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In this work, we study the dynamics of an infinite array of nonlinear dimer oscillators which are linearly coupled as in the classical model of Su, Schrieffer and Heeger (SSH). The ratio of in-cell and out-of-cell couplings of the SSH model defines distinct $\textit{phases}$: topologically trivial and topologically non-trivial. We first consider the case of weak out-of-cell coupling, corresponding to the topologically trivial regime for linear SSH; for any prescribed isolated dimer frequency, $\omega_b$, which satisfies non-resonance and non-degeneracy assumptions, we prove that there are discrete breather solutions for sufficiently small values of the out-of-cell coupling parameter. These states are $2\pi/\omega_b$- periodic in time and exponentially localized in space. We then study the global continuation with respect to this coupling parameter. We first consider the case where $\omega_b$, the seeding discrete breather frequency, is in the (coupling dependent) phonon gap of the underlying linear infinite array. As the coupling is increased, the phonon gap decreases in width and tends to a point (at which the topological transition for linear SSH occurs). In this limit, the spatial scale of the discrete breather grows and its amplitude decreases, indicating the weakly nonlinear long wave regime. Asymptotic analysis shows that in this regime the discrete breather envelope is determined by a vector gap soliton of the limiting envelope equations. We use the envelope theory to describe discrete breathers for SSH- coupling parameters corresponding to topologically trivial and, by exploiting an emergent symmetry, topologically nontrivial regimes, when the spectral gap is small. Our asymptotic theory shows excellent agreement with extensive numerical simulations over a wide range of parameters.

Consider the tight binding model of graphene, sharply terminated along an edge ${\bf l}$ parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges ${\bf l}$ into those of "zigzag type" and those of "armchair type", generalizing the classical zigzag and armchair edges. We prove that zero energy/flat band edge states arise for edges of zigzag type, but never for those of armchair type. We exhibit explicit formulas for flat band edge states when they exist. We produce strong evidence for the existence of dispersive (non flat) edge state curves of nonzero energy for most ${\bf l}$., Comment: 64 pages, to appear in Communications on Pure and Applied Mathematics

The principal use of photonic crystals is to engineer the photonic density of states, which controls light-matter coupling. We theoretically show that strained 2D photonic crystals can generate artificial electromagnetic fields and highly degenerate Landau levels. Since photonic crystals are not described by tight-binding, we employ a multiscale expansion of the full wave equation. Using numerical simulations, we observe dispersive Landau levels which we show can be flattened by engineering a pseudoelectric field. Artificial fields yield a design principle for aperiodic nanophotonic systems., Comment: See ancillary files for supplemental material providing detailed derivations

We study energy propagation along line-defects (edges) in 2D continuous, energy preserving periodic media. The unperturbed medium (bulk) is modeled by a honeycomb Schroedinger operator, which is periodic with respect to the triangular lattice, invariant under parity, P, and complex-conjugation, C. A honeycomb operator has Dirac points: two dispersion surfaces touch conically at an energy level, $E_D$ [25,27]. Periodic perturbations which break P or C open a gap in the essential spectrum about energy $E_D$. Such operators model an insulator near energy $E_D$. Our edge operator is a small perturbation of the bulk and models a transition (via a domain wall) between distinct periodic, P or C breaking perturbations. The edge operator permits energy transport along the line-defect. The associated energy channels are called edge states. They are time-harmonic solutions which are localized near and propagating along the line-defect. We completely characterize the edge state spectrum within the bulk spectral gap about $E_D$. At the center of our analysis is an expansion of the edge operator resolvent for energies near $E_D$. The leading term features the resolvent of an effective Dirac operator. Edge state eigenvalues are poles of the resolvent, which bifurcate from the Dirac point. The corresponding eigenstates have the multiscale structure identified in [23]. We extend earlier work on zigzag-type edges [14] to all rational edges. We elucidate the role in edge state formation played by the type of symmetry-breaking and the orientation of the edge. We prove the resolvent expansion by a new direct and transparent strategy. Our results provide a rigorous explanation of the numerical observations [22,38}; see also the photonic experimental study in [42]. Finally, we discuss implications for the Valley Hall Effect, which concerns quantum Hall-like energy transport in honeycomb structures., Comment: 43 pages, 6 figures

