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The Hirota equation and the Sasa-Satsuma equation are U(1)-invariantintegrable generalizations of the modified Korteweg-de Vries equation. Thesetwo generalizations admit oscillatory solitons, which describe harmonicallymodulated complex solitary waves parameterized by their speed, modulationfrequency, and phase. Depending on the modulation frequency, the speeds ofoscillatory waves (1-solitons) can be positive, negative, or zero, in contrastto the strictly positive speed of ordinary solitons. When the speed is zero, anoscillatory wave is a time-periodic standing wave. Oscillatory 2-solitons withnon-zero wave speeds are shown to describe overtake collisions of a fast waveand a slow wave moving in the same direction, or head-on collisions of twowaves moving in opposite directions. When one wave speed is zero, oscillatory2-solitons are shown to describe collisions in which a moving wave overtakes astanding wave. An asymptotic analysis using moving coordinates is carried outto show that, in all collisions, the speeds and modulation frequencies of theindividual waves are preserved, while the phases and positions undergo a shiftsuch that the center of momentum of the two waves moves at a constant speed.The primary constants of motion as well as some other features of the nonlinearinteraction of the colliding waves are discussed.

Oscillatory solitons of U(1)-invariant mKdV equations II: Asymptotic behavior and constants of motion