Long time decay and asymptotics for the complex mKdV equation

TL;DR

This paper establishes the long-time decay and asymptotic behavior of solutions to the complex mKdV equation, extending known results from the real-valued case and introducing a decomposition method to handle non-derivative nonlinearities.

Contribution

The authors develop a novel decomposition approach for the complex mKdV, enabling control of low-frequency dynamics and extending asymptotic analysis techniques to complex-valued solutions.

Findings

Solutions exhibit modified scattering similar to the real case.

The decomposition $u = S + w$ improves low-frequency control.

Method is robust and adaptable to other equations.

Abstract

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.