We study the single electron model of a semi-infinite graphene sheet interfaced with the vacuum and terminated along a zigzag edge. The model is a Schroedinger operator acting on $L^2(\mathbb{R}^2)$: $H^\lambda_{\rm edge}=-\Delta+\lambda^2 V_\sharp$, with a potential $V_\sharp$ given by a sum of translates an atomic potential well, $V_0$, of depth $\lambda^2$, centered on a subset of the vertices of a discrete honeycomb structure with a zigzag edge. We give a complete analysis of the low-lying energy spectrum of $H^\lambda_{\rm edge}$ in the strong binding regime ($\lambda$ large). In particular, we prove scaled resolvent convergence of $H^\lambda_{\rm edge}$ acting on $L^2(\mathbb{R}^2)$, to the (appropriately conjugated) resolvent of a limiting discrete tight-binding Hamiltonian acting in $l^2(\mathbb{N}_0;\mathbb{C}^2)$. We also prove the existence of {\it edge states}: solutions of the eigenvalue problem for $H^\lambda_{\rm edge}$ which are localized transverse to the edge and pseudo-periodic (propagating or plane-wave like) parallel to the edge. These edge states arise from a "flat-band" of eigenstates the tight-binding Hamiltonian., Comment: Revised version -- 89 pages, 2 figures; new title and abstract, revised introduction. In addition to a construction of the nearly flat band of edge states, the article now includes a proof of scaled resolvent convergence in a neighborhood of the low-lying spectrum

Consider electromagnetic waves in two-dimensional {\it honeycomb structured media}. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator $\LA=-\nabla_\bx\cdot A(\bx) \nabla_\bx$, where $A(\bx)$ is $\Lambda_h-$ periodic ($\Lambda_h$ denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, $A(\bx)$ is $\mathcal{P}\mathcal{C}-$ invariant ($A(\bx)=\overline{A(-\bx)}$) and $120^\circ$ rotationally invariant ($A(R^*\bx)=R^*A(\bx)R$, where $R$ is a $120^\circ$ rotation in the plane). We first obtain results on the existence, stability and instability of Dirac points, conical intersections between two adjacent Floquet-Bloch dispersion surfaces. We then show that the introduction through small and slow variations of a {\it domain wall} across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) {\it edge states}. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. These results imply the existence of {\it uni-directional} propagating edge states for two classes of time-reversal invariant media in which $\mathcal{C}$ symmetry is broken: magneto-optic media and bi-anisotropic media. Our analysis applies and extends the tools previously developed in the context of honeycomb Schr\"odinger operators., Comment: 65 pages, 8 figures, To appear in Archive for Rational Mechanics and Analysis

In this article, we study the Schr\"odinger operator for a large class of periodic potentials with the symmetry of a hexagonal tiling of the plane. The potentials we consider are superpositions of localized potential wells, centered on the vertices of a regular honeycomb structure corresponding to the single electron model of graphene and its artificial analogues. We consider this Schr\"odinger operator in the regime of strong binding, where the depth of the potential wells is large. Our main result is that for sufficiently deep potentials, the lowest two Floquet-Bloch dispersion surfaces, when appropriately rescaled, converge uniformly to those of the two-band tight-binding model (Wallace, 1947). Furthermore, we establish as corollaries, in the regime of strong binding, results on (a) the existence of spectral gaps for honeycomb potentials that break $\mathcal{P}\mathcal{T}$ symmetry and (b) the existence of topologically protected edge states -- states which propagate parallel to and are localized transverse to a line-defect or "edge" - for a large class of rational edges, and which are robust to large localized perturbations of the edge. We believe that the ideas of this article may be applicable in other settings for which a tight-binding model emerges in an extreme parameter limit., Comment: To appear in Communications on Pure and Applied Mathematics; Theorem 6.2 of the revised version establishes scaled norm convergence of the resolvent of -\Delta + \lambda^2 V(x) to the resolvent of the tight-binding Hamiltonian

This paper summarizes and extends the authors' work on the bifurcation of topologically protected edge states in continuous two-dimensional honeycomb structures. We consider a family of Schr\"odinger Hamiltonians consisting of a bulk honeycomb potential and a perturbing edge potential. The edge potential interpolates between two different periodic structures via a domain wall. We begin by reviewing our recent bifurcation theory of edge states for continuous two-dimensional honeycomb structures. The topologically protected bifurcation of edge states is seeded by the zero-energy eigenstate of a one-dimensional Dirac operator. We contrast these protected bifurcations with (more common) non-protected bifurcations from spectral band edges, which are induced by bound states of an effective Schr\"odinger operator. Numerical simulations for honeycomb structures of varying contrasts and "rational edges" (zigzag, armchair and others), support the following scenario: (a) For low contrast, under a sign condition on a distinguished Fourier coefficient of the bulk honeycomb potential, there exist topologically protected edge states localized transverse to zigzag edges. Otherwise, and for general edges, we expect long lived {\it edge quasi-modes} which slowly leak energy into the bulk. (b) For an arbitrary rational edge, there is a threshold in the medium-contrast (depending on the choice of edge) above which there exist topologically protected edge states. In the special case of the armchair edge, there are two families of protected edge states; for each parallel quasimomentum (the quantum number associated with translation invariance) there are edge states which propagate in opposite directions along the armchair edge., Comment: 12 pages, 9 figures

How does one search for a needle in a multi-dimensional haystack without knowing what a needle is and without knowing if there is one in the haystack? This kind of problem requires a paradigm shift - away from hypothesis driven searches of the data - towards a methodology that lets the data speak for itself. Dynamic Quantum Clustering (DQC) is such a methodology. DQC is a powerful visual method that works with big, high-dimensional data. It exploits variations of the density of the data (in feature space) and unearths subsets of the data that exhibit correlations among all the measured variables. The outcome of a DQC analysis is a movie that shows how and why sets of data-points are eventually classified as members of simple clusters or as members of - what we call - extended structures. This allows DQC to be successfully used in a non-conventional exploratory mode where one searches data for unexpected information without the need to model the data. We show how this works for big, complex, real-world datasets that come from five distinct fields: i.e., x-ray nano-chemistry, condensed matter, biology, seismology and finance. These studies show how DQC excels at uncovering unexpected, small - but meaningful - subsets of the data that contain important information. We also establish an important new result: namely, that big, complex datasets often contain interesting structures that will be missed by many conventional clustering techniques. Experience shows that these structures appear frequently enough that it is crucial to know they can exist, and that when they do, they encode important hidden information. In short, we not only demonstrate that DQC can be flexibly applied to datasets that present significantly different challenges, we also show how a simple analysis can be used to look for the needle in the haystack, determine what it is, and find what this means., Comment: 37 pages, 22 figures, 1 Table

We consider the discrete eigenvalues of the operator $H_\eps=-\Delta+V(\x)+\eps^2Q(\eps\x)$, where $V(\x)$ is periodic and $Q(\y)$ is localized on $\R^d,\ \ d\ge1$. For $\eps>0$ and sufficiently small, discrete eigenvalues may bifurcate (emerge) from spectral band edges of the periodic Schr\"odinger operator, $H_0 = -\Delta_\x+V(\x)$, into spectral gaps. The nature of the bifurcation depends on the homogenized Schr\"odinger operator $L_{A,Q}=-\nabla_\y\cdot A \nabla_\y +\ Q(\y)$. Here, $A$ denotes the inverse effective mass matrix, associated with the spectral band edge, which is the site of the bifurcation., Comment: 26 pages, 3 figures, to appear SIAM J. Math. Anal

Consider the dynamics of a gas bubble in an inviscid, compressible liquid with surface tension. Kinematic and dynamic boundary conditions couple the bubble surface deformation dynamics with the dynamics of waves in the fluid. This system has a spherical equilibrium state, resulting from the balance of the pressure at infinity and the gas pressure within the bubble. We study the linearized dynamics about this equilibrium state in a center of mass frame: 1) We prove that the velocity potential and bubble surface perturbation satisfy point-wise in space exponential time-decay estimates. 2) The time-decay rate is governed by scattering resonances, eigenvalues of a non-selfadjoint spectral problem. These are pole singularities in the lower half plane of the analytic continuation of a resolvent operator from the upper half plane, across the real axis into the lower half plane. 3) The time-decay estimates are a consequence of resonance mode expansions for the velocity potential and bubble surface perturbations. 4) For small compressibility (Mach number, a ratio of bubble wall velocity to sound speed, \epsilon), this is a singular perturbation of the incompressible limit. The scattering resonances which govern the anomalously slow time-decay, are {\it Rayleigh resonances}. Asymptotics, supported by high-precision numerical studies, indicate that the Rayleigh resonances which are closest to the real axis satisfy | \frac{\Im \lambda_\star(\epsilon)}{\Re \lambda_\star(\epsilon)} | = {\cal O} (\exp(-\kappa\ \We\ \epsilon^{-2})), \kappa>0. Here, \We denotes the Weber number, a dimensionless ratio comparing inertia and surface tension. 5) To obtain the above results we prove a general result, of independent interest, estimating the Neumann to Dirichlet map for the wave equation, exterior to a sphere., Comment: to appear in SIAM Journal on Mathematical Analysis

The large class of moving boundary processes in the plane modeled by the so-called Laplacian growth, which describes, e.g., solidification, electrodeposition, viscous fingering, bacterial growth, etc., is known to be integrable and to exhibit a large number of exact solutions. In this work, the boundaries are assumed to be in the class of lemniscates with all zeros inside the bounded component of the complex plane. We prove that for any initial boundary taken from this class, the evolving boundary instantly stops being in the class, or else Laplacian growth destroys lemniscates instantly.

