Equations for Deep Water Counter Streaming Waves and New Integrals of Motion

The waves on a free surface of 2D deep water can be split into two groups: the waves moving to the right, and the waves moving to the left. A specific feature of the four-wave interactions of water waves allows to describe the evolution of the two groups as a system of two equations. The fundamental consequence of this decomposition is the conservation of the “number of waves” in each particular group. The envelope approximation for the waves in each group of counter streaming waves is obtained.