This paper constructs a fast Fourier spectral algorithm for the wave kinetic equation arising as a kinetic limit of the cubic Schrödinger equation. The resonant four-wave collision operator is first rewritten as a spherical integral analogous to the classical Boltzmann operator. Fourier approximation then exposes double-convolution structures that can be evaluated with fast Fourier transforms, avoiding large precomputed operator tensors and their storage. Spherical-design quadrature handles angular integration in three dimensions. The authors test stationary distributions and time-dependent, isotropic, anisotropic, continuous, and discontinuous data in two and three dimensions. The computations exhibit spectral convergence for smooth cases, retain the expected mass and energy behavior, and resolve non-isotropic wave spectra. Timing studies show the practical gain from the FFT formulation. Relative to discrete-interaction and direct quadrature approaches, the method provides a more accurate and scalable route for simulating high-dimensional wave-turbulence kinetics.
- Recast the resonant wave collision operator as a Boltzmann-like spherical integral.
- Derived double-convolution formulas suitable for fast Fourier transform evaluation.
- Eliminated large operator precomputation and storage requirements.
- Combined Fourier discretization with spherical-design quadrature for three-dimensional problems.
- Verified accuracy, conservation behavior, anisotropic dynamics, and efficiency in two and three dimensions.