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Travelling waves and conservation laws are studied for a wide class ofU(1)-invariant complex mKdV equations containing the two known integrablegeneralizations of the ordinary (real) mKdV equation. The main results ontravelling waves include deriving new complex solitary waves and kinks thatgeneralize the well-known mKdV $\sech$ and $\tanh$ solutions. The main resultson conservation laws consist of explicitly finding all 1st order conserveddensities that yield phase-invariant counterparts of the well-known mKdVconserved densities for momentum, energy, and Galilean energy, and a newconserved density describing the angular twist of complex kink solutions

Travelling waves and conservation laws for complex mKdV-type equations