Scattering of the defocusing Calogero--Moser derivative nonlinear Schrödinger equation

Technical Summary: Scattering of the Defocusing Calogero–Moser Derivative Nonlinear Schrödinger Equation

Problem Statement
This paper investigates the long-time asymptotic behavior of solutions to the defocusing Calogero–Moser derivative nonlinear Schrödinger (CM-DNLS) equation:
$$i\partial_t u + \partial_x^2 u - 2\Pi D(|u|^2)u = 0, \quad (t, x) \in \mathbb{R} \times \mathbb{R},$$
where $\Pi$ is the Riesz-Szegő orthogonal projector onto the Hardy space $L^2_+(\mathbb{R})$ (functions with Fourier support in $[0, \infty)$). While global well-posedness for this equation in $H^s_+(\mathbb{R})$ ($s \ge 0$) has been established under small-mass assumptions or specific conditions, the scattering behavior for a broad class of initial data remains a central question. Specifically, the authors address whether solutions scatter to free Schrödinger solutions as $t \to \pm \infty$ for initial data in the weighted space $H^{1,\alpha}_+(\mathbb{R}) := \{u \in H^1_+(\mathbb{R}) : |x|^\alpha u \in L^2(\mathbb{R})\}$ with $\alpha > 0$.

Methodology
The analysis relies on the integrable structure of the equation, specifically the Lax pair formulation and the associated distorted Fourier transform. The core methodological framework involves three main components:

1. Gérard-type Explicit Formula: The authors utilize an explicit formula for the solution derived by Killip–Laurens–Vişan, which expresses the solution $u(t, z)$ in the upper half-plane via the resolvent of the Lax operator $L_{u_0}$ and the adjoint multiplication operator $X^*$:
   $$u(t, z) = \frac{1}{2i\pi} I_+ \left[ (X^* + 2tL_{u_0} - z\text{Id})^{-1} u_0 \right].$$
2. Distorted Fourier Transform: A significant portion of the work is dedicated to constructing the distorted Fourier transform $\mathcal{F}_{d, u_0}$ associated with the Lax operator $L_{u_0}$. This involves:
   • Proving the non-existence of eigenvalues for $L_{u_0}$ when $u \in H^{1,\alpha}_+$.
   • Constructing normalized generalized eigenfunctions $m_-(x, \lambda)$ and establishing their uniform $L^\infty$ bounds with respect to the spectral parameter $\lambda$.
   • Demonstrating that $\mathcal{F}_{d, u_0}$ is a unitary operator from $L^2_+(\mathbb{R})$ onto $L^2(0, \infty)$, effectively diagonalizing the Lax operator.
3. Spectral Representation of the Defect Term: The authors rewrite the explicit formula in distorted Fourier variables. A critical step is characterizing the action of the operator $X^*$ in these variables. They derive that $X^*$ acts as a differential operator $i\partial_\lambda$ plus a non-local potential term involving the half-line Hilbert transform $H$ and the potential $u$. This allows the "defect term" (the difference between the nonlinear solution and the free linear evolution) to be expressed as an oscillatory integral involving the distorted spectral data.

Key Contributions and Results
The primary result is the proof of scattering for the defocusing CM-DNLS equation for initial data in $H^{1,\alpha}_+(\mathbb{R})$ with $\alpha > 0$.

• Theorem 1.5: For any initial data $u_0 \in H^{1,\alpha}_+(\mathbb{R})$, there exist scattering states $v^\pm_0 \in H^1_+(\mathbb{R})$ such that:
  $$\lim_{t \to \pm \infty} \| u(t) - e^{it\partial_x^2} v^\pm_0 \|_{H^1(\mathbb{R})} = 0.$$
• Characterization of Scattering States: The scattering states are explicitly characterized using the distorted Fourier transform. Specifically, the negative-time scattering state $v^-_0$ satisfies $\widehat{v^-_0}(\xi) = \mathcal{F}_{d, u_0} u_0(\xi) \mathbf{1}_{\xi > 0}$. The positive-time state is derived via a symmetry argument involving the reflection of the initial data.
• Extension Beyond Rational Cases: The authors extend the application of the Gérard-type explicit formula beyond the previously studied rational initial data cases (which often rely on soliton resolution for specific finite-gap potentials) to a broad class of rougher, weighted Sobolev initial data.
• Technical Innovation: The paper introduces a novel approach to analyzing the long-time behavior of integrable equations. Instead of directly analyzing the explicit formula in physical space (as done in previous works on the Benjamin-Ono equation), the authors first transform the formula into distorted Fourier variables. They then analyze the resulting distorted representation of the defect term using oscillatory integral estimates. This spectral approach allows for the handling of initial data with limited spatial decay ($|x|^\alpha u \in L^2$ with $\alpha > 0$ rather than $\alpha \ge 1$).

Significance and Claims
The paper claims to be one of the first works to successfully apply the Gérard-type explicit formula to study the long-time behavior of an integrable equation for a broad class of initial data beyond the rational cases.

• Generalization of Scattering Results: While Kim–Kwon previously established scattering for the focusing CM-DNLS in $H^{1,1}(\mathbb{R})$ (requiring $xu \in L^2$), this work extends the chiral scattering result to the more singular space $H^{1,\alpha}_+(\mathbb{R})$ with $\alpha > 0$. This is significant because it includes ground state solutions of the focusing equation that do not belong to $H^{1,1}$.
• Spectral Characterization: The work provides a precise spectral characterization of the scattering term via the distorted Fourier transform associated with the Lax operator, offering a deeper understanding of the asymptotic dynamics in terms of the spectral data of the initial condition.
• Methodological Bridge: By combining the explicit formula with the distorted Fourier transform and oscillatory estimates, the paper establishes a framework that could potentially be adapted to other integrable systems where direct physical-space analysis of the explicit formula is intractable.

The authors note that their approach differs essentially from direct analysis of the explicit formula by first characterizing the defect term spectrally. They also mention that their method, combined with analogous results for the focusing case, suggests the possibility of establishing scattering for the focusing CM-DNLS under small-mass assumptions in the same weighted spaces, though the primary focus of this paper remains the defocusing case.