Long time decay and asymptotics for the complex mKdV equation

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 $|x|\ge t^{1/3}$ and behave self-similarly for $|x|\le t^{1/3}$. 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.

59 pages; expanded discussion on self-similar solutions to create Section 4

References in corpus (3)

• Few-cycle optical rogue waves:complex modified Korteweg-de Vries equation
• Spectral analysis and long-time asymptotics of complex mKdV equation
• The Einstein-Klein-Gordon coupled system: global stability of the Minkowski solution

Cited by in corpus (1)

• Asymptotics for small data solutions of the Ablowitz-Ladik equation