Arctic and Antarctic Research Institute,

St.Petersburg, Russia, e-mail:<lavren@aari.nw.ru>

Road, Vicksburg, MS 39180,USA

(1) Landau Institute for Theoretical Physics, Moscow. 117334, Russia

(2) Department of Mathematics, University of Arisona, Tucson, AZ 857 21, USA

(3) Waves and Solitons LLC, 918 W.Windsong Dr., Phoenix, AZ 85045,USA

1. INTRODUCTION

Gravity waves in water surface are characterized by a small steepness value. It makes possible to apply the weak-

turbulence approach in order to find out the p roblem solution with the help of kinetic equation for energy spectrum.

This equation was formulated by K.Hasselmann (1962,1963) and V.Zakharov (1968). In terms of wave action

spectrum the equation is as follows:

= *G*nl = ∫∫∫ T (*k,k * 1 ,*k * 2 ,k 3 )*δ*(*k *+*k * 1 - *k * 2 -*k * 3 )*δ *(*ω *+ *ω *- *ω*2 - *ω*3 )

1

{N 2 N 3 ( N + *N * 1 ) - *N*1 N ( N 2 + *N * 3 )}d**k**1 d**k ** 2 d**k ** 3

(1)

where N i = *N *( *k*i ) is a spectral density of wave action; T ( *k,k * 1 , k 2 , k3 ) is a kernel function of non-linear wave

δ (ω) are the delta -functions describing a resonance interaction between four wave

interaction; δ (*k *) and

components:

(2a)

ω + ω1 = ω2 + ω3

(2b)

.

For gravity waves ω(*k*) =

explicit expression for

Zakharov 1999).

The equation (1) has been a subject of numerical modeling for almost three decades (Webb 1978, K.Hasselmann

1985; Resio and Perrie 1991; Polnikov 1993; Komen et al., 1994; Komatsu and Masuda 1996, Zakharov 1999,

Lavrenov 1998, 2001 etc.). However, some basic properties of this equation are not clarified till now.

Usually the equation (1) is considered to preserve standard constants of motion wave action, energy and

momentum: