Willy A Hereman
Colorado School of Mines, Math And Computer Science, Faculty Member
Research Interests:
Research Interests:
Research Interests:
Research Interests:
Algorithms for the symbolic computation of conserved densities, fluxes, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations are presented. In the algorithms we use discrete versions... more
Algorithms for the symbolic computation of conserved densities, fluxes, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations are presented. In the algorithms we use discrete versions of the Frechet and variational derivatives, as well as discrete Euler and homotopy operators. The algorithms are illustrated for prototypical nonlinear lattices, including the Kac-van Moerbeke (Volterra) and Toda lattices. Results are shown for the modified Volterra and Ablowitz-Ladik lattices.
Research Interests:
Research Interests:
The supercritical composition of a plasma model with cold positive ions in the presence of a two-temperature electron population is investigated, initially by a reductive perturbation approach, under the combined requirements that there... more
The supercritical composition of a plasma model with cold positive ions in the presence of a two-temperature electron population is investigated, initially by a reductive perturbation approach, under the combined requirements that there be neither quadratic nor cubic nonlinearities in the evolution equation. This leads to a unique choice for the set of compositional parameters and a modified Korteweg-de Vries equation (mKdV) with a quartic nonlinear term. The conclusions about its one-soliton solution and integrability will also be valid for more complicated plasma compositions. Only three polynomial conservation laws can be obtained. The mKdV equation with quartic nonlinearity is not completely integrable, thus precluding the existence of multi-soliton solutions. Next, the full Sagdeev pseudopotential method has been applied and this allows for a detailed comparison with the reductive perturbation results. This comparison shows that the mKdV solitons have slightly larger amplitudes and widths than those obtained from the more complete Sagdeev solution and that only slightly superacoustic mKdV solitons have acceptable amplitudes and widths, in the light of the full solutions.
Research Interests:
The partial differential equation for the wave function in the problem of the diffraction of light by an ultrasonic wave is derived as the result of transformations and related approximations of the electrical field equation itself. It is... more
The partial differential equation for the wave function in the problem of the diffraction of light by an ultrasonic wave is derived as the result of transformations and related approximations of the electrical field equation itself. It is shown that this way of treating the diffraction problem is equivalent to the generating function method applied to the Raman-Nath system of difference-differential equations for the amplitudes of the diffracted waves. This general method is worked out here for diffraction due to an amplitude-modulated ultrasonic wave, for which the symmetry properties of the diffraction pattern are also investigated with the help of the transformed wave equation. Afterwards the wave function is considered as a generating function for the amplitudes. The corresponding partial differential equation for this generator is solved exactly in the case of large ultrasonic wave lengths at oblique incidence of the light.
The intensities of the diffraction pattern of light in a liquid, disturbed by an amplitude-modulated ultrasonic wave, have been calculated on the hand of Raman-Nath's elementary theory. The results obtained by Mertens are extended to the... more
The intensities of the diffraction pattern of light in a liquid, disturbed by an amplitude-modulated ultrasonic wave, have been calculated on the hand of Raman-Nath's elementary theory. The results obtained by Mertens are extended to the case of a multiple frequency transducer-output. The intensity expressions obtained here for oblique incidence of the light beam, include interference cases neglected by Aggarwal et al. Finally the symmetry of the various types of diffraction lines is investigated: in the large wavelength approximation the principal lines remain symmetric with respect to the zero order central line, but there is no longer symmetry of the satellite lines with respect to a principal line.
A survey of acoustooptical research since more than fifty years is presented. After the description of the phenomenon of diffraction of light by an ultrasonic wave in a liquid, the general theoretical analysis is given. In the linear case... more
A survey of acoustooptical research since more than fifty years is presented. After the description of the phenomenon of diffraction of light by an ultrasonic wave in a liquid, the general theoretical analysis is given. In the linear case Raman-Nath and Bragg diffraction regimes are discussed. Diffraction of light by adjacent ultrasonic beams and by surface acoustic waves are further treated. Several applications, such as ultrasonic light modulators, light deflectors, radar signal processors and sound field analysis are briefly reviewed.
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... more
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 tanh-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 equation.
