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The statistical initial value problem for the envelope equation \[ \frac{{\partial w}}{{\partial t}} - \gamma \frac{{\partial ^2 w}}{{\partial x^2 }} = \chi w - \beta w^2 w^ * ,\quad \gamma ,\beta \,{\text{complex}} \], is discussed. It is shown that there exists an instability (stability) criterion \[ \beta _r \gamma _r + \beta _i \gamma _i \gtreqless 0,\quad \beta _r > 0 \], which determines whether the system underlying the above equation achieves or does not achieve a monochromatic state.

A fit was made to Serpukhov and NAL data on proton--proton interactions at 50 to 303 GeV/c using the charged-prong distribution given by the Gram-- Charlier asymptotic expansion. Moment ratios are also calculated using the universal fit and the results are found to be in excellent agreement with experiment. (LBS)

A number of properties of the Michaelis-Menten kinetic equations are derived. To that end the enzyme concentration as a function of time is written as a sum of two terms. The first one, ƒ, is chosen such that it depends exclusively on the concentration of the substrate. The second term, ξ, turns out to be given by a decreasing exponential and describes the transient phase. It is not given by the solution of the homogeneous part of the kinetic equation of the enzyme, as has been supposed by Wong. An asymptotic series expansion of ƒ is obtained, the first term of which equals the steady- state expression of the enzyme concentration.

Ray techniques are used to construct an approximate solution for radio waves propagating in the ionosphere when the loss of energy due to collisions is not negligible. For the time-harmonic problem, the solution is in the form of an asymptotic expansion in a large parameter scaling the operating frequency and the natural frequencies characterizing the ionospheric properties. Complex rays are introduced as the characteristics of the resulting eikonal equation, analytically extended to complex space points. The rapidly varying exponential factor and a series for the wave amplitude are both determined by integrating along the rays. The propagation of narrow-band modulation is also determined by integrating along the same complex rays. It is shown that the nature of the initial disturbance plays a vital role in determining the character of the rays required for the solution of a particular problem.

In the contour integralf and g are regular functions of z in a neighbourhood of the contour C and of the complex parameters (α1, α2, …, αp) = α in a domain of . N is a positive parameter and asymptotic expansions are considered as N → + ∞. The method of steepest descents provides an asymptotic expansion for each fixed value of α, but this is non-uniform with respect to α near critical values α0 at which certain saddle points tend to coincidence. A more complicated expansion, involving Airy functions or generalizations thereof, is valid and uniform for α near α0. This expansion is known to be valid in a neighbourhood of α0 which does not depend on N, in contrast to certain other expansions having regions of validity which contract to a point or surface as N → ∞. It is thereby suggested that the actual region of validity has a definite geometrical shape determined by the functions f and g and the contour C, just as steepest descents expansions are valid in regions bounded by Stokes' lines. In this paper, procedures are derived for the determination of such regions, subject to assumptions concerning f, g and C. The validity of the expansion is established in regions so determined.

Let be the Fourier coefficients of the solution of the Cauchy problem for a nonlinear parabolic equation of the form where is a linear elliptic operator of order and is the nonlinear part of the equation. Then are the moment functions of the equation, i.e. the average of the function with respect to a probability measure , where characterizes the degree of concentration of the measure. In this paper we give an asymptotic expansion for the functions as .Bibliography: 8 items.

LetS i have the Wishart distributionW p(∑i,ni) fori=1,2. An asymptotic expansion of the distribution of\( - 2\log \lambda = - 2\log \left[ {\prod\limits_{\alpha = 1}^2 {\left| {{{S_\alpha } \mathord{\left/ {\vphantom {{S_\alpha } {n_\alpha }}} \right. \kern-\nulldelimiterspace} {n_\alpha }}} \right|^{{{n_\alpha } \mathord{\left/ {\vphantom {{n_\alpha } 2}} \right. \kern-\nulldelimiterspace} 2}} } \cdot \left| {{{\left( {S_1 + S_2 } \right)} \mathord{\left/ {\vphantom {{\left( {S_1 + S_2 } \right)} n}} \right. \kern-\nulldelimiterspace} n}} \right|^{ - {n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} } \right]\) for large n=n1+n2 is derived, when∑ 1∑ 2 −1 =I+n−1/2θ, based on an asymptotic solution of the system of partial differential equations for the hypergeometric function2 F 1, obtained recently by Muirhead [2]. Another asymptotic formula is also applied to the distributions of −2 log λ and −log|S 2(S 1+S 2)−1| under fixed∑ 1∑ 2 −1 , which gives the earlier results by Nagao [4]. Some useful asymptotic formulas for1 F 1 were investigated by Sugiura [7].

