Instability of periodic wavetrains in nonlinear dispersive systems

Abstract

The defining property of the class of physical systems under consideration herein is that, by striking a balance between nonlinear and frequency-dispersive effects, they can transmit periodic waves of finite amplitude but constant form. For any such system, therefore, in respect of propagation in the x direction relative to a state of rest, the dynamical equations have exact periodic solutions of the form $\eta$(x, t) = H(x-ct), say, where c is a constant phase velocity depending on wave amplitude as well as on frequency or wavelength. This paper is concerned with the proposition that in many cases these uniform wavetrains are unstable to small disturbances of a certain kind, so that in practice they will disintegrate if the attempt is made to send them over great distances. The outstanding example only recently brought to light is that finite gravity waves on deep water are unstable: unmistakable experimental evidence of this property is now available, and it has also been demonstrated analytically. In $\S$2 the essential factors leading to instability are explained in general terms. A disturbance capable of gaining energy from the primary wave motion consists of a pair of wave modes at side-band frequencies and wavenumbers fractionally different from the fundamental frequency and wavenumber. In consequence of a nonlinear effect on these modes counteracting the detuning effect of dispersion on them, they are forced into resonance with second-harmonic components of the primary motion and thereafter their amplitudes grow mutually at a rate that is exponential in time or distance. In $\S$3 a detailed stability analysis is presented for wavetrains on water of arbitrary depth h, and it is shown that they are unstable if the fundamental wavenumber k satisfies kh > 1.363, but are otherwise stable. Finally, in $\S$4, some experimental results regarding the instability of deep-water waves are discussed, and a few prospective applications to other specific systems are reviewed.