Spectral Algorithms for 3-Wave Kinetic and $C_{12}$ Quantum Boltzmann Equations with General Resonance Manifolds in $\mathbb{R}^d$

Abstract

Following recent developments in numerical schemes for 3-wave kinetic equations [2, 7, 42, 44, 43], we develop spectral algorithms for multidimensional 3-wave kinetic equations and $C_{12}$ quantum Boltzmann equations with general polynomial dispersion relations. The principal numerical difficulty arises from the resonance constraint, supported on a nonlinear manifold in wave-vector space. We approximate the Dirac distribution by a truncated Fourier representation and derive two spectral discretizations of the collision operator. The first is a direct spectral method with complexity $\mathcal{O}\big(L(2N)^{3d}\big)$, while the second exploits multidimensional FFTs to reduce the complexity to $\mathcal{O}\big(L(2N)^{2d}\log(2N)\big)$. Numerical tests show excellent agreement between the two methods, with the fast algorithm providing substantial computational savings. To suppress unresolved high-frequency modes, we combine the classical $2/3$-rule with exponential spectral filtering. Simulations in two and three dimensions capture the gain--loss dynamics of the $C_{12}$ quantum Boltzmann equation for both rapidly and algebraically decaying initial data. For the 3-wave kinetic equation, the computations exhibit strong oscillations and rapid spectral broadening, providing numerical evidence of an apparent energy cascade toward high frequencies. The results also show that the dispersion relation and spatial dimension strongly influence the transient resonant dynamics.