It is shown that highly localized discrete modes may exist on a nonlin
ear standing background wave. In the framework of the rotating wave ap
proximation, a few types of such modes are found analytically and nume
rically for the chain of particles interacting through harmonic and qu
artic anharmonic potentials in the case of the so-called ''hard'' nonl
inearity. The case of ''soft'' nonlinearity is also briefly discussed.