Studies in Applied Mathematics

Chen, B.; Sallman, P. G.

doi: 10.1002/sapm19806211pmid: N/A

Numerical results are presented for the shape of finite‐amplitude steady irrotational inviscid gravity waves obtained by solving an integrodifferential equation. It is found that the family of solutions giving the wave as a function of height or equivalent parameter has bifurcation points for h/λ≈0.13, where h is waveheight and λ is wavelength. Two bifurcation points and the branches emanating from them are found specifically, corresponding to a doubling and tripling of the wavelength. Solutions on the new branches are calculated.

Kevorkian, J.

doi: 10.1002/sapm198062123pmid: N/A

We survey multiple‐variable expansion procedures appropriate for nonlinear systems in resonance using the model of two coupled weakly nonlinear oscillators with either constant or slowly varying frequencies. In the autonomous problem we show that an n‐variable expansion (where n depends on the order of accuracy desired) yields uniformly valid results. We also consider the problem of passage through resonance for the nonautonomous problem and describe the solution by constructing a sequence of three expansions. The solution before resonance is developed as a generalized multiple‐variable expansion and is matched with an inner expansion valid during resonance. This latter is then matched with a postresonance solution and determines it completely. Numerical integrations are used to substantiate the theoretical results. The dominant effect of passage through resonance is shown to be the excitation of a higher‐order oscillation beyond resonance. Contrary to the claim in a recent work, the total action of the system does not remain constant if one accounts for the leading perturbation terms in the postresonance solution. Instead, the total action goes from one constant value to another.

Peck, G. W.

doi: 10.1002/sapm198062169pmid: N/A

A disturbance spreads in a rectilinear n dimensional grid moving from each affected point to at most one neighbor in each unit of time. The question how many points can be affected in N units of time is answered to the two leading orders of N.

Kaup, D. J.

doi: 10.1002/sapm198062175pmid: N/A

Based on the recent solution of the three‐dimensional three‐wave interaction by H. Cornille, we have found a means for solving the separable initial‐value problem of this interaction, which is when the envelopes are initially nonoverlapping. Explicit means for calculating the kernels directly from the initial pulse profiles are given.

Morris, Robert A.

doi: 10.1002/sapm198062185pmid: N/A

Frobenius operators Fn are introduced on sequences of binomial type. The Laguerre polynomials are essentially characterized by the property that Fn coincides with n‐fold binomial convolution.

Schay, Geza

doi: 10.1002/sapm198062193pmid: N/A