Nonlinear Waves on Water — Solitons

Abstract

There is one kind of wave mentioned in the introduction to Chap. 1 which we have so far pointedly ignored — waves on water. These waves must be among the earliest recorded illustrations of nonlinear behavior, breaking noticeably at all but the smallest amplitudes. In previous chapters the equations of motion were linear. Of course, it has been understood that linear equations are idealizations. No matter how small the displacement of a spring, if finite, it must be that its elastic limit is in some fashion exceeded as discussed, for example, by Erber et al., listed under Further Reading. Idealization implies that, although something has been left out for simplicity, the idealized model reflects the essentials of the physical situation. But water waves are different. Results obtained by linearizing, that is, by ignoring the nonlinear parts, are most frequently too far from reality to be useful. In particular, linearization misses a central phenomenon, solitons, which are isolated waves or pulses which maintain their identity indefinitely just when we most expect that dispersion effects will lead to their rapid disappearance. Further, water waves act as prototypes for many other nonlinear physical phenomena which, as technology develops are becoming ever more important.