On the spectrum of the Lax operator of the Benjamin-Ono equation on the torus. (English) Zbl 1452.37070
The aim of this paper is to investigate the spectrum of the Lax operator \(L_u\) of the Benjamin-Ono equation [T. B. Benjamin, J. Fluid Mech. 29, 559–592 (1967; Zbl 0147.46502); H. Ono, J. Phys. Soc. Japan 39, No. 4, 1082–1091 (1975; Zbl 1334.76027)]:
\[
\partial_tu=H\partial^2-xu-\partial_x(u^2),\quad (t,x)\in\mathbb{\times\mathbb{T}},\quad \mathbb{T}=\mathbb{R}/2\pi\mathbb{Z},
\]
for complex-valued potentials \(u\) in the Sobolev space \(H^{-s}(\mathbb{T}, \mathbb{C})\), \(0\leq s<1/2\), with small imaginary part. They prove some analytic properties of the moment map defined in terms of spectral data of \(L_u\).
Reviewer: Ahmed Lesfari (El Jadida)
MSC:
| 37K20 | Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions |
| 37K15 | Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems |
References:
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