David W. McLaughlin

The soliton: A new concept in applied science

The term soliton has recently been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed. To date at least seven distinct wave systems, representing a wide range of applications in applied science, have been found to exhibit such solutions. This review paper covers the current status of soliton research, paying particular attention to the very important inverse method whereby the initial value problem for a nonlinear wave system can be solved exactly through a succession of linear calculations.

Canonically Conjugate Variables for the Korteweg-de Vries Equation and the Toda Lattice with Periodic Boundary Conditions*)

A new set of canonically conjugate variables 1s introduced for the penoclic Korteweg-de Vries equation and the periodic Toda lattice. These vanables are used for reducing both equations to a nonlinear system which can be integrated in terms of theta functions. It becomes clear that the discrete and the continuous problems are, in a sense, Jsomorph1c. Action variables are defined by loop integrals, and the basic oscillation frequencies are computed. In the infinite-period limit, these action variables tend to the ones used in the canonical description of the inverse-scattering solution method.

Geometry of the modulational instability III. Homoclinic orbits for the periodic sine-Gordon equation

In this paper the homoclinic geometric structure of the integrable sine-Gordon equation under periodic boundary conditions is developed. Specifically, focus is given to orbits homoclinic to N-tori. Simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given. A labeling is provided which identifies and catalogues all such orbits. These orbits are related in a one-to-one manner to linearized instabilities. Explicit formulas for all homoclinic orbits are given in terms of Backlund transformations.

Correlations between chaos in a perturbed sine-Gordon equation and a truncated model system

The purpose of this paper is to present a first step toward providing coordinates and associated dynamics for low-dimensional attractors in nearly integrable partial differential equations (pdes), in particular, where the truncated system reflects salient geometric properties of the pde. This is achieved by correlating: (i) Numerical results on the bifurcations to temporal chaos with spatial coherence of the damped, periodically forced sine-Gordon equation with periodic boundary conditions; (ii) An interpretation of the spatial and temporal bifurcation structures of this perturbed integrable system with regard to the exact structure of the sine-Gordon phase space; (iii) A model dynamical systems problem, which is itself a perturbed integrable Hamiltonian system, derived from the perturbed sine-Gordon equation by a finite mode Fourier truncation in the nonlinear Schrodinger limit; and (iv) The bifurcations to chaos in the truncated phase space.In particular, a potential source of chaos in both the pde and ...

A modal representation of chaotic attractors for the driven, damped pendulum chain

Abstract In this Letter we introduce a two-mode Fourier truncation of the damped driven nonlinear Schrodinger equation which captures the nature of one type of chaotic attractor for this pde. More specifically, this truncation correctly describes homoclinic crossings which are sources of temporal sensitivity along this chaotic attractor. The truncation provides a simple and explicit model dynamical system which is of interest in its own right and which can be used as a guide for analytical studies of chaotic attractors for the full pde. The most important feature of the model is that it faithfully represents the homoclinic structures of the full pde.

Attractors and transients for a perturbed periodic KdV equation: A nonlinear spectral analysis

SummaryIn this paper we rigorously show the existence and smoothness inε of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parameterε. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.

Coherence and chaos in the driven damped sine-Gordon equation: measurement of the soliton spectrum

Abstract A numerical procedure is developed which measures the sine-Gordon soliton and radiation content of any field (φ, φt) which is periodic in space. The procedure is applied to the field generated by a damped, driven sine-Gordon equation. This field can be either temporally periodic (locked to the driver) or chaotic. In either case the numerical measurement shows that the spatial structure can be described by only a few spatially localized (soliton wave-train) modes. The numerical procedure quantitatively identifies the presence, number and properties of these soliton wave-trains. For example, an increase of spatial symmetry is accompanied by the injection of additional solitons into the field.

Four examples of the inverse method as a canonical transformation

The Toda lattice, the nonlinear Schrodinger equation, the sine−Gordon equation, and the Korteweg−de Vries equation are four nonlinear equations of physical importance which have recently been solved by the inverse method. For these examples, this method of solution is interpreted as a canonical transformation from the initial Hamiltonian dynamics to an ’’action−angle’’ form. This canonical structure clarifies the independence of an infinite number of constants of the motion and indicates the special nature of the solution by the inverse method.

Quasi-periodic route to chaos in a near-integrable PDE: Homoclinic crossings

Abstract A new numerical experiment is discussed which shows a quasi-periodic route to intermittent chaos - typical for near-conservative, dispersive waves of small amplitude (nonlinear Schrodinger regime) in one spatial dimension. This route has: temporally one frequency, then two, then chaos; associated spatial symmetry changes; a low dimensional intermittent strange attractor. A nonlinear spectral transform has been used to show: a small number of nonlinear modes in the chaotic state; interaction of coherent modes with radiation modes: and, most importantly, that (unperturbed) homoclinic states are crossed repeatedly in these regimes. These homoclinic states: (a) separate spatially localized modes from radiation modes, and (b) act as sources of extreme sensitivity which can produce temporal chaos: for the first time homoclinic states have been simultaneously associated with both spatial patterns and temporal chaos.

Solitary waves as fixed points of infinite-dimensional maps for an optical bistable ring cavity: Analysis

The transverse behavior of a laser beam propagating through a bistable optical cavity is investigated analytically and numerically. Numerical experiments that study the (one‐dimensional) transverse structure of the steady state profile are described. Mathematical descriptions of (i) an infinite‐dimensional map that models the situation, (ii) the solitary waves that represent the transverse steady state structures, (iii) a projection formalism that reduces the infinite‐dimensional map to a finite‐dimensional one, and (iv) the theoretical analysis of this reduced map are presented in detail. The accuracy of this theoretical analysis is established by comparing its predictions to numerical observations.