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Long time decay and asymptotics for the complex mKdV equation

Gavin Stewart

TL;DR

The paper analyzes the long-time behavior of the complex mKdV equation with small localized data, proving that its asymptotics match those known for the real case. A key innovation is decomposing the solution as $u=S+w$, with $S$ a self-similar profile whose mean tracks the evolving zero-mode, and $w$ a remainder whose low-frequency content decays faster thanks to cancellations in $|S|^2\partial_x S$. The authors develop a robust, largely algebra-structure–independent approach via the space-time resonance method, obtaining precise region-wise asymptotics: rapid decay to the right, modified scattering on the left, and self-similar behavior in the core region, all governed by a Painlevé II–type self-similar equation. They further establish weighted energy estimates and a careful Fourier-space analysis to close a bootstrap and prove global existence for small data. The techniques are adaptable to other complex mKdV–type equations and highlight a robust framework for Painlevé-type long-time asymptotics beyond complete integrability.

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.

Long time decay and asymptotics for the complex mKdV equation

TL;DR

The paper analyzes the long-time behavior of the complex mKdV equation with small localized data, proving that its asymptotics match those known for the real case. A key innovation is decomposing the solution as , with a self-similar profile whose mean tracks the evolving zero-mode, and a remainder whose low-frequency content decays faster thanks to cancellations in . The authors develop a robust, largely algebra-structure–independent approach via the space-time resonance method, obtaining precise region-wise asymptotics: rapid decay to the right, modified scattering on the left, and self-similar behavior in the core region, all governed by a Painlevé II–type self-similar equation. They further establish weighted energy estimates and a careful Fourier-space analysis to close a bootstrap and prove global existence for small data. The techniques are adaptable to other complex mKdV–type equations and highlight a robust framework for Painlevé-type long-time asymptotics beyond complete integrability.

Abstract

We study the asymptotics of the complex modified Korteweg-de Vries equation , 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 and behave self-similarly for . 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 , where is a self-similar solution with the same mean as and is a remainder that has better decay. By using the explicit expression for , we are able to get better low-frequency behavior for 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.

Paper Structure

This paper contains 56 sections, 11 theorems, 395 equations.

Table of Contents

  1. Introduction
  2. Known results
  3. Main results
  4. $x \geq t^{1/3}$
  5. $x \leq -t^{-1/3}$
  6. $|x| \leq t^{1/3 + 4\beta}$
  7. Main difficulties
  8. Plan of the proof
  9. Overview of the space-time resonance method
  10. Step 1: Stationary phase estimate for high frequencies
  11. Step 2: Modulation analysis in the self-similar region
  12. Step 3: Weighted bounds for w
  13. On the structure of the nonlinearity
  14. Organization of the paper
  15. Preliminaries
  16. ...and 41 more sections

Key Result

Theorem 1.1

There exists an $\epsilon_0 > 0$ such that if $\epsilon < \epsilon_0$, and $u_* \in H^2$ satisfies then the solution $u$ to eqn:cmkdv-t-1 exists on $[1,\infty)$ and has the following asymptotics in different regions of physical space:

Theorems & Definitions (33)

  • Theorem 1.1
  • Remark 1
  • Remark 2
  • Corollary 1.2
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 2.1
  • ...and 23 more