Spectral method for the Zakharov equation with periodic boundary conditions (Q1262104)

The author treats the mixed periodic problem for the Zakharov equation (1) \(\eta_{tt}-\eta_{xx}=| \epsilon |^ 2_{xx}\), \(i\epsilon_ t+\epsilon_{xx}-\eta \epsilon =0\). Motivated by the existence (and uniqueness) of global smooth solutions of (1) the spectral method (in the space-variable) is applied to the equation. It is proved that the method is stable, and converges in sup-norm. An estimate for the rate of convergence is also given.