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Well-posedness and invariant measures for complex valued modified KdV equation

Carlos E. Kenig, Andrea R. Nahmod, Nataša Pavlović, Gigliola Staffilani, Nicola Visciglia

TL;DR

This work analyzes the 1D periodic complex-valued mKdV equation in the NLS hierarchy, proving a deterministic well-posedness theory for $s>\tfrac{4}{3}$ and constructing a sequence of invariant, higher-regularity Gaussian measures tied to the conservation laws $E_{2n+1}$. The authors develop a finite-dimensional approximation $\Phi_N(t)$, establish uniform bounds, and prove convergence to the full flow $\Phi(t)$, enabling almost sure globalization on large Sobolev spaces. The probabilistic results show the existence of full-measure sets $\Sigma^s$ on which global solutions exist with at most logarithmic growth in $H^s$, and the associated weighted Gaussian measures $\rho_{n,R}$ are invariant under the nonlinear flow on these sets. The methodology extends Zhidkov’s approach to the complex-valued context and generalizes to higher conservation laws, offering a robust probabilistic framework for invariant measures beyond Gibbs, with potential implications for the statistical mechanics of the NLS/mKdV hierarchy.

Abstract

We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported on Sobolev spaces with increasing regularity. The fact that we work with complex valued functions makes the analysis of the invariance much harder compared to the real valued case, that can be handled instead following the ideas used by Zhidkov [73].

Well-posedness and invariant measures for complex valued modified KdV equation

TL;DR

This work analyzes the 1D periodic complex-valued mKdV equation in the NLS hierarchy, proving a deterministic well-posedness theory for and constructing a sequence of invariant, higher-regularity Gaussian measures tied to the conservation laws . The authors develop a finite-dimensional approximation , establish uniform bounds, and prove convergence to the full flow , enabling almost sure globalization on large Sobolev spaces. The probabilistic results show the existence of full-measure sets on which global solutions exist with at most logarithmic growth in , and the associated weighted Gaussian measures are invariant under the nonlinear flow on these sets. The methodology extends Zhidkov’s approach to the complex-valued context and generalizes to higher conservation laws, offering a robust probabilistic framework for invariant measures beyond Gibbs, with potential implications for the statistical mechanics of the NLS/mKdV hierarchy.

Abstract

We consider the one dimensional periodic complex valued mKdV, which corresponds to the first equation above cubic NLS in the associated integrable hierarchy. Our main result is the construction of a sequence of invariant measures supported on Sobolev spaces with increasing regularity. The fact that we work with complex valued functions makes the analysis of the invariance much harder compared to the real valued case, that can be handled instead following the ideas used by Zhidkov [73].
Paper Structure (22 sections, 28 theorems, 297 equations)

This paper contains 22 sections, 28 theorems, 297 equations.

Table of Contents

  1. Introduction
  2. Our results in a nutshell
  3. The deterministic result
  4. The probabilistic result
  5. Organization of the paper
  6. Notation
  7. Finite dimensional approximation and proof of Theorem \ref{['Main']}
  8. Energy and dispersive estimate for the flow $\Phi_N(t)$
  9. Proof of Theorem \ref{['Main']}
  10. Long time convergence of $\Phi_N(t)$ to $\Phi(t)$
  11. Weighted Gaussian measures, pairing and multilinear gaussian estimates
  12. Gaussian measures and basic properties
  13. Weighted Gaussian measures
  14. Pairing and multilinear products of Gaussians
  15. Almost invariance of $\rho_{2,N,R}$ along the flow $\Phi_N(t)$
  16. ...and 7 more sections

Key Result

Theorem 1.2

Assume $u_0\in H^s({\mathbb{T}})$ for $s>4/3$. Then the Cauchy problem data admits a unique local strong solution $u\in {\mathcal{C}}([-T,T],H^s({\mathbb{T}}))$, where $T=T(\|u_0\|_{H^s({\mathbb{T}})})>0$.

Theorems & Definitions (61)

  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Proposition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 51 more