We present a systematic and formal approach toward finding solitary wave solutions of non-linear evolution and wave equations from the real exponential solutions of the underlying linear equations. The physical concept is one of the... more
We present a systematic and formal approach toward finding solitary wave solutions of non-linear 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 non-linearities in the system. In the present paper the emphasis is, however, on the mathematical aspects, i.e. the formal procedure necessary to find single solitary wave solutions. By means of examples we show 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. We also outline the extension of our technique to construct N-soliton solutions and indicate connections with other existing methods.
It is argued from a physical point of view that the criteria for acousto-optic Bragg diffraction (characterized by only two orders of light being present) must ultimately include the strength of the sound field. This follows because the... more
It is argued from a physical point of view that the criteria for acousto-optic Bragg diffraction (characterized by only two orders of light being present) must ultimately include the strength of the sound field. This follows because the scattering effect that is due to the sidelobes of a rectangular transducer, although negligible at low power levels where the Bragg criterion is indeed sound-level independent, becomes of increasing importance at high power levels and causes additional orders to be generated. In this paper we demonstrate this numerically by reduction of the sidelobes through Hamming apodization of the sound field. The results clearly demonstrate a significant reduction in the light powers of the spurious orders, thereby extending Bragg operation to higher sound intensities than normally feasible.
The Harry Dym equation, which is related to the classical string problem, is derived in three different ways. An implicit cusp solitary-wave solution is constructed via a simple direct method. The existing connections between the Harry... more
The Harry Dym equation, which is related to the classical string problem, is derived in three different ways. An implicit cusp solitary-wave solution is constructed via a simple direct method. The existing connections between the Harry Dym and the Kortweg-de Vries equations are uniformised and simplified, and transformations between their respective solutions are carried out explicitly. Whenever possible, physical insights are provided.
The construction of Lie symmetry vector fields of a partial differential equation is well described in literature. In this note we evaluate the Lie symmetry vector fields for the Zakharov equation. We also consider an extended Zakharov... more
The construction of Lie symmetry vector fields of a partial differential equation is well described in literature. In this note we evaluate the Lie symmetry vector fields for the Zakharov equation. We also consider an extended Zakharov equation.
We comment on the Lie point symmetries for the Khokholov-Zabolotskaya equations as calculated by Roy Chowdhury and Nasker, and demonstrate that their result for the coefficients of the vector field is correct but incomplete.
A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briefly outline. Examples illustrate the use of the... more
A survey of symbolic programs for the determination of Lie symmetry groups of systems of differential equations is presented. The purpose, methods and algorithms of symmetry analysis are briefly outline. Examples illustrate the use of the software. Directions for further research and development are indicated.
The solitary wave solution is given for nonlinear equations, generalizing the standard and modified Korteweg-de Vries and Schamel equations, as recently investigated by Xiao. A search for conservation laws of a slightly more general class... more
The solitary wave solution is given for nonlinear equations, generalizing the standard and modified Korteweg-de Vries and Schamel equations, as recently investigated by Xiao. A search for conservation laws of a slightly more general class of nonlinear evolution equations reveals that the generalized Schamel equations can have no more than three polynomial invariants. The method is based on obtaining suitable building blocks for conserved densities under scalings which leave the evolution equations invariant.
Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are briefly outlined. Examples illustrate the use of the... more
Computer algebra packages and tools that aid in the computation of Lie symmetries of differential equations are reviewed. The methods and algorithms of Lie symmetry analysis are briefly outlined. Examples illustrate the use of the symbolic software.
Trilateration techniques use distance measurements to survey the spatial coordinates of unknown positions. In practice, distances are measured with error, and statistical methods can quantify the uncertainty in the estimates of the... more
Trilateration techniques use distance measurements to survey the spatial coordinates of unknown positions. In practice, distances are measured with error, and statistical methods can quantify the uncertainty in the estimates of the unknown location. Three methods for estimating the three-dimensional position of a point via trilateration are presented: a linear least squares estimator, an iteratively reweighted least squares estimator, and a nonlinear least squares technique. In general, the nonlinear least squares technique performs best, but in some situations a linear estimator could in theory be constructed that would outperform it. By eliminating the need to measure angles, trilateration facilitates the implementations of fully automated real-time positioning systems similar to the global positioning system (GPS). The methods presented in this paper are tested in the context of a realistic positioning problem that was posed by the Thunder Basin Coal Company in Wright, Wyoming.
This is a 2-page review paper on the Korteweg-de Vries equation prepared for the The Princeton Companion to Applied Mathematics, Princeton University Press, 2015. Editor: Nicholas J. Higham.