Abstract

DO FREE SURFACE POTENTIAL GRAVITY WAVES ON THE SURFACE OF DEEP FLUID FORM AN INTEGRABLE SYSTEM?

Vladimir Zakharov

ABSTRACT

The question formulated in the title of this talk is a conjecture. However, there are very strong reasons to believe that this conjecture is correct. Let us enlist three of them:
1. Euler equation for potential flow of deep incompressible fluid could have extra constants of motion. The number of constants is finite and depends on initial data. However, this number can be arbitrary large.
2. Certain coefficient functions describing processes of nonlinear wave interaction are identically equal to zero.
3. Giant "breathers", analogs of envelop solitons do exist up to very high level of steepness (up to mu=0.7 or more). This is a very natural explanation for formation singular freak waves. The results presented in my talk, are supported by analytical considerations and massive numerical simulations.