Evolution in space and time of resonant wave triads - II. A class of exact solutions

Abstract

A class of exact solutions is obtained of the nonlinear evolution equations for resonant wave triads. These solutions, which exhibit dependence on two spatial coordinates and on time, are deduced by elementary analysis. This class turns out to be identical with that found by Zakharov (1976) as a special case of his general ‘inverse-scattering’ solution. Under certain circumstances, these solutions develop singularities, or ‘burst’, after a finite time. The criterion for the development of a ‘burst’ is determined and interpreted in terms of initial wave energies. Some particular solutions, representative of initially localized disturbances, are examined in detail with particular reference to the nature of the singularities which may develop.