Galiev S., Mallinson G.

Wave nonlinearity is an important element of Nature. This nonlinearity is often focused near and at critical points where the behavior of greatly different systems can exhibit strong similarities. These points have been called resonances. In the resonant bands around these points, nonlinear effects can increase strongly and some first-order linear effects drop to zero. As a result the nonlinear effects strongly localize in space and time, and catastrophic natural phenomena may be generated. In this paper the concept of resonance is used to explain catastrophic coast amplification and evolution of tsunami. It was shown that the coast zone forms the resonant band where the tsunami speed is very close to or coincides with the wave phase speed of very long waves. The catastrophic evolution takes place as the tsunami passes this band. The propagation of tsunami in the deep ocean have been studied in many publications. For example, Titov developed the numerical method which can solve problems connected with the generation and the propagation of tsunamis. In contrast, the coastal evolution of tsunami is more difficult to simulate since this evolution is governed by many parameters. Which parameters are the most important is difficult to determine. Here we introduce a transresonant parameter R. The nonlinear catastrophic amplification takes place if R<1. By considering unidirectional waves, an ordinary nonlinear equation for ocean coast waves is derived. According to this equation the amplification and the evolution of the waves may be connected with linear or nonlinear resonance of ocean waves near the coast line. Tsunami is considered as an earthquake-induced solitary-like wave. The analytic solutions are presented which describe data of experiments and numerical calculations, and the evolution of tsunami from deep ocean to the coast line. It is found that the nonlinear resonance can explain the catastrophic coast amplification of