The localized Zakharov equation: Derivation and validation

In Zakharov’s equation, the spectral function represents the entire horizontal plane. In practical applications, one often has to use a finite number of Fourier modes that are determined for limited regions of the horizontal plane, but vary from region to region.To overcome this shortcoming, we utilize a discrete windowed Fourier transform to obtain a new localized Zakharov equation (LZE), which can handle spatial variations in a more transparent way. The LZE is successfully validated by comparing different aspects of its performance with results from the Zakharov equation and a modified nonlinear Schrödinger equation.