Properties of time-dependent solutions of the Hasselmann kinetic equation

The Hasselmann kinetic equation for deep-water nonlinear gravity waves is s tudied analytically and numerically to elucidate the existence of a self-si milar form of the spectrum resulting from a time-dependent solution of this equation on large time scales. It is shown analytically that, due to the p resence of three integrals of motion, the complete self-similarity of solut ions to the kinetic equation is impossible. However, this fact does not for bid the existence of an "incomplete self-similarity," which is defined as t he establishment of fixed values of the integral parameters of the spectrum in the course of its long-term evolution. This inference is confirmed by t he results of a numerical study carried out by invoking both an improved me thod of calculating the kinetic integral and modern schemes for numerically solving the kinetic equation. The results of numerical experiments are use d to determine the limiting characteristics of the form of the energy spect rum of waves on evolutionary scales when the form of the spectrum is comple tely controlled by nonlinear processes in waves.