Asymptotics of spectral quantities of Zakharov-Shabat operators

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.

Estimates for periodic Zakharov-Shabat operators

We consider the periodic Zakharov-Shabat operators on the real line. The spectrum of this operator consists of intervals separated by gaps with the lengths $|g_n|\ge 0, n\in \Z$. Let $\m_n^\pm$ be the corresponding effective masses and let $h_n$ be heights of the corresponding slits in the quasimomentum domain. We obtain a priