Undelayed self-focusing of nondispersive waves with a broad spectrum

The self-focusing of wave beams in a nondispersive medium with a cubic nonlinearity is analyzed. The analysis is based on a field equation analogous to a nonlinear Schrödinger equation. Waves whose temporal profile is a periodic train of shock waves are studied. Calculations are carried out in the aberrationless approximation. The cubic nonlinearity causes not only a self-focusing but also, because of the shock fronts, an absorption of a wave. The self-focusing, however, is only slight. Although the beam does contract, the wave amplitudes undergo essentially no increase.