Estimates for periodic Zakharov-Shabat operators

Asymptotics of spectral quantities of Zakharov-Shabat operators

In this paper we provide new asymptotic estimates of various spectral quantities of Zakharov-Shabat operators on the circle. These estimates are uniform on bounded subsets of potentials in Sobolev spaces.

Generic non-selfadjoint Zakharov-Shabat operators

In this paper we develop tools to study families of non-selfadjoint operators $L(φ), φ\in P$, characterized by the property that the spectrum of $L(φ)$ is (partially) simple. As a case study we consider the Zakharov-Shabat operators $L(φ)$ appearing in the Lax pair of the focusing NLS on the circle. The main result says that the set of potentials $φ$ of Sobolev class $H^N, N \geq 0$, so that all small eigenvalues of $L(φ)$ are simple, is path connected and dense.

The WKB approximation of semiclassical eigenvalues of the Zakharov-Shabat problem

We numerically compute eigenvalues of the non-self-adjoint Zakharov--Shabat problem in the semiclassical regime. In particular, we compute the eigenvalues for a Gaussian potential and compare the results to the corresponding (formal) WKB approximations used in the approach to the semiclassical or zero-dispersion limit of the focusing nonlinear Schroedinger equation via semiclassical soliton ensembles. This numerical experiment, taken together with recent numerical experiments [17,18], speaks directly to the viability of this approach; in particular, our experiment suggests a value for the rate…

周期聚焦Zakharov-Shabat算子的谱

On the spectrum of the periodic focusing Zakharov-Shabat operator

研究圆上聚焦Zakharov-Shabat算子的谱，证明预解集由两个连通分量组成，获得Floquet与Dirichlet谱的新界，并揭示半经典极限下谱局部化为“十字”形。

Discrete skew selfadjoint canonical systems and the isotropic Heisenberg magnet model

A discrete analog of a skew selfadjoint canonical (Zakharov-Shabat or AKNS) system with a pseudo-exponential potential is introduced. For the corresponding Weyl function the direct and inverse problem are solved explicitly in terms of three parameter matrices. As an application explicit solutions are obtained for the discrete integrable nonlinear equation corresponding to the isotropic Heisenberg magnet model. State space techniques from mathematical system theory play an important role in the proofs.