Abstract

Breaking of water waves is a dramatic phenomenon, which often occurs in nature. Wave breaking can characterized by finite time singularity formation in solutions of dynamic equations. In the talk we give three examples of wave breaking. The first is related to foam formation on crests of sea waves due to Kelvin-Helmholtz instability of the interface between two ideal fluids. Evolution of interface surface is described by a nonlinear (2+1)-dimensional Klein-Gordon equation. A proof of singularity formation in a finite time is given. Our results agree with the sharp dependence on wind velocity of the fraction of sea surface area covered by foam as obtained from satellite and airplane observations. The second example concerns the integrable dynamics of the interface between a light viscous fluid with Stokes flow and a heavy ideal fluid. Surface evolution is determined from the motion of complex singularities (poles) of two complex Burgers equations. The interface loses its smoothness if poles reach the interface. In the third example we show that sometimes wave breaking does not really occur. We consider the Hamiltonian form of the water wave equations for the free surface motion and show that they are ill-posed and formally wave breaking should happen in arbitrary small time. However we found that these equations become well-posed after a canonical transformation to new variables and no wave breaking actually occurs. Implications of the new variables for numerical simulations of ocean dynamics are discussed.