A new class of solutions to Laplacian growth with zero surface tension is presented and shown to contain all other known solutions as special or limiting cases. These solutions, which are time-dependent conformal maps with branch cuts inside the unit circle, are governed by a nonlinear integral equation and describe oil fjords with non-parallel walls in viscous fingering experiments in Hele-Shaw cells. Integrals of motion for the multi-cut Laplacian growth solutions in terms of singularities of the Schwarz function are found, and the dynamics of densities (jumps) on the cuts are derived. The subclass of these solutions with linear Cauchy densities on the cuts of the Schwarz function is of particular interest, because in this case the integral equation for the conformal map becomes linear. These solutions can also be of physical importance by representing oil/air interfaces, which form oil fjords with a constant opening angle, in accordance with recent experiments in a Hele-shaw cell., Comment: 19 pages, 3 figures, improved order of exposition

We present a unified approach for qualitative and quantitative analysis of stability and instability dynamics of positive bright solitons in multi-dimensional focusing nonlinear media with a potential (lattice), which can be periodic, periodic with defects, quasiperiodic, single waveguide, etc. We show that when the soliton is unstable, the type of instability dynamic that develops depends on which of two stability conditions is violated. Specifically, violation of the slope condition leads to an amplitude instability, whereas violation of the spectral condition leads to a drift instability. We also present a quantitative approach that allows to predict the stability and instability strength.

We consider a class of nonlinear Schrodinger / Gross-Pitaveskii (NLS-GP) equations, i.e. NLS with a linear potential. We obtain conditions for a symmetry breaking bifurcation in a symmetric family of states as N, the squared L^2 norm (particle number, optical power), is increased. In the special case where the linear potential is a double-well with well separation L, we estimate N_{cr}, the symmetry breaking threshold. Along the ``lowest energy'' symmetric branch, there is an exchange of stability from the symmetric to asymmetric branch as N is increased beyond N_{cr}., Comment: 38 pages, 6 figures

We consider Laplacian Growth of self-similar domains in different geometries. Self-similarity determines the analytic structure of the Schwarz function of the moving boundary. The knowledge of this analytic structure allows us to derive the integral equation for the conformal map. It is shown that solutions to the integral equation obey also a second order differential equation which is the one dimensional Schroedinger equation with the sinh inverse square potential. The solutions, which are expressed through the Gauss hypergeometric function, characterize the geometry of self-similar patterns in a wedge. We also find the potential for the Coulomb gas representation of the self-similar Laplacian growth in a wedge and calculate the corresponding free energy., Comment: 16 pages, 9 figures

A theory of time dependent nonlinear dispersive equations of the Schroedinger / Gross-Pitaevskii and Hartree type is developed. The short, intermediate and large time behavior is found, by deriving nonlinear Master equations (NLME), governing the evolution of the mode powers, and by a novel multi-time scale analysis of these equations. The scattering theory is developed and coherent resonance phenomena and associated lifetimes are derived. Applications include BEC large time dynamics and nonlinear optical systems. The theory reveals a nonlinear transition phenomenon, ``selection of the ground state'', and NLME predicts the decay of excited state, with half its energy transferred to the ground state and half to radiation modes. Our results predict the recent experimental observations of Mandelik et. al. in nonlinear optical waveguides.

