Nowhere continuity of the flow map of an integrable derivative nonlinear Schrödinger system on the torus

Technical Summary: Nowhere Continuity of the Flow Map of an Integrable Derivative Nonlinear Schrödinger System on the Torus

Problem Statement
This paper investigates the well-posedness of the Cauchy problem for a completely integrable derivative nonlinear Schrödinger (DNLS) system of the Chen-Lee-Liu type on the torus $\mathbb{T} = \mathbb{R}/2\pi\mathbb{Z}$. The system is given by:
$$\begin{cases} i\partial_t u - \partial_x^2 u - iuv\partial_x u = 0, \\ i\partial_t v + \partial_x^2 v - iuv\partial_x v = 0, \end{cases}$$
with initial data $(u_0, v_0) \in H^s(\mathbb{T}) \times H^s(\mathbb{T})$ for $s > 3/2$. While the system is known to be completely integrable with infinite conserved quantities (e.g., mass $M(u,v) = \int_{\mathbb{T}} uv \, dx$), the regularity of the flow map (the mapping from initial data to solutions) in Sobolev spaces remains the central question. The author addresses whether the flow map is continuous in $H^s(\mathbb{T})$ and identifies conditions under which well-posedness holds.

Methodology
The analysis relies on gauge transformations and the exploitation of smoothing effects associated with Cauchy-Riemann-type elliptic operators.

1. Gauge Transformation: To handle the derivative loss in the nonlinear term $uv\partial_x u$, the author applies a gauge transformation:
   $$U = \exp\left(-\frac{i}{2}\partial_x^{-1}(uv)\right)u, \quad V = \exp\left(\frac{i}{2}\partial_x^{-1}(uv)\right)v.$$
   This transformation converts the original system into a new system for $(U, V)$ containing a term proportional to $\partial_x U$ (and $\partial_x V$) with a coefficient determined by the conserved mass $M(u,v)$. Specifically, the transformed equation includes a term $i M(u,v) \partial_x U$.

2. Analysis of the Transformed System: The transformed system reveals that if the imaginary part of the conserved mass, $\text{Im } M(u,v)$, is non-zero, the system formally contains a Cauchy-Riemann-type operator $i\partial_t + a\partial_x$ (where $a \in \mathbb{R}$). The author utilizes the smoothing properties of this operator to derive necessary conditions on the initial data for a solution to exist.

3. Construction of Counterexamples: To prove discontinuity, the author constructs sequences of initial data converging to a given point in $H^s(\mathbb{T})$ but possessing properties (specifically regarding the Fourier support of the data relative to the sign of $\text{Im } M$) that violate the necessary conditions derived from the smoothing effect. This leads to the non-existence of solutions for these sequences.

4. Contraction Mapping: For the well-posedness result, the author restricts the initial data to the subspace where $\text{Im } M(u_0, v_0) = 0$. In this subspace, the problematic linear term in the gauge-transformed system vanishes, allowing the application of a standard contraction mapping argument in a suitable Banach space.

Key Contributions and Results

• Theorem 1.2 (Nowhere Continuity): The flow map of the system (1.1) is discontinuous at every point in $H^s(\mathbb{T})$ for $s > 3/2$. Specifically, for any initial data $(u_0, v_0) \in H^s(\mathbb{T})$, there exists a sequence $\{(u_{0,n}, v_{0,n})\}$ converging to $(u_0, v_0)$ such that no corresponding solution exists in $H^s(\mathbb{T})$ on any time interval $[0, T]$ or $[-T, 0]$. This result holds regardless of the integrability of the system.

  • Mechanism: The proof (Theorem 2.1) establishes that if a solution exists and $\text{Im } M(u_0, v_0) > 0$ (or $<0$), the initial data must satisfy specific regularity conditions on its negative (or positive) frequency components (i.e., $(P_- u_0, P_- v_0) \in H^{s+\delta}$). By constructing sequences that converge in $H^s$ but violate these higher regularity conditions, the author demonstrates non-existence.

• Theorem 1.3 (Conditional Well-Posedness): The Cauchy problem is well-posed in the subspace $H^s_0(\mathbb{T}) := \{(f, g) \in H^s(\mathbb{T}) \mid \text{Im } M(f, g) = 0\}$.

  • Result: For initial data in this subspace, there exists a unique local solution, and the flow map is continuous.
  • Significance: This condition $\text{Im } M = 0$ is shown to be the precise requirement to eliminate the ill-posedness caused by the Cauchy-Riemann-type term in the gauge-transformed equation.

• Applications to Specific Models:

  • The result implies that the standard DNLS equation (obtained by setting $v=u$) is well-posed in $H^s(\mathbb{T})$ because for $v=u$, $M(u,u) = \int |u|^2 dx$ is real, satisfying the condition.
  • The nonlocal DNLS equation (derived via $v = u^*$ where $u^*(t,x) = u(t, -x)$) is well-posed in the space of functions with Fourier support in the positive half-line ($H^s_+(\mathbb{T})$), as this also ensures $\text{Im } M = 0$.

Significance and Claims
The paper claims to resolve the well-posedness of this specific integrable system in Sobolev spaces by demonstrating that integrability alone does not guarantee the continuity of the flow map. The author emphasizes that while the system possesses infinite conserved quantities, the presence of the term $\text{Im } M(u,v) \partial_x$ in the gauge-transformed variables creates a mechanism for ill-posedness (specifically, the failure of the flow map to be continuous) whenever $\text{Im } M \neq 0$.

The work highlights a sharp dichotomy: the system is ill-posed in the full space $H^s(\mathbb{T})$ but becomes well-posed when restricted to the subspace defined by the vanishing of the imaginary part of the mass. The author notes that the regularity condition $s > 3/2$ is not claimed to be optimal but is sufficient for the arguments presented. The findings suggest that for periodic derivative nonlinear Schrödinger systems, the "integrable" structure requires careful handling of gauge transformations and conservation laws to establish well-posedness, as the standard gauge transformation used in the real line setting does not fully remove the constant term in the periodic case without imposing the $\text{Im } M = 0$ condition.