In a previous paper (F. Calogero and A. Degasperis, J. Math. Phys. 9 (1968), 90) an approach to the inverse problem for the nonrelativistic S -wave potential scattering has been developed with the aim of obtaining explicit and exact expressions of the potential in terms of the phase shift and of the bound states parameters. Since this method, based on the high energy asymptotic expansion, yields the expression of the potential and of all its derivative at the origin only if these quantities are bounded, physically interesting interactions such as the Yukawa potential could not be considered. In this paper this approach has been generalized to potentials with a simple pole at the origin and the expression of the Laurent coefficients of the potential in a neighborhood of the origin has been obtained as an explicit functional of the experimental data. Furthermore the general structure of the high energy expansion of the phase shift has been derived and discussed. The requirement that the potential has only a simple pole at the origin turns out to imply strong constraints on the experimental data. These conditions on the experimental data have been explicitely expressed as a set of infinitely many nonlinear sum rules involving only the phase shift and the energies of the bound states.

The signaling problem for the one dimensional Klein-Gordon equation with spatially varying coefficients is analyzed. A formal, uniformly valid, asymptotic expansion of the solution is obtained with the help of two families of rays, and involving four functions : two successive Bessel functions of integer order and two new functions which we call the diffraction functions. The validity of the expansion is established when the coefficients in the Klein-Gordon equation are constants, and the results are applied to a signaling problem for a class of acoustic wave guides.

For pt.II, see abstr. A3682 of 1974. The theory developed in two previous papers, for studying the Weaire model in terms of returning walks, is used here to study the density of electronic states at the band edges. The relationship between topological disorder and the shape of the band edges is considered and the shape of the inner band edge is shown to depend on oscillating terms in the asymptotic expansion of the number of returning walks. The work suggests the possibility of a band gap in a fourfold coordinated structure with mod V1/V2 mod >1/2.

An incompressible fluid of constant thermal diffusivity k , flows with velocity u = Sy in the x -direction, where S is a scaling factor for the velocity gradient at the wall y = 0. The region — L ≤ x ≤ 0 is occupied by a heated film of temperature T 1 , the rest of the wall being insulated. Far from the film the fluid temperature is T 0 < T 1 . The finite heated film is approximated by a semi-infinite half-plane x < 0 by assuming that the boundary-layer solution is valid somewhere on the finite region upstream of the trailing edge. Exact solutions in terms of Fourier inverse integrals are obtained by using the Wiener-Hopf technique for the dimensionless temperature distribution on the half-plane x > 0 and the heat transfer from the heated film. An asymptotic expansion is made in inverse powers of x and the coefficient of the leading term is used to calculate the exact value of the total heat-transfer as a function of the length L . It is shown that the boundary layer solution differs from the exact solution by a term of order L -1/3 for large L . An expansion in powers of x for the heat transfer upstream of the trailing edge is also found. Application of the theory, together with that of Springer & Pedley (1973), to hot films used in experiments are discussed for the range of values of L(S/K) ½ , up to 20.

An initial value investigation is made of the propagation of capillary-gravity waves generated by an oscillating pressure distribution acting at the free surface of a running stream of finite, infinite, and shallow depth. The solution for the free surface elevation is obtained explicitly by using the generalized Fourier transform and its asymptotic expansion. It is found that the solution consists of both the steady state and the transient components. The latter decays asymptotically as t → ∞ and the ultimate steady state is attained. It is shown that the steady state consists of two or four progressive capillary-gravity waves travelling both upstream and downstream according as the basic stream velocity is less or greater than the critical speed. Special attention is given to the existence of the critical values associated with the running stream of finite, infinite, and shallow depth. A comparison is made between the unsteady wave motions in an inviscid fluid with or without surface tension.

A numerical simulation concerning the spatial evolution of electron plasma waves and electron distributions in collisionless plasmas is performed. Only those particles in and around the resonant region are followed numerically, while the bulk plasma is treated analytically. Our results are in good agreement with existing theoretical results as well as experimental observations. The method introduced here is valid for waves with arbitrary amplitude. It does not depend on either a large-amplitude asymptotic expansion or a small-amplitude perturbation scheme.

In a recent paper in this Journal (7, 1973, 1–6) Tam suggested that early work by Batchelor on the decay of a laminar trailing vortex was in error. Tam's claim was that the decay of axial velocity defect at the center of the vortex went asx−1 instead ofx−1 logx. The purpose of this note is to refute Tam's claim. A further term in the asymptotic expansion for the circulation in the vortex is also obtained.