The focusing Calogero--Moser derivative nonlinear Schrödinger equation in the Hardy--Zhidkov space: explicit flow formula and dark multi-solitons
Abstract
This paper studies the focusing Calogero--Moser derivative nonlinear Schrödinger (CMdNLS) equation under the nonzero boundary condition |u(t,x)|→ 1 as |x|→+∞. In sharp contrast to the classical paradigm for the cubic nonlinear Schrödinger equation, namely that focusing nonlinearities support only bright solitons on a zero background, we show that the focusing CMdNLS equation admits a rich family of \emph{dark multi-soliton} solutions on a nonzero background, whose existence is driven by a spectral gap [−1,0] in the point spectrum of the Lax operator, together with the Hardy space structure. The main results are threefold. First, we establish an explicit formula for general solutions in the Hardy--Zhidkov space 𝒵2+, via a \emph{semigroup method} based on new commutator estimates between the Lax--Beurling shift semigroup and Toeplitz operators. This approach circumvents the non-commutativity of the unbounded generator with the time derivative and requires no weighted hypothesis on solutions or on the domain of validity of commutator formulas. Second, the algebraic definition of dark multi-soliton potentials is shown to be equivalent to two spectral conditions on the Lax operator. Together with the conservation of a generating functional, this yields the invariance of the dark N-soliton set 𝒰N under the CMdNLS flow. Third, the action-angle variables on 𝒰N are constructed, which establishes the complete integrability of the focusing CMdNLS equation on 𝒰N. Combining these results, we obtain an inverse spectral formula, from which it follows that the poles of dark multi-soliton solutions satisfy a complexified Calogero--Moser system. Long time asymptotics and uniform Sobolev estimates are also derived.
Related
- Soliton resolution conjecture for the Calogero--Moser derivative nonlinear Schrödinger equation in the scaling-critical space
- Threshold Scattering for the Three-Dimensional Focusing Energy-Critical Nonlinear Schrödinger Equation
- Global well-posedness for the derivative nonlinear Schrödinger equation on the circle
- Explicit quasi-periodic solutions to the discrete derivative nonlinear Schrödinger equation
- Slow blowup solutions for the focusing energy supercritical nonlinear Schrödinger equation
BibTeX
@misc{sun2026focusing,
title = {{The focusing Calogero--Moser derivative nonlinear Schrödinger equation in the Hardy--Zhidkov space: explicit flow formula and dark multi-solitons}},
author = {Ruoci Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00955},
eprint = {2610.00955},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00955},
}
RIS
TY - UNPB
TI - The focusing Calogero--Moser derivative nonlinear Schrödinger equation in the Hardy--Zhidkov space: explicit flow formula and dark multi-solitons
AU - Ruoci Sun
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00955
UR - https://arxiv.org/abs/2610.00955
AB - This paper studies the focusing Calogero--Moser derivative nonlinear Schrödinger (CMdNLS) equation under the nonzero boundary condition |u(t,x)|→ 1 as |x|→+∞. In sharp contrast to the classical paradigm for the cubic nonlinear Schrödinger equation, namely that focusing nonlinearities support only bright solitons on a zero background, we show that the focusing CMdNLS equation admits a rich family of \emph{dark multi-soliton} solutions on a nonzero background, whose existence is driven by a spectral gap [−1,0] in the point spectrum of the Lax operator, together with the Hardy space structure. The main results are threefold. First, we establish an explicit formula for general solutions in the Hardy--Zhidkov space 𝒵^2_+, via a \emph{semigroup method} based on new commutator estimates between the Lax--Beurling shift semigroup and Toeplitz operators. This approach circumvents the non-commutativity of the unbounded generator with the time derivative and requires no weighted hypothesis on solutions or on the domain of validity of commutator formulas. Second, the algebraic definition of dark multi-soliton potentials is shown to be equivalent to two spectral conditions on the Lax operator. Together with the conservation of a generating functional, this yields the invariance of the dark N-soliton set 𝒰_N under the CMdNLS flow. Third, the action-angle variables on 𝒰_N are constructed, which establishes the complete integrability of the focusing CMdNLS equation on 𝒰_N. Combining these results, we obtain an inverse spectral formula, from which it follows that the poles of dark multi-soliton solutions satisfy a complexified Calogero--Moser system. Long time asymptotics and uniform Sobolev estimates are also derived.
ER -