In the beat-wave accelerator [Phys. Rev. Lett. 43, 267 (1979)], a large-amplitude Langmuir wave is produced by the beating of two laser beams whose frequencies differ by approximately the plasma frequency. The relativistic equations governing this three-wave interaction are shown to admit two types of solitary-wave solutions. Temporal solitary waves propagate at speeds greater than the speed of light and carry no information. Spatial solitary waves propagate at speeds less than the speed of light and do carry information. Analytic expressions are obtained for the envelopes of these waves and for the relationship between their speed and maximum amplitude. For the limit in which the propagation speed of the solitary wave is equal to the speed of light, there is no distinction between a temporal solitary wave and a spatial solitary wave. However, it can be shown that the solitary wave is unstable in this limit. The potential of the spatial solitary waves for particle acceleration [Phys. Rev. Lett. 57, 1421 (1986)] is studied. Although the spatial solitary waves are capable of accelerating particles to high energy, for typical beat-wave parameters the laser–plasma coupling efficiency is too low for this scheme to be practical.