We examine an example of spontaneous symmetry breaking in a double-well waveguide with a symmetric potential. The ground state of the system beyond a critical power becomes asymmetric. The effect is illustrated numerically, and quantitatively analyzed via a Galerkin truncation that clearly shows the bifurcation from a symmetric to an asymmetric steady state. This phenomenon is also demonstrated experimentally when a probe beam is launched appropriately into an optically induced photonic lattice in a photorefractive material., Comment: 4 pages, 3 figures

We study the evolution of the energy (mode-power) distribution for a class of randomly perturbed Hamiltonian partial differential equations and derive {\it master equations} for the dynamics of the expected power in the discrete modes. In the case where the unperturbed dynamics has only discrete frequencies (finitely or infinitely many) the mode-power distribution is governed by an equation of discrete diffusion type for times of order $\cO(\ve^{-2})$. Here $\ve$ denotes the size of the random perturbation. If the unperturbed system has discrete and continuous spectrum the mode-power distribution is governed by an equation of discrete diffusion-damping type for times of order $\cO(\ve^{-2})$. The methods involve an extension of the authors' work on deterministic periodic and almost periodic perturbations, and yield new results which complement results of others, derived by probabilistic methods., Comment: 51 pages LaTex

The Laplacian growth (the Hele-Shaw problem) of multi-connected domains in the case of zero surface tension is proven to be equivalent to an integrable systems of Whitham equations known in soliton theory. The Whitham equations describe slowly modulated periodic solutions of integrable hierarchies of nonlinear differential equations. Through this connection the Laplacian growth is understood as a flow in the moduli space of Riemann surfaces., Comment: 33 pages, 7 figures, typos corrected, new references added

We prove for a class of nonlinear Schr\"odinger systems (NLS) having two nonlinear bound states that the (generic) large time behavior is characterized by decay of the excited state, asymptotic approach to the nonlinear ground state and dispersive radiation. Our analysis elucidates the mechanism through which initial conditions which are very near the excited state branch evolve into a (nonlinear) ground state, a phenomenon known as {\it ground state selection}. Key steps in the analysis are the introduction of a particular linearization and the derivation of a normal form which reflects the dynamics on all time scales and yields, in particular, nonlinear Master equations. Then, a novel multiple time scale dynamic stability theory is developed. Consequently, we give a detailed description of the asymptotic behavior of the two bound state NLS for all small initial data. The methods are general and can be extended to treat NLS with more than two bound states and more general nonlinearities including those of Hartree-Fock type., Comment: Revision of 2001 preprint; 108 pages TeX

In this paper we investigate the emergence of time-periodic and and time-quasiperiodic (sometimes infinitely long lived and sometimes very long lived or metastable) solutions of discrete nonlinear wave equations: discrete sine Gordon, discrete $\phi^4$ and discrete nonlinear Schr\"odinger. The solutions we consider are periodic oscillations on a kink or standing wave breather background. The origin of these oscillations is the presence of internal modes, associated with the static ground state. Some of these modes are associated with the breaking of translational invariance, in going from a spatially continuous to a spatially discrete system. Others are associated with discrete modes which bifurcate from the continuous spectrum. It is also possible that such modes exist in the continuum limit and persist in the discrete case. The regimes of existence, stability and metastability of states as the lattice spacing is varied are investigated analytically and numerically. A consequence of our analysis is a class of spatially localized, time quasiperiodic solutions of the discrete nonlinear Schr\"odinger equation. We demonstrate, however, that this class of quasiperiodic solution is rather special and that its natural generalizations yield only metastable quasiperiodic solutions., Comment: 10 pages, 7 figures, to appear in the Special Issue of Mathematics and Computers in Simulation dedicated to the 2nd IMACS Nonlinear Waves Conference

We consider the interaction of a nonlinear Schrodinger soliton with a localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Following Forinash et al, we derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The system thus derived is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations., Comment: 21 pages, 14 figures, Invited paper for Mathematics as a Guide to the Understanding of Applied Nonlinear Problems, a conference in honor of Klaus Kirchgassner's 70th birthday, Kloster Irsee, Germany, Jan 6-10, 2002

We discuss the optical and radio properties of 30,000 FIRST sources positionally associated with an SDSS source in 1230 deg$^2$ of sky. The majority (83%) of the FIRST sources identified with an SDSS source brighter than r=21 are optically resolved. We estimate an upper limit of 5% for the fraction of quasars with broad-band optical colors indistinguishable from those of stars. The distribution of quasars in the radio flux -- optical flux plane supports the existence of the "quasar radio-dichotomy"; 8% of all quasars with i<18.5 are radio-loud and this fraction seems independent of redshift and optical luminosity. The radio-loud quasars have a redder median color by 0.08 mag, and a 3 times larger fraction of objects with red colors. FIRST galaxies represent 5% of all SDSS galaxies with r<17.5, and 1% for r<20, and are dominated by red galaxies. Magnitude and redshift limited samples show that radio galaxies have a different optical luminosity distribution than non-radio galaxies selected by the same criteria; when galaxies are further separated by their colors, this result remains valid for both blue and red galaxies. The distributions of radio-to-optical flux ratio are similar for blue and red galaxies in redshift-limited samples; this similarity implies that the difference in their luminosity functions, and resulting selection effects, are the dominant cause for the preponderance of red radio galaxies in flux-limited samples. We confirm that the AGN-to-starburst galaxy number ratio increases with radio flux, and find that radio emission from AGNs is more concentrated than radio emission from starburst galaxies (abridged)., Comment: submitted to AJ, color gif figures, PS figures available from ivezic@astro.princeton.edu

