Canonical system of equations for 1D water waves

One of the essential tasks of the theory of water waves is a construction of simplified mathematical models, which are applied to the description of complex events, such as wave breaking, appearing of freak waves in the assumption of weak nonlinearity. The Zakharov equation and its simplification, such as nonlinear Schrodinger equations and Dysthe equations, are among them. Recently, for unidirectional waves, the so-called super compact equation was derived in Ref. 1. In the present article, the waves moving in both directions are considered. Namely, 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 describing the evolution of these two groups as a system of two equations. One of the significant consequences of this decomposition is the conservation of the number of waves in each particular group. To derive this system of equations, a particular canonical transformation is used. This transformation is possible due to the miraculous cancellation of the four-wave interaction for some groups of waves in the one-dimensional wave field. The obtained equations are remarkably simple. They can be called a canonical system of equations. They include a nonlinear wave term together with an advection term that can describe the initial stage of wave-breaking. They also include interaction terms (between counter-streaming waves). It is also suitable for analytical study as well as for numerical simulation.