摘要:

Nonlinear wave groups in deep water consist of wave modes for which nonlinear interactions and dispersion are in balance. The evolution equations for the wave modes are derived, and properties of nonlinear wave groups are found from these equations. It is shown that the nonlinear wave groups are linearly unstable to sideband modulations in the sense that the linearized perturbation theory, in providing a good fit over the initial time interval, predicts that the growth of the modulations is exponential. Instead the perturbed wave group is shown to return cyclically to a state close to its initial state. The cyclic recurrence is demonstrated analytically for the simpler wave groups and numerically otherwise. The interactions between nonlinear wave groups of the same and of nearly the same central wavenumbers are calculated.

ISSN:0022-2526    

DOI:10.1002/sapm19796111

年代:1979

数据来源: WILEY

摘要:

With Bäcklund transformations, we construct explicit solutions of Painlevé equations 2 and 4. Independently, we find solutions of degenerate cases of equations 3 and 5. The six Painlevé transcendents are referred to as 1

ISSN:0022-2526    

DOI:10.1002/sapm197961131

年代:1979

数据来源: WILEY

摘要:

A dynamical system is considered whose normal frequencies and normal modes vary slowly with time in such a way that two frequencies come into close coincidence. When this occurs the corresponding normal modes undergo a drastic change in their physical properties. Away from coincidence, each normal mode conserves its action. A multiple‐time‐scale asymptotic procedure is employed to derive equations which describe the mode coupling at coincidence. These equations are solved exactly using parabolic cylinder functions. It is found that in general, action is exchanged between modes at coincidence, but that except for very strong coupling the amount of action exchanged is quite sm

ISSN:0022-2526    

DOI:10.1002/sapm197961155

年代:1979

数据来源: WILEY

摘要:

Letπ= (π(1),π(2),...,π(n)) be a permutation of the arbitraryn‐setSof postive integers. Asuccessioninπis any pairπ(i),π(i+ 1) withπ(i+ 1) =π(i) + 1,i= 1,2...,n−1. We show that the number of permutations ofSwhich have preciselyksuccessions depends only uponn,kandb, wherebis the number of maximal disjoint intervals in the set [n+m]\S, andn+mis the largest element inS. We derive a linear recurrence relation for this number, which we call thesuccession numberσ(n,k;b), as well as an explicit formula in terms of derangement numbers. The linear recurrence is used to derive the generating function for succession numbers. is also derived by formal power series methods from a well‐known generating function for succession in general in

ISSN:0022-2526    

DOI:10.1002/sapm197961173

年代:1979

数据来源: WILEY

摘要:

The equation which governs the stability of plane Couette flow admits solutions of either well‐balanced or dominant‐recessive type. As is well known, the solutions of well‐balanced type can be expressed in terms of simple exponential or hyperbolic functions. The main aim of this paper, therefore, is to show that the solutions of dominant‐recessive type can be expressedexactlyas the sum of three products which involve certain rapidly and slowly varying functions. The rapidly varying functions are simply Airy functions, together with their first integrals and first derivatives. Of the three slowly varying functions, one is expressible in terms of a hyperbolic function and the other two have simple integral representations which, for bounded values of the wavenumberα, haveconvergentexpansions in powers of (iαR)−1. An application of these results is also made to the problem of semibounded plane Couette flow, and it is shown that the eigenvalue relation for this problem can be expressed in an extremely

ISSN:0022-2526    

DOI:10.1002/sapm197961183

年代:1979

数据来源: WILEY