  We study the asymptotics of the complex modified Korteweg-de Vries equation
$\partial_t u + \partial_x^3 u = -|u|^2 \partial_x u$, which can be used to
model vortex filament dynamics. In the real-valued case, it is known that
solutions with small, localized initial data exhibit modified scattering for
$|x| \geq t^{1/3}$ and behave self-similarly for $|x| \leq t^{1/3}$. We prove
that the same asymptotics hold for complex mKdV. The major difficulty in the
complex case is that the nonlinearity cannot be expressed as a derivative,
which makes the low-frequency dynamics harder to control. To overcome this
difficulty, we introduce the decomposition $u = S + w$, where $S$ is a
self-similar solution with the same mean as $u$ and $w$ is a remainder that has
better decay. By using the explicit expression for $S$, we are able to get
better low-frequency behavior for $u$ than we could from dispersive estimates
alone. An advantage of our method is its robustness: It does not depend on the
precise algebraic structure of the equation, and as such can be more readily
adapted to other contexts.
