Complete integrability of the Calogero--Sutherland DNLS equation on L2+(𝕋)
Abstract
We establish the complete integrability of the defocusing Calogero--Sutherland derivative nonlinear Schrödinger equation labelCS−abstracttagCS i∂t u +∂x2 u + 2i u ∂x Π (|u|2)=0, x∈𝕋, by constructing a nonlinear Fourier transform Φ: u∈ L2+(𝕋)↦ (ζn(u))n≥ 0∈ ℓ2(ℕ0), ζn(u)=√(γn(u)) e iarg⟨ u| fn⟩, built from the eigenvalues (λn), the spectral gaps γn:=λn−λn−1−1≥ 0 and the eigenfunctions (fn) of the Lax operator Lu. We prove that Φ is a norm-preserving homeomorphism. In these coordinates, the flow becomes the explicit rotation ζn(t)=e−iωn tζn(0) for all n, whose frequencies ωn depend only on the conserved spectrum (λn). As consequences, every orbit of \eqref{CS-abstract} is precompact in L2+(𝕋), and t↦ u(t) is Bohr almost periodic. For finite-gap data, we obtain quasi-periodicity in time and a sharp criterion for genuine periodicity.
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BibTeX
@misc{badreddine2026complete,
title = {{Complete integrability of the Calogero--Sutherland DNLS equation on $L^2_+(\mathbb T)$}},
author = {Rana Badreddine},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23625},
eprint = {2609.23625},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05776004v1},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Complete integrability of the Calogero--Sutherland DNLS equation on L^2_+(𝕋)
AU - Rana Badreddine
PY - 2026
DA - 2026/09/20
PB - HAL
DO - 10.48550/arxiv.2609.23625
UR - https://hal.science/hal-05776004v1
AB - We establish the complete integrability of the defocusing Calogero--Sutherland derivative nonlinear Schrödinger equation labelCS−abstracttagCS i∂_t u +∂_x^2 u + 2i u ∂_x Π (|u|^2)=0, x∈𝕋, by constructing a nonlinear Fourier transform Φ: u∈ L^2_+(𝕋)↦ (ζ_n(u))_n≥ 0∈ ℓ^2(ℕ_0), ζ_n(u)=√(γ_n(u)) e^ iarg⟨ u| f_n⟩, built from the eigenvalues (λ_n), the spectral gaps γ_n:=λ_n−λ_n−1−1≥ 0 and the eigenfunctions (f_n) of the Lax operator L_u. We prove that Φ is a norm-preserving homeomorphism. In these coordinates, the flow becomes the explicit rotation ζ_n(t)=e^−iω_n tζ_n(0) for all n, whose frequencies ω_n depend only on the conserved spectrum (λ_n). As consequences, every orbit of \eqref{CS-abstract} is precompact in L^2_+(𝕋), and t↦ u(t) is Bohr almost periodic. For finite-gap data, we obtain quasi-periodicity in time and a sharp criterion for genuine periodicity.
ER -