Modulation instability of spiral density waves: high-order nonlinearity effects

The evolution of the envelope of a tightly wound spiral density wave in a gravitating gaseous disk is investigated, including nonlinear terms to fifth order in the amplitude. These terms in the expansion of the fundamental harmonic set limits on the applicability of the weakly nonlinear approximation, and demonstrate that the group velocity depends on the wave amplitude. For a disk with solar-neighborhood parameters and a spiral pattern speed Ω/sub p/ = 20 km sec-1 kpc-1, the fifth-order nonlinearity will be significant if the wave amplitude is not too small (sigma-tilde/sigma0>30%). One also must allow for fifth-order nonlinearity if the coefficient of the cubic nonlinearity vanishes