations with quadratic coupling describing the time evolution of the complex amplitudes of three resonant wave modes. Collisions of wave packets induce energy transfer between different modes via pumping and decay. We analyze the collision of two or three packets in the semiclassical limit by applying the inverse-scattering transform. Using WKB analysis, we construct an associated semiclassical soliton ensemble, a family of reflectionless solutions defined through their scattering data, intended to accurately approximate the initial data in the semiclassical limit. The map from the initial packets to the soliton ensemble is explicit and amenable to asymptotic and numerical analysis. Plots of the soliton ensembles indicate the space-time plane is partitioned into regions containing either quiescent, slowly varying, or rapidly oscillatory waves. This behavior resembles the well-known generation of dispersive shock waves in equations such as the Korteweg-de Vries and nonlinear Schrodinger equations, although the physical mechanism must be different in the absence of dispersion., 82 pages, 14 figures

We study the Cauchy problem for the sine-Gordon equation in the semiclassical limit with pure-impulse initial data of sufficient strength to generate both high-frequency rotational motion near the peak of the impulse profile and also high-frequency librational motion in the tails. Subject to suitable conditions of a general nature, we analyze the fluxon condensate solution approximating the given initial data for small time near points where the initial data crosses the separatrix of the phase portrait of the simple pendulum. We show that the solution is locally constructed as a universal curvilinear grid of superluminal kinks and grazing collisions thereof, with the grid curves being determined from rational solutions of the Painleve-II system.

We study the Benjamin-Ono hierarchy with positive initial data of a general type, in the limit when the dispersion parameter tends to zero. We establish simple formulae for the limits (in appropriate weak or distributional senses) of an infinite family of simultaneously conserved densities in terms of alternating sums of branches of solutions of the inviscid Burgers hierarchy., 12 pages, 2 figures. Submitted to Comm. Math. Sci. special issue in honor of C. D. Levermore's 60th birthday

Generic wave train solutions to the complex Ablowitz-Ladik equations are developed using methods of algebraic geometry. The inverse spectral transform is used to realize these solutions as potentials in a spatially discrete linear operator. The manifold of wave trains is infinite-dimensional, but is stratified by finite-dimensional submanifolds indexed by nonnegative integers g. Each of these strata is a foliation whose leaves are parametrized by the moduli space of (possibly singular) hyperelliptic Riemann surfaces of genus g. The generic leaf is a g-dimensional complex torus. Thus, each wave train is constructed from a finite number of complex numbers comprising a set of spectral data, indicating that the wave train has a finite number of degrees of freedom. Our construction uses a new Lax pair differing from that originally given by Ablowitz and Ladik. This new Lax pair allows a simplified construction that avoids some of the degeneracies encountered in previous analyses that make use of the original discretized AKNS Lax pair. Generic wave trains are built from Baker-Akhiezer functions on nonsingular Riemann surfaces having distinct branch points, and the construction is extended to handle singular Riemann surfaces that are pinched off at a coinciding pair of branch points. The corresponding solutions in the pinched case may also be derived from wave trains belonging to nonsingular surfaces using Backlund transformations. The problem of reducing the complex Ablowitz-Ladik equations to the focusing and defocusing versions of the discrete nonlinear Schrodinger equation is solved by specifying which spectral data correspond to focusing or defocusing potentials. Within the class of finite genus complex potentials, spatially periodic potentials are isolated, resulting in a formula for the solution to the spatially periodic initial-value problem. Formal modulation equations governing slow evolution of (g + 1)-phase wave trains are developed, and a gauge invariance is used to simplify the equations in the focusing and defocusing cases. In both of these cases, the modulation equations can be either hyperbolic (suggesting modulational stability) or elliptic (suggesting modulational instability), depending upon the local initial data. As has been shown to be the case with modulation equations for other integrable systems, hyperbolic data will remain hyperbolic under the evolution, at least until infinite derivatives develop.

