Nonlinear dynamical and kinetic edge-wave equations

We derive a nonlinear dynamical equation for edge-wave spectrum over a cylindrical (laterally uniform) beach based on Zakharov's Hamiltonian formalism. An extension of the phase-averaged (kinetic) equation has also been developed. Analytical solution and numerical implementations of the dynamical model have been developed for an isolated triad. The resonances condition can now been identified automatically; and the scale-invariant properties of the resonance manifolds for plane and exponential beaches have been discussed. Numerical results of butterfly clusters have been presented for the kinetic equation. We can also observe a cascade behavior when the simulation involves multiple resonance manifolds.