Relaxation properties of (1+1)-dimensional driven interfaces in disordered media

We use the Kardar-Parisi-Zhang equation with quenched noise in order to study the relaxation properties of driven interfaces in disordered media. For lambda not equal 0 this equation belongs to the directed percolation depinning universality class and for gimel = 0 it belongs to the quenched Edwards-Wilkinson universality class. We study the Fourier transform of the two-time autocorrelation function of the interface height C-k(t', t). These functions depend on the difference of times t-t' in the steady-state regime. We find a two-step relaxation decay in this regime for both universality classes. The long time tail can be fitted by a stretched exponential function, where the exponent beta depends on the universality class. The relaxation time and the wavelength of the Fourier transform, where the two-step relaxation is lost, are related to the length of the pinned regions. The stretched exponential relaxation is caused by the existence of pinned regions which is a direct consequence of the quenched noise.