Peter D. Miller

Exploiting discreteness for switching in waveguide arrays

A new approach to multiport switching in arrays of nonlinear waveguides is proposed. Whereas other schemes have relied on suppressing the inherent transverse discreteness of these arrays, this approach takes advantage of that feature. 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 use of a controlled perturbation to displace such a trapped beam in the transverse direction. This displacement is quantized to an integer number of waveguides, thus permitting unambiguous selection of the output channel.

On the semiclassical limit of the focusing nonlinear Schrödinger equation

Abstract 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.

The quantum discrete self-trapping equation in the Hartree approximation

Abstract We show how the Hartree approximation (HA) can be used to study the quantum discrete self-trapping (QDST) equation, which - in turn - provides a model for the quantum description of several interesting nonlinear effects such as energy localization, soliton interactions, and chaos. The accuracy of the Hartree approximation is evaluated by comparing results with exact quantum mechanical calculations using the number state method. Since the Hartree method involves solving a classical DST equation, two classes of solutions are of particular interest: (i) Stationary solutions, which approximate certain energy eigenstates, and (ii) Time dependent solutions, which approximate the dynamics of wave packets of energy eigenstates. Both classes of solution are considered for systems with two and three degrees of freedom (the dimer and the trimer), and some comments are made on systems with an arbitrary number of freedoms.

On the generation of solitons and breathers in the modified Korteweg–de Vries equation

We consider the evolution of an initial disturbance described by the modified Korteweg-de Vries equation with a positive coefficient of the cubic nonlinear term, so that it can support solitons. Our primary aim is to determine the circumstances which can lead to the formation of solitons and/or breathers. We use the associated scattering problem and determine the discrete spectrum, where real eigenvalues describe solitons and complex eigenvalues describe breathers. For analytical convenience we consider various piecewise-constant initial conditions. We show how complex eigenvalues may be generated by bifurcation from either the real axis, or the imaginary axis; in the former case the bifurcation occurs as the unfolding of a double real eigenvalue. A bifurcation from the real axis describes the transition of a soliton pair with opposite polarities into a breather, while the bifurcation from the imaginary axis describes the generation of a breather from the continuous spectrum. Within the class of initial conditions we consider, a disturbance of one polarity, either positive or negative, will only generate solitons, and the number of solitons depends on the total mass. On the other hand, an initial disturbance with both polarities and very small mass will favor the generation of breathers, and the number of breathers then depends on the total energy. Direct numerical simulations of the modified Korteweg-de Vries equation confirms the analytical results, and show in detail the formation of solitons, breathers, and quasistationary coupled soliton pairs. Being based on spectral theory, our analytical results apply to the entire hierarchy of evolution equations connected with the same eigenvalue problem. (c) 2000 American Institute of Physics.

Self-similar evolution of self-written waveguides.

Numerical simulations show that channel waveguides can be self-written in photosensitive materials. As the waveguide evolves, its shape remains approximately constant, even though its depth and width change. We find an exact solution that describes this evolution, which we show to be self-similar. A wide variety of single-peaked beams form waveguides that converge to this solution.

Graphical Krein Signature Theory and Evans--Krein Functions

Two concepts, evidently very different in nature, have proved to be useful in analytical and numerical studies of spectral stability in nonlinear wave theory: (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, for example, by studying derivatives of the latter. 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. Once esta...

Macroscopic lattice dynamics

Abstract Fully nonlinear modulation equations are presented that govern the slow evolution of single-phase harmonic wavetrains under a large family of spatially discrete nonintegrable flows, among which is the discrete nonlinear Schrodinger equation (DNLS). These modulation equations are a pair of partial differential equations in conservation form that are hyperbolic for some data and elliptic for other data. In some cases these equations are capable of dynamically changing type from hyperbolic to elliptic, a phenomenon that has been associated with the modulational instability of the underlying wavetrain. By putting the modulation equations in Riemann invariant form, one can select initial data that avoid this dynamic change of type. Numerical experiments demonstrating the theoretical results are presented.

Large-degree asymptotics of rational Painlevé-II functions: critical behaviour

This paper is a continuation of our analysis, begun in Buckingham and Miller (2014 Nonlinearity 27 2489–577), of the rational solutions of the inhomogeneous Painleve-II equation and associated rational solutions of the homogeneous coupled Painleve-II system in the limit of large degree. In this paper we establish asymptotic formulae valid near a certain curvilinear triangle in the complex plane that was previously shown to separate two distinct types of asymptotic behaviour. Our results display both a trigonometric degeneration of the rational Painleve-II functions and also a degeneration to the tritronquee solution of the Painleve-I equation. Our rigorous analysis is based on the steepest descent method applied to a Riemann–Hilbert representation of the rational Painleve-II functions, and supplies leading-order formulae as well as error estimates.

BINDING-ENERGIES FOR DISCRETE NONLINEAR SCHRODINGER-EQUATIONS

The standard quantum discrete nonlinear Schrodinger equation with periodic boundary conditions and an arbitrary number of freedoms (f) is solved exactly at the second and third quantum levels. If f → ∞ at a sufficiently small level of anharmonicity (γ), the value for soliton binding energy from quantum field theory (QFT) in the continuum limit is recovered. For fixed f, however, the QFT result always fails for γ sufficiently large and also for γ sufficiently small. Corresponding calculations are discussed for the quantized Ablowitz-Ladik equation at the second quantum level with periodic boundary conditions.

Metastability of breather modes of time-dependent potentials

We study the solutions of linear Schr ¨ odinger 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 Schr ¨ odinger 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 2 , where is a measure of the perturbation size, the decay is exponential, with a rate of decay given by an analogue of Fermis golden rule. For times of order 1 the breather modes are frequency shifted. This behaviour 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 behaviour 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.