Equations for 1D waves on the surface of deep water
Abstract
We apply a canonical transformation to a water wave equations to remove cubic nonlinear terms and to drastically simplify fourthorder terms in the Hamiltonian. This transformation from natural Hamiltonian variables η,ψ to new complex normal variables c,c∗ explicitly uses the fact of vanishing exact fourwave interaction for water gravity waves for a 2D potential fluid. The new variable is the sum c(x,t) = c+(x,t) + c(x,t) of two analytic functions: c+(x,t)  is analytic in the upper halfplane, c(x,t)  is analytic in the lowerplane. We obtained system of two coupled differential equations for c+ and c which is very suitable for analytical studies and numerical simulations: [ ] ∂c++ iωˆc+ = ∂+ i(c+2  c 2)c++ c+ˆk (c+2  c 2) ic+c c∗ c∗ˆk(c+c ) , ∂t x[ x x ] ∂c+ iωˆc = ∂ i(c 2  c+2)c c ˆk (c 2  c+ 2) ic c+c+∗+ c+∗ˆk (c+c )(1) ∂t x x x Here ωˆ and ˆk are correspond to the multiplication by √gk and k in the Fourier space, ∗ denotes complex conjugation, the subscript x is the derivative with respect to the variable x, the differentiation operators ∂x+ and ∂x are ikΘ(k) and ikΘ(k), where Θ(k) is the Heaviside step function. Physical variables η(x,t) and ψ(x,t) can be restored from complex variable c(x,t). The system (1) has the simple solution: c+ = AeikAx iωAt, c = Be ikBxiωBt, where 2 2 2 2 2 ωA = ωkA + kA2(A2 B)2  kAkBB2+ kAkA  kBB2 ωB = ωkB +kB (B  A ) kAkBA + kBkA  kB A We performed numerical simulation of system (1) for water waves moving in opposite directions.
 Publication:

EGU General Assembly Conference Abstracts
 Pub Date:
 April 2018
 Bibcode:
 2018EGUGA..2013067K