ABSTRACT
The unusual waves are studied using analytic solutions of nonlinear wave equations and experimental data. It is shown that in the case of unidirectional one-dimensional waves, the equation for long waves and the basic nonlinear wave equations of physics are reduced to the same equation having solutions in the form of elliptic integrals. Thus, it is shown that a wide spectrum of nonlinear wave equations has a solution in the form of elliptic integrals which describe extreme waves. The resonant nature of the occurrence of extreme waves is hidden more deeply. The amplification can occur because of resonant interaction of nonlinear waves or wind action in the ocean. The wave evolution is described by highly nonlinear wave equations. These equations contain the d’Alembert operator and the terms taking into account viscous, dispersive, and highly nonlinear effects. Interaction of these effects near resonances may be very complex. However, very close to resonances, the influence of nonlinearity becomes the most important.
