Properties of Time-Dependent Solutionsof the Hasselmann Kinetic Equation amg

The Hass­elm­ann kinetic equa­tion for deep-water nonl­inear gravity waves is studied anal­ytic­ally and nume­rica­lly to eluc­idate the exis­tence of a self-similar 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 pres­ence of three inte­grals of motion, the comp­lete self-simi­lar­ity of solu­tions to the kinetic equa­tion is impo­ssi­ble. However, this fact does not forbid the exis­tence of an «inco­mpl­ete self-simi­lar­ity,» which is defined as the esta­blis­hment of fixed values of the inte­gral para­met­ers of the spec­trum in the course of its long-term evol­ution. This infe­rence is conf­irmed by the results of a nume­rical study carried out by invo­king both an impr­oved method of calc­ulat­ing the kinetic inte­gral and modern schemes for nume­rica­lly solving the kinetic equa­tion. The results of nume­rical expe­rime­nts are used to dete­rmine the limi­ting char­acte­rist­ics of the form of the energy spec­trum of waves on evol­utio­nary scales when the form of the spec­trum is comp­let­ely cont­rol­led by nonl­inear proc­esses in waves.