The use of the sine-Gordon equation as a model of magnetic flux propagation in Josephson junctions motivates studying the initial-value problem for this equation in the semiclassical limit in which the dispersion parameter e tends to zero. Assuming natural initial data having the profile of a moving − 2 π kink at time zero, we analytically calculate the scattering data of this completely integrable Cauchy problem for all e > 0 sufficiently small, and further we invert the scattering transform to calculate the solution for a sequence of arbitrarily small e . This sequence of exact solutions is analogous to that of the well-known N -soliton (or higher-order soliton) solutions of the focusing nonlinear Schrodinger equation. We then use plots obtained from a careful numerical implementation of the inverse-scattering algorithm for reflectionless potentials to study the asymptotic behavior of solutions in the semiclassical limit. In the limit e ↓ 0 one observes the appearance of nonlinear caustics, i.e. curves in space–time that are independent of e but vary with the initial data and that separate regions in which the solution is expected to have different numbers of nonlinear phases. In the appendices, we give a self-contained account of the Cauchy problem from the perspectives of both inverse scattering and classical analysis (Picard iteration). Specifically, Appendix A contains a complete formulation of the inverse-scattering method for generic L 1 -Sobolev initial data, and Appendix B establishes the well-posedness for L p -Sobolev initial data (which in particular completely justifies the inverse-scattering analysis in Appendix A).

In this contribution, we summarize recent results [8, 9] on the stability analysis of periodicwavetrains for the sine-Gordon and general nonlinearKlein-Gordon equations. Stability is considered both from the point of view of spectral analysis of the linearized problem and from the point of view of the formal modulation theory of Whitham [12]. The connection between these two approaches is made through a modulational instability index [9], which arises from a detailed analysis of the Floquet spectrum of the linearized perturbation equation around the wave near the origin. We analyze waves of both subluminal and superluminal propagation velocities, as well as waves of both librational and rotational types. Our general results imply in particular that for the sine-Gordon case only subluminal rotationalwaves are spectrally stable. Our proof of this fact corrects a frequently cited one given by Scott [11].

We present a detailed analysis of the solution of the focusing nonlinear Schrodinger equation with initial condition (x,0) = N sech(x) in the limit N ! 1. We begin by presenting new and more accurate numerical reconstructions of the N-soliton by inverse scattering (numerical linear algebra) for N = 5, 10, 20, and 40. We then recast the inverse-scattering problem as a Riemann-Hilbert problem and provide a rigorous asymptotic analysis of this problem in the large-N limit. For those (x,t) where results have been obtained by other authors, we improve the error estimates from O(N 1/3 ) to O(N 1 ). We also analyze the Fourier power spectrum in this regime and relate the results to the optical phenomenon of supercontinuum generation. We then study the N-soliton for values of (x,t) where analysis has not been carried out before, and we uncover new phenomena. The main discovery of this paper is the mathematical mechanism for a secondary caustic (phase transition), which turns out to differ from the mechanism that generates the primary caustic. The mechanism for the generation of the secondary caustic depends essentially on the discrete nature of the spectrum for the N-soliton, and more significantly, cannot be recovered from an analysis of an ostensibly similar Riemann-Hilbert problem in the conditions of which a certain formal continuum limit is taken on an ad hoc basis.

Using exact formulae for the scattering data of the Benjamin-Ono equation valid for general rational potentials recently obtained by Miller and Wetzel (2015), we rigorously analyze the scattering data in the small-dispersion limit. In particular, we deduce precise asymptotic formulae for the reflection coefficient, the location of the eigenvalues and their density, and the asymptotic dependence of the phase constant (associated with each eigenvalue) on the eigenvalue itself. Our results give direct confirmation of conjectures in the literature that have been partly justified by means of inverse scattering, and they also provide new details not previously reported in the literature., 29 pages, 10 figures

Figure 1.6. The contour N of discontinuity for the sectionally analytic function N(w). With the exception of the two components of R+ \ I which are oriented leftto-right, the contour arcs are oriented with the regions labeled “+” on the left and with the regions labeled “−” on the right. The image of N \R+ under w → E(w) with E defined by (1.10) consists of the four straight line segments: (i) {E} = −δ , 0 0 is a sufficiently small number. The self-intersection point w = −1 is the critical point of E .

We present the results of a large number of careful numerical experiments carried out to investigate the way that the solution of the integrable focusing nonlinear Schroodinger equation with fixed initial data, when taken to be close to the semiclassical limit, depends on analyticity properties of the data. In particular, we study a family of initial data that have complex singularities at an adjustable distance from the real axis. We also make use of a simple relation that provides the exact solution, at the centre of the wave field, of the elliptic quasilinear system that appears formally as a leading–order model for the semi–classical dynamics. Among other things, we conclude that the semi–classical limit cannot be expected to be continuous with respect to the initial data, even for real analytic data, if there are certain complex singularities present. We argue that, in order to have well–posedness of the semiclassical limit, the correct setting is a physically relevant space of functions with compactly supported Fourier transforms.

A formal method for approximating eigenvalues of the nonselfadjoint Zakharov–Shabat eigenvalue problem in the semiclassical scaling is described. Analyticity of the potential is assumed and appears to be crucial. The method involves finding appropriate paths between pairs of complex turning points, and reproduces the Y-shaped spectra observed by Bronski [Physica D 97 (1996) 376]. An application to all-optical ultrashort pulse generation is briefly described, and the kind of tools that are required to make the results rigorous are indicated.

A linear Y-junction waveguide device is designed using a generalization of the theory of solitonic potentials for the linear Schrodinger equation. This Y-junction device, unlike other adiabatic Y-junctions, has the advantage that it may be directly written into a planar medium with homogeneous saturable nonlinearity by a strong light beam. The generalized theory provides the error terms that are introduced when the parameters of a solitonic potential are allowed to vary in the propagation direction, and shows that under certain adiabaticity conditions the error is small although the deformation of the potential is significant. At the operating wavelength for which the device is designed to function optimally, the Y-junction has two approximate bound modes that we find explicitly. Each mode has the property that when it is excited at the neck of the junction, it exits in only one of the two output ports. In this way, the device functions like a standard modal splitter in a multimode slab waveguide. When the wavelength is detuned, modal beating is introduced that degrades the optimal switching characteristics. We describe this effect in terms of four universal coupling functions using perturbation theory.

We report on a transmission electron microscopy (TEM) technique that can be used to measure strain due to individual nanometer-scale coherent heterostructures such as quantum dots or inclusions. The measurement relies on two-beam imaging and on an approximation that employs a universal model for lattice plane bending. We demonstrate that analysis is simple and accurate. Using this method, we measured the average strain in dome-shaped Ge islands grown on Si (0 0 1). We found that the method of specimen preparation can significantly affect the observed strain in these islands.

We use multiscale perturbation theory in conjunction with the inverse scattering transform to study the interaction of a number of solitons of the cubic nonlinear Schroedinger equation under the influence of a small correction to the nonlinear potential. We assume that the solitons are all moving with the same velocity at the initial instant; this maximizes the effect each soliton has on the others as a consequence of the perturbation. Over the long time scales that we consider, the amplitudes of the solitons remain fixed, while their center of mass coordinates obey Newton's equations with a force law for which we present an integral formula. For the interaction of two solitons with a quintic perturbation term we present more details since symmetries -- one related to the form of the perturbation and one related to the small number of particles involved -- allow the problem to be reduced to a one-dimensional one with a single parameter, an effective mass. The main results include calculations of the binding energy and oscillation frequency of nearby solitons in the stable case when the perturbation is an attractive correction to the potential and of the asymptotic "ejection" velocity in the unstable case. Numerical experiments illustrate the accuracy of the perturbative calculations and indicate their range of validity., 28 pages, 7 figures, Submitted to Phys Rev E Revised: 21 pages, 6 figures, To appear in Phys Rev E (many displayed equations moved inline to shorten manuscript)

We study the solutions of linear Schroedinger equations in which the potential energy is a periodic function of time and is sufficiently localized in space. We consider the potential to be close to one that is time periodic and yet explicitly solvable. A large family of such potentials has been constructed and the corresponding Schroedinger equation solved by Miller and Akhmediev. Exact bound states, or breather modes, exist in the unperturbed problem and are found to be generically metastable in the presence of small periodic perturbations. Thus, these states are long-lived but eventually decay. On a time scale of order $\epsilon^{-2}$, where $\epsilon$ is a measure of the perturbation size, the decay is exponential, with a rate of decay given by an analogue of Fermi's golden rule. For times of order $\epsilon^{-1}$ the breather modes are frequency shifted. This behavior is derived first by classical multiple-scale expansions, and then in certain circumstances we are able to apply the rigorous theory developed by Soffer and Weinstein and extended by Kirr and Weinstein to justify the expansions and also provide longer-time asymptotics that indicate eventual dispersive decay of the bound states with behavior that is algebraic in time. As an application, we use our techniques to study the frequency dependence of the guidance properties of certain optical waveguides. We supplement our results with numerical experiments., Comment: 69 pages, 13 figures, to appear in Nonlinearity

We consider the steady propagation of a light beam in a planar waveguide whose width and depth are periodically modulated in the direction of propagation. Using methods of soliton theory, a class of periodic potentials is presented for which the complete set of Floquet solutions of the linear Schrodinger equation can be found exactly at a particular optical frequency. For potentials in this class, there are exactly two bound Floquet solutions at this frequency, and they are degenerate, having the same Floquet multiplier. We study analytically the behavior of the waveguide under small changes in the frequency and observe a breaking of the degeneracy in the Floquet multiplier at first order. We predict and observe numerically the disappearance of both bound states at second order. These results suggest applications to spectral filtering.

With the use of a variant of the method of separation of variables, the initial value problem for the time-dependent linear Schrodinger equation is solved exactly for a large class of potential functions related to multisoliton interactions in the vector nonlinear Schrodinger equation. Completeness of states is proved for absolutely continuous initial data in L1.

We present numerical experiments that provide new strong evidence of the existence of the semiclassical limit for the focusing nonlinear Schrodinger equation in one space dimension. Our experiments also address the spatiotemporal structure of the limit. Like in the defocusing case, the semiclassical limit appears to be characterized by sharply delimited regions of space-time containing multiphase wave microstructure. Unlike in the defocusing case, the macroscopic dynamics seem to be governed by elliptic partial differential equations. These equations can be integrated for analytic initial data, and in this connection, we interpret the caustics separating the regions of smoothly modulated microstructure as the boundaries of domains of analyticity of the solutions of the macroscopic model. For more general initial data in common function spaces, the initial value problem is ill-posed. Thus the semiclassical limit of a sequence of well-posed initial value problems is an ill-posed initial value problem.

We have explored quantitative methods to study projected density fluctuations on the scale 0.5–2 nm in amorphous thin-films using weak phase object transmission electron microscopy. A useful method includes quantitative analysis of electron images and their Fourier amplitudes, essentially combining small-angle scattering analysis and high-resolution imaging from the same microscopic region. By comparing the Fourier spectra of differently prepared SiO 2 specimens, we show that clear differences between similar-looking samples can be detected quantitatively, and defect sizes and densities can be measured. Further development of this technique may hold significant value for researchers working to understand small-angle diffraction and for those investigating the performance of microelectronic thin oxides.

Fully nonlinear modulation equations are obtained for plane waves in a discrete system with quadratic nonlinearity, in the limit when the modulational scales are long compared to the wavelength and period of the modulated wave. The discrete system we study is a model for second-harmonic generation in nonlinear optical waveguide arrays and also for exciton waves at the interface between two crystals near Fermi resonance. The modulation equations predict their own breakdown by changing type from hyperbolic to elliptic. Modulational stability (hyperbolicity of the modulation equations) is explicitly shown to be implied by linear stability but not vice versa. When the plane-wave parameters vary slowly in regions of linear stability, the modulation equations are hyperbolic and accurately describe the macroscopic behavior of the system whose microscopic dynamics is locally given by plane waves. We show how the existence of Riemann invariants allows one to test modulated wave initial data to see whether the modulating wave will avoid all linear instabilities and ultimately resolve into simple disturbances that satisfy the Hopf or inviscid Burgers equation. We apply our general results to several important limiting cases of the microscopic model in question.

