Adrianus Korpel

Exact solitary wave solutions of nonlinear evolution and wave equations using a direct algebraic method

The authors present a systematic and formal approach toward finding solitary wave solutions of nonlinear evolution and wave equations from the real exponential solutions of the underlying linear equations. The physical concept is one of the mixing of these elementary solutions through the nonlinearities in the system. The emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples it is shown how various cases of pulse-type and kink-type solutions are to be obtained by this method. An exhaustive list of equations so treated is presented in tabular form, together with the particular intermediate relations necessary for deriving their solutions. The extension of the technique to construct N-soliton solutions and indicate connections with other existing methods is outlined.

Acousto-optics—A review of fundamentals

The paper first reviews the historical development of acousto-optics and its applications. Following this, a heuristic explanation of acousto-optic effects is presented, with the emphasis on the plane wave model of interaction. Finally, there is a discussion of some basic configurations of relevance to signal processing.

Explicit formalism for acousto-optic multiple plane-wave scattering

In a previous article, a plane-wave model for strong acousto-optic interaction was developed. In this earlier model, any particular order is contributed to by a multiple-scattering process, involving all other orders. The model involves “direct” scattering paths and “feedback” paths and, in general, provides an implicit solution. The present article describes a variant of this multiple-scattering formalism, in which feedback paths and direct paths are combined so that all orders are explicitly derivable from the incident-light field. We show the usefulness and validity of this approach by rigorously deriving the well-known expressions for Raman-Nath and Bragg diffraction.

A general physical approach to solitary wave construction from linear solutions

We simplify the physical approach of constructing solitary wave solutions of nondissipative evolution and wave equations from the physical mixing of the real, rather than complex, exponential solutions of the linear equation, in two separate regions. In our new approach, we use mixing in one region only to construct a closed form for the solitary wave solution valid in both regions. Moreover, we extend the approach to deal with equations whose solutions (like tanh2-type) have a constant term in their expansion into real exponentials, and with equations whose linear part allows more than two exponential solutions. Finally, we also demonstrate the application of our technique to a typical dissipative equation, e.g., the Burgers

Two-dimensional plane wave theory of strong acousto-optic interaction in isotropic media

Strong interaction is the preferred (high-efficiency) mode of operation in acousto-optic devices. Yet most theories of strong interaction use simplifying assumptions, such as sharply bounded sound columns, which are often unrealistic. The theory presented in this paper makes no such assumptions; it describes strong acousto-optic interaction as a multiple scattering of plane waves.

A heuristic guide to nonlinear dispersive wave equations and soliton-type solutions

In this paper we present a heuristic way of constructing nonlinear dispersive equations that lead to soliton or soliton-type solutions. We assume only that a general knowledge of the dispersion relation of the system is known, together with some insight into the effect of nonlinearity on wave speed. We show that such knowledge is sufficient to derive most known soliton equations and thus to provide the engineer with a quick way to assess whether or not his particular system is likely to exhibit soliton behavior. Naturally, a more detailed description requires knowledge of the basic equations governing the system and special techniques to handle initial conditions. To that purpose we provide the reader with ample references which he may want to consult in order to augment the information gained by the method outlined in this paper.

Feynman diagram approach to acousto-optic scattering in the near-Bragg region

Plane-wave multiple-scattering theory is used to analyze scattering in the near-Bragg region with Feynman diagram techniques. Approximate amplitudes of scattered orders are obtained as power series in 1/Q, where Q is the Klein–Cook Bragg parameter. Results are compared with the Klein–Cook computer solution.

Subharmonic generation by resonant three‐wave interaction of deep‐water capillary waves

Subharmonic generation has been observed during the propagation of deep‐water capillary waves. The observations are shown to be in agreement with the theory of degenerate resonant noncollinear three‐wave interaction in a nonlinear, dispersive medium.

Nonlinear echoes, phase conjugation, time reversal, and electronic holography

In this paper we review the experiments on nonlinear echo phenomena during the last three decades, from spin echoes to echoes in piezoelectric powders. We show how the common principle is one of a physical Fourier transform space in which time reversal is brought about through phase conjugation. It will be seen how this leads to intriguing applications in signal processing, signal storage, and electronic holography.

Convenient operator formalism for Fourier optics and inhomogeneous and nonlinear wave propagation

A simple operator symbolism is proposed for describing paraxial propagation with a combination of plane-wave spectra and Fresnel diffraction concepts. When it is applied to typical optical configurations, the symbolism permits rapid calculation of the operator characterizing the overall system. Applied to inhomogeneous and nonlinear media, it describes the various split-step wave-propagation algorithms conveniently and plausibly.