Showing papers on "Multiple-scale analysis published in 1977"

Note on Higher Order Terms in Reductive Perturbation Theory

Abstract: It is remarked that the second order correction given by Ichikawa et al. for a single ion acoustic K-dV (Korteweg-de Vries) soliton contains a secular term. To eliminate the secularity, the method of multiple scales combined with the reductive perturbation method is proposed. It is then found that not only the second order correction is modified so as to be secular-free but also the phase factor of the lowest K-dV soliton suffers a modification proportional to its amplitude.

Singular Perturbation Techniques: A Comparison of the Method of Matched Asymptotic Expansions with that of Multiple Scales

Abstract: A comparison is made between the uniformly valid asymptotic representations which can be developed for the solution of a singular perturbation boundary value problem involving a linear second order differential equation by using both the technique of matched asymptotic expansions and the method of multiple scales. Next, there is a discussion of some of the subtle features as well as the relative advantages, limitations, logical extensions, and typical applications of these two methods of obtaining uniformly valid asymptotic representations when applied to slightly more general singular perturbation problems which arise from investigations of various phenomena in the natural sciences. Finally, a parameter identification example relevant to biological population dynamics is presented to illustrate the fact that the mere knowledge of these singular perturbation techniques can be a powerful analytical tool.

Nonlinear mode coupling of elastic waves

Abstract: The theory of nonlinear elasticity is applied to a study of mode coupling of elastic waves at a particular frequency, called the critical frequency, in a long circular wire. It is assumed that the wire is of homogeneous isotropic elastic material and that the nonlinearity of medium (the effects of higher‐order elasticity) is primary rather than that involved in the Lagrangian stress and strain tensors. The latter is suggested from the fact that the third‐order elastic constants are of larger order of magnitude than the Lame constants. The method of multiple scales is employed to obtain a system of equations which describes the behavior of the amplitudes involved in the mode coupling. The analysis of the equations shows that nonlinear mode coupling can occur at the critical frequency and that, except at this frequency, the wave undergoes only a phase shift. Further, progressive wave solutions show that the two wave amplitudes can be expressed in terms of Jacobian elliptic functions, and energy exchange between two modes takes place. Under special conditions, these periodic solutions degenerate to the solitary or shocklike solutions.

Spin‐frame independent variables in general relativity

Abstract: The purpose of this paper is to present the spin‐frame independent variables in general relativity. The work is based on the fact that the tetrad Newman–Penrose form of Einstein’s equations can be put into the Yang–Mills form with the group SL(2,C) as the gauge group. The set of Mandelstam path‐dependent dynamical variables for such a theory forms spin‐frame independent variables in general relativity. The empty‐space field equations for spin‐frame independent variables have formally the same form as Maxwell’s equations. In addition, the full set of field equations for spin‐frame independent variables have formally the same form as the equations of nonlinear electrodynamics.