The phenomenon of spatio-temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem. A simple family of exact solutions to the Ablowitz-Ladik equations is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions, which determines two wavenumbers. Within the family of solutions, the frequencies do not depend on amplitude; this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being actions that foliate the phase space, and two being macroscopically observable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, which is consistent with the full set of four genus 1 modulation equations. The modulation of the phase information, easily observed due to the presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical experiments.

The linear Schr\"odinger equation is explicitly solved for the case of a refractive index profile created by the interaction of $N$ bright solitons in a Kerr medium. The bound states give exact solutions to the problem of determining how a beam of light is split as it passes through the collision of $N$ solitons; the main result of this work is an analytical formula for the power transmission matrix of this linear problem. As specific examples, the cases of the two-soliton and three-soliton linear couplers are considered, and the power transmission characteristics for bound states are explicitly calculated in both cases. It is further shown that the behavior of linear waves propagating through an arbitrary collision of $N$ solitons can be completely understood in terms of the scattering of linear waves from a soliton X junction. The unbound states of an $N$-soliton waveguide are also explicitly computable and describe the evolution of radiation modes.

The bound state solutions of the linear Schrodinger equation describing steady propagation of a signal in a planar waveguide are explicitly obtained for the case of a refractive index profile created by the interaction of N arbitrary dark solitons in a defocusing Kerr medium. Analysis of these solutions shows that linear signals confined to the arms of the waveguide prior to the collision do not interfere with each other at all as they pass through the interaction region of the dark solitons. Each signal emerges from the collision with exactly the same power and angle as it had at the input, thus behaving as though all solitons other than the one in which it was initially confined were altogether absent.

A new approach to multiport switching in arrays of nonlinear waveguides is proposed. While other schemes have relied on suppressing the inherent transverse discreteness, this approach takes advantage of this feature of the array. One of the effects of discreteness is to keep intense beams trapped in a single waveguide for the length of the array. Switching may be achieved by using a controlled perturbation to displace such a trapped beam in the transverse direction. This displacement is quantized to an integer number of waveguides, thus allowing unambiguous selection of the output channel.

Two concepts, very different in nature, have proved to be useful in analytical and numerical studies of spectral stability: (i) the Krein signature of an eigenvalue, a quantity usually defined in terms of the relative orientation of certain subspaces that is capable of detecting the structural instability of imaginary eigenvalues and hence their potential for moving into the right half-plane leading to dynamical instability under perturbation of the system, and (ii) the Evans function, an analytic function detecting the location of eigenvalues. One might expect these two concepts to be related, but unfortunately examples demonstrate that there is no way in general to deduce the Krein signature of an eigenvalue from the Evans function. The purpose of this paper is to recall and popularize a simple graphical interpretation of the Krein signature well-known in the spectral theory of polynomial operator pencils. This interpretation avoids altogether the need to view the Krein signature in terms of root subspaces and their relation to indefinite quadratic forms. To demonstrate the utility of this graphical interpretation of the Krein signature, we use it to define a simple generalization of the Evans function -- the Evans-Krein function -- that allows the calculation of Krein signatures in a way that is easy to incorporate into existing Evans function evaluation codes at virtually no additional computational cost. The graphical Krein signature also enables us to give elegant proofs of index theorems for linearized Hamiltonians in the finite dimensional setting: a general result implying as a corollary the generalized Vakhitov-Kolokolov criterion (or Grillakis-Shatah-Strauss criterion) and a count of real eigenvalues for linearized Hamiltonian systems in canonical form. These applications demonstrate how the simple graphical nature of the Krein signature may be easily exploited., 43 pages, 5 figures, submitted to SIAM Review

We study the spectral stability properties of periodic traveling waves in the sine-Gordon equation, including waves of both subluminal and superluminal propagation velocities as well as waves of both librational and rotational types. We prove that only subluminal rotational waves are spectrally stable and establish exponential instability in the other three cases. Our proof corrects a frequently cited one given by Scott., Comment: 22 pages, 6 figures