  Harmonically modulated complex solitary waves which are a generalized type of
envelope soliton (herein coined oscillatory solitons) are studied for the two
U(1)-invariant integrable generalizations of the modified Korteweg-de Vries
equation, given by the Hirota equation and the Sasa-Satsuma equation. A
bilinear formulation of these two equations is used to derive the oscillatory
1-soliton and 2-soliton solutions, which are then written out in a physical
form parameterized in terms of their speed, modulation frequency, and phase.
Depending on the modulation frequency, the speeds of oscillatory waves
(1-solitons) can be positive, negative, or zero, in contrast to the strictly
positive speed of ordinary solitons. When the speed is zero, an oscillatory
wave is a time-periodic standing wave. Properties of the amplitude and phase of
oscillatory 1-solitons are derived. Oscillatory 2-solitons are graphically
illustrated to describe collisions between two oscillatory 1-solitons in the
case when the speeds are distinct. In the special case of equal speeds,
oscillatory 2-solitons are shown to reduce to harmonically modulated breather
waves.

Exactly Solvable and Integrable Systems