  The derivative nonlinear Schrodinger (DNLS) equation is the canonical model
for dynamics of nonlinear waves in plasma physics and optics. We study exact
solutions describing rogue waves on the background of periodic standing waves
in the DNLS equation. We show that the space-time localization of a rogue wave
is only possible if the periodic standing wave is modulationally unstable. If
the periodic standing wave is modulationally stable, the rogue wave solutions
degenerate into algebraic solitons propagating along the background and
interacting with the periodic standing waves. Maximal amplitudes of rogue waves
are found analytically and confirmed numerically.
