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On ill-posedness for the Gabitov--Turitsyn equation

Matthew Kowalski

TL;DR

The Gabitov--Turitsyn equation, modeling dispersion-managed optical-fiber pulse dynamics, exhibits two scaling pseudo-symmetries that define critical regularities $s_m$ and $s_i$. The authors establish local well-posedness with analytic dependence for $s \ge \max(s_m,0)$ and show ill-posedness in the analytic sense for $s < s_m$ via a power-series framework. They further demonstrate norm inflation in Sobolev spaces for $s < s_i$ in several regimes: mass-subcritical in the 1D cubic GT and energy-supercritical with $1 \le s < s_i$, using a combination of multilinear estimates and a virial-identity–driven energy equipartition mechanism. A key contribution is the use of two distinct scaling pseudo-symmetries to chart the boundary between stable and unstable dynamics, with implications for both the mathematical theory and fiber-optic modeling of long-time pulse evolution.

Abstract

We investigate the well- and ill-posedness theory for the Gabitov--Turitsyn equation, which models the long-time dynamics of pulses in dispersion-managed optical fibers. We identify two critical regularities, corresponding to two scaling pseudo-symmetries, that demarcate regimes of ill-posedness. First, we identify $s_m = \tfrac{d}{2} - \tfrac{2}{p},$ coinciding with the monomial NLS. For $s \geq \max(s_m,0)$, local well-posedness is known to hold in $H^s$, while for $s < s_m$, we show that the data-to-solution map fails to be $C^{p+1}$ in $H^s$. Second, we identify $s_i = \tfrac{d}{2} - \tfrac{4}{p},$ below which we conjecture that norm inflation occurs. We resolve this conjecture in $H^s$ in the case $s < \min (s_i, 0)$ -- specifically for the one-dimensional cubic model -- and in the case $1 \leq s < s_i$. In the case $s_i \geq 1$, we establish norm inflation by showing that suitable solutions undergo {\em energy equipartition}: a rapid renormalization of kinetic and potential energy.

On ill-posedness for the Gabitov--Turitsyn equation

TL;DR

The Gabitov--Turitsyn equation, modeling dispersion-managed optical-fiber pulse dynamics, exhibits two scaling pseudo-symmetries that define critical regularities and . The authors establish local well-posedness with analytic dependence for and show ill-posedness in the analytic sense for via a power-series framework. They further demonstrate norm inflation in Sobolev spaces for in several regimes: mass-subcritical in the 1D cubic GT and energy-supercritical with , using a combination of multilinear estimates and a virial-identity–driven energy equipartition mechanism. A key contribution is the use of two distinct scaling pseudo-symmetries to chart the boundary between stable and unstable dynamics, with implications for both the mathematical theory and fiber-optic modeling of long-time pulse evolution.

Abstract

We investigate the well- and ill-posedness theory for the Gabitov--Turitsyn equation, which models the long-time dynamics of pulses in dispersion-managed optical fibers. We identify two critical regularities, corresponding to two scaling pseudo-symmetries, that demarcate regimes of ill-posedness. First, we identify coinciding with the monomial NLS. For , local well-posedness is known to hold in , while for , we show that the data-to-solution map fails to be in . Second, we identify below which we conjecture that norm inflation occurs. We resolve this conjecture in in the case -- specifically for the one-dimensional cubic model -- and in the case . In the case , we establish norm inflation by showing that suitable solutions undergo {\em energy equipartition}: a rapid renormalization of kinetic and potential energy.

Paper Structure

This paper contains 18 sections, 10 theorems, 109 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Monomial scaling pseudo-symmetry
  3. Integrated scaling pseudo-symmetry
  4. Connection to fiber-optics
  5. Preliminaries and linear theory
  6. Analytic local well-posedness
  7. Definition of the power series
  8. Quantitative local well-posedness
  9. Global existence for small initial data
  10. Failure of analytic well-posedness
  11. Mass sub-critical norm inflation
  12. Multilinear estimates
  13. Proof of Theorem \ref{['intro/inflation-negative']}
  14. Energy super-critical norm inflation
  15. Virial identity
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $u_0 \in \dot{H}^{s_m \vee 0}$ and $T \sim \|u_0\|^{-p}_{\dot{H}^{s_m \vee 0}}$, there exists a unique solution to GT: Moreover, the data-to-solution map is real analytic on a neighborhood of $u_0 = 0$: for $\|u_0\|_{\dot{H}^{s_m \vee 0}} \leq R$ and $T \sim R^{-p}$ the data-to-solution map satisfies a power series expansion; see lwp/power-series. In addition, there exists $\delta = \delta(p

Figures (1)

  • Figure 1: The change of variables $\tau = \sigma + s$ and $\rho = s - \sigma$ in the case $T < 1$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.3: Norm inflation in $H^s$
  • Theorem 1.4
  • Proposition 1.5: Virial identity
  • Proposition 1.6: Energy equipartition
  • Theorem 1.7
  • Definition 2.1: Schrödinger-admissible
  • Proposition 2.2: Shifted Strichartz estimates
  • Proposition 3.1: Quantitative well-posedness
  • ...and 11 more