[Paper Review] On the global well-posedness of the Calogero-Sutherland derivative nonlinear Schrödinger equation

Rana Badreddine|arXiv (Cornell University)|Mar 2, 2023

Advanced Mathematical Physics Problems20 references4 citations

TL;DR

This paper establishes global well-posedness for the Calogero-Sutherland derivative nonlinear Schrödinger equation on the torus, using a Lax pair formulation to derive an explicit solution. It proves global existence and relative compactness of trajectories in all Hardy-Sobolev spaces $ H^s_+(\mathbb{T}) $ for $ s \geq 0 $, with small $ L^2 $-data in the focusing case and arbitrary $ L^2 $-data in the defocusing case.

ABSTRACT

We consider the Calogero-Sutherland derivative nonlinear Schrödinger equation in the focusing (with sign $+$) and defocusing case (with sign $-$) $$ i\partial_tu+\partial_x^2u\,\pm\,\frac2i\,\partial_xΠ(|u|^2)u=0\,,\qquad (t,x)\in\mathbb{R} imes\mathbb{T}, $$ where $Π$ is the Szegő projector $Π\left(\sum_{n\in \mathbb{Z}}\widehat{u}(n)\mathrm{e}^{inx} ight)=\sum_{n\geq 0 }\widehat{u}(n)\mathrm{e}^{inx}$. Thanks to a Lax pair formulation, we derive the explicit solution to this equation. Furthermore, we prove the global well-posedness for this $L^2$-critical equation in all the Hardy Sobolev spaces $H^s_+(\mathbb{T}),$ $s\geq0\,,$ with small $L^2$-initial data in the focusing case, and for arbitrarily $L^2$-data in the defocusing case. In addition, we establish the relative compactness of the trajectories in all $H^s_+(\mathbb{T}),$ $s\geq0\,.$

Motivation & Objective

• To establish global well-posedness for the Calogero-Sutherland derivative nonlinear Schrödinger equation on the torus in both focusing and defocusing cases.
• To extend the flow of the equation from high-regularity spaces to $ L^2_+(\mathbb{T}) $ and $ H^s_+(\mathbb{T}) $ for $ s \geq 0 $, ensuring strong convergence and mass conservation.
• To prove relative compactness of solution trajectories in all $ H^s_+(\mathbb{T}) $ spaces for $ s \geq 0 $, leveraging conservation laws and spectral properties.
• To provide an explicit solution formula using the Lax pair structure, enabling extension of the flow to low-regularity initial data.
• To address the open problem of global well-posedness in the focusing case without smallness assumptions, using the explicit solution and spectral estimates.

Proposed method

• Utilizes a Lax pair formulation to derive an explicit solution for the Calogero-Sutherland DNLS equation, based on the operator $ L_{u_0} $ and the Szegö projector $ \Pi $.
• Employs spectral theory of $ L_{u_0} $ for $ u_0 \in L^2_+(\mathbb{T}) $ to characterize the solution limit and prove weak convergence in $ L^2_+(\mathbb{T}) $.
• Establishes strong convergence in $ L^2_+(\mathbb{T}) $ and conservation of $ L^2 $-mass by analyzing the resolvent $ (\mathrm{Id} - z e^{-it} e^{-2itL_{u_0}} S^*)^{-1} $.
• Applies Young's and Sobolev inequalities to control the growth of Sobolev norms in the defocusing case, relying on conservation laws.
• Uses the explicit solution formula $ u(t,z) = \left\langle (\mathrm{Id} - z e^{-it} e^{-2itL_{u_0}} S^*)^{-1} u_0 \mid 1 \right\rangle $ to extend the flow to $ H^s_+(\mathbb{T}) $ for $ s \geq 0 $.
• Applies spectral estimates and Parseval's identity to control $ \|u(t)\|_{L^2} $ and infer relative compactness of trajectories in $ H^s_+(\mathbb{T}) $.

Experimental results

Research questions

• RQ1 Can the Calogero-Sutherland DNLS equation be globally well-posed in $ H^s_+(\mathbb{T}) $ for $ s \geq 0 $, even for large initial data in the focusing case?
• RQ2 Does the explicit solution derived from the Lax pair structure allow for the extension of the flow to $ L^2_+(\mathbb{T}) $ and low-regularity spaces?
• RQ3 What is the role of the Szegö projector and the Hardy space $ H^s_+(\mathbb{T}) $ in ensuring the integrability and global dynamics of the equation?
• RQ4 How do conservation laws and spectral properties of $ L_{u_0} $ control the growth of Sobolev norms in the defocusing case?
• RQ5 Under what conditions does the solution trajectory remain relatively compact in $ H^s_+(\mathbb{T}) $? Is this true for all $ s \geq 0 $?

Key findings

• The Calogero-Sutherland DNLS equation is globally well-posed in $ H^s_+(\mathbb{T}) $ for all $ s \geq 0 $ in the defocusing case, with arbitrary $ L^2 $-initial data.
• For the focusing case, global well-posedness holds in $ H^s_+(\mathbb{T}) $ for $ s \geq 0 $ with small $ L^2 $-initial data, extending previous results.
• The solution is explicitly given by $ u(t,z) = \left\langle (\mathrm{Id} - z e^{-it} e^{-2itL_{u_0}} S^*)^{-1} u_0 \mid 1 \right\rangle $, which enables flow extension to low-regularity spaces.
• The trajectories $ \{ \mathcal{S}^{\pm}(t)u_0 \mid t \in \mathbb{R} \} $ are relatively compact in $ H^s_+(\mathbb{T}) $ for all $ s \geq 0 $, due to uniform bounds on Sobolev norms.
• In the defocusing case, conservation laws imply uniform control of $ \|u(t)\|_{H^s} $ for $ s > \frac{3}{2} $, independent of initial data size.
• For the focusing case, the explicit solution and spectral estimates suggest that global well-posedness may hold for arbitrary initial data if the initial function lies in the cyclic subspace generated by the flow operators.