Equation in Frequency Space
The integration term in the evolution equation (19) makes it inefficient to solve it numerically in the real space. An alternative and more efficient method is to solve the equation in the reciprocal space by the Fourier transformation, which converts the integral-differential equation into a regular partial differential equation. The integration operation, as well as the differentiation over space is removed and the evolution equation can be dramatically simplified.
Denote the Fourier transform of  by  , where  and  are the coordinates in the reciprocal space. That is
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(21) |
Regard P as a function  , and transform it to  . Take the Fourier transform on both sides of Eq. (19), and we obtain that
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(22) |
where
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(23) |
Because  is a nonlinear function, amplitudes  for various modes  are coupled.
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