  Using the Darboux transformation for the Korteweg-de Vries equation, we
construct and analyze exact solutions describing the interaction of a solitary
wave and a traveling cnoidal wave. Due to their unsteady, wavepacket-like
character, these wave patterns are referred to as breathers. Both elevation
(bright) and depression (dark) breather solutions are obtained. The nonlinear
dispersion relations demonstrate that the bright (dark) breathers propagate
faster (slower) than the background cnoidal wave. Two-soliton solutions are
obtained in the limit of degeneration of the cnoidal wave. In the small
amplitude regime, the dark breathers are accurately approximated by dark
soliton solutions of the nonlinear Schr\"odinger equation. These results
provide insight into recent experiments on soliton-dispersive shock wave
interactions and soliton gases.
