Long Time Decay and Asymptotics for the Complex mKdV Equation

We study the asymptotics of the complex modified Korteweg-de Vries equation {equation*} ∂_t u + ∂_x^3 u = ± |u|^2 ∂_x u {equation*} In the real valued case, it is known that solutions with small, localized initial data exhibit modified scattering for |x| ≥ t1/3, and behave self-similarly for |x| ≤ t1/3. 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 prevents us from using the scaling vector field to get control in weighted L² spaces. Instead, we must argue carefully about how wave packets at different frequencies interact in physical space while exploiting cancellations to prevent a loss of derivatives. A key ingredient in our argument is 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.