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Super compact spatial and temporal equations for water waves

Name
Alexander
Surname
Dyachenko
Scientific organization
Landau Institute for Theoretical physics, Novosibirsk State University
Academic degree
Doctor of Science
Position
Senior Researcher
Scientific discipline
Mathematics & Mechanics
Topic
Super compact spatial and temporal equations for water waves
Abstract
We derive very simple super compact equations for unidirectional gravity water waves. Zakharov equation was essentially simplified
by some canonical transformation. We suggest a specific form of such transformation that allows one to derive a remarkably simple form of temporal and spatial equatios. One can name it the super compact equation. This equation is very straightforward and
includes a nonlinear wave term and advection term. Moreover, this equation also allows one to derive a spatial version of the water waves equation which can be used to describe experiments it the flume.
Keywords
surface waves, freak waves, hamiltonian approach, numerical simulation
Summary

We derive very simple compact equations for unidirectional gravity water waves. For such waves, as it is well-known,
the coefficient of nontrivial four-wave interaction is identically zero. This fact allows one to essentially simplify
the Zakharov equation, applying a canonical transformation. Obviously this transformation is not unique. In this paper
we suggest a specific form of such transformation that allows one to derive a remarkably simple form of the Zakharov
equation. One can name it the super compact equation. This equation is very straightforward and
includes a nonlinear wave term (\`{a} la NLSE) and advection term. Moreover, this equation also  allows one to derive a
spatial version of the water waves equation which can be used to describe experiments it the flume.