Gap solitons are localized nonlinear coherent states which have been shown both theoretically and experimentally to propagate in periodic structures. Although theory allows for their propagation at any speed $v$, $0\le v\le c$, they have been observed in experiments at speeds of approximately 50% of $c$. It is of scientific and technological interest to trap gap solitons. We first introduce an explicit multiparameter family of periodic structures with localized defects, which support linear defect modes. These linear defect modes are shown to persist into the nonlinear regime, as {\it nonlinear defect modes}. Using mathematical analysis and numerical simulations we then investigate the capture of an incident gap soliton by these defects. The mechanism of capture of a gap soliton is resonant transfer of its energy to nonlinear defect modes. We introduce a useful bifurcation diagram from which information on the parameter regimes of gap soliton capture, reflection and transmission can be obtained by simple conservation of energy and resonant energy transfer principles., Comment: 45 pages, Submitted to Journal of the Optical Society B

We review the concept of $\tau$-function for simple analytic curves. The $\tau$-function gives a formal solution to the 2D inverse potential problem and appears as the $\tau$-function of the integrable hierarchy which describes conformal maps of simply-connected domains bounded by analytic curves to the unit disk. The $\tau$-function also emerges in the context of topological gravity and enjoys an interpretation as a large $N$ limit of the normal matrix model., Comment: 13 pages, no figures, prepared for Proccedings of MSRI Workshop ``Matrix Models and Painlev\'e Equations'', Berkeley (USA) 1999

In this paper we consider two models of soliton dynamics (the sine Gordon and the \phi^4 equations) on a 1-dimensional lattice. We are interested in particular in the behavior of their kink-like solutions inside the Peierls- Nabarro barrier and its variation as a function of the discreteness parameter. We find explicitly the asymptotic states of the system for any value of the discreteness parameter and the rates of decay of the initial data to these asymptotic states. We show that genuinely periodic solutions are possible and we identify the regimes of the discreteness parameter for which they are expected to persist. We also prove that quasiperiodic solutions cannot exist. Our results are verified by numerical simulations., Comment: 50 pages, 10 figures, LaTeX document

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 present a theory of resonances for a class of non-autonomous Hamiltonians to treat the structural instability of spatially localized and time-periodic solutions associated with an unperturbed autonomous Hamiltonian. The mechanism of instability is radiative decay, due to resonant coupling of the discrete modes to the continuum modes by the time-dependent perturbation. This results in a slow transfer of energy from the discrete modes to the continuum. The rate of decay of solutions is slow and hence the decaying bound states can be viewed as metastable. The ideas are closely related to the authors' work on (i) a time dependent approach to the instability of eigenvalues embedded in the continuous spectra, and (ii) resonances, radiation damping and instability in Hamiltonian nonlinear wave equations. The theory is applied to a general class of Schr\"odinger equations. The phenomenon of ionization may be viewed as a resonance problem of the type we consider and we apply our theory to find the rate of ionization, spectral line shift and local decay estimates for such Hamiltonians., Comment: To appear in Journal of Statistical Physics

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field., Comment: To appear in Inventiones Mathematicae

An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photon-radiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (time-dependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (non-vanishing of the {\it Fermi golden rule}) and (ii) local decay estimates for the unperturbed dynamics with initial data consisting of continuum modes associated with an interval containing the embedded eigenvalue of the unperturbed Hamiltonian. No assumption of dilation analyticity of the potential is made. Our method explicitly demonstrates the flow of energy from the resonant discrete mode to continuum modes due to their coupling. The approach is also applicable to nonautonomous linear problems and to nonlinear problems. We derive the time behavior of the resonant states for intermediate and long times. Examples and applications are presented. Among them is a proof of the instability of an embedded eigenvalue at a threshold energy under suitable hypotheses., Comment: to appear in Geometrical and Functional Analysis