The Fourier transform of the function v(t,⋅) = P (t,⋅,∂āˆ•∂x)u(t,⋅) is formally given by
⌊ ⌋ 1 ∑n ˆv(t,⋅) = ---nāˆ•2⌈ ˆAj(t,⋅)∗ikjˆu(t,⋅)+ ˆB(t,⋅)∗ ˆu(t,⋅)⌉ , (2π) j=1
where Aˆj(t,⋅), ˆB (t,⋅) denote the Fourier transform of Aj (t,⋅) and B (t,⋅), respectively, and where the star denotes the convolution operator. Unless Aj and B are independent of x, the different k-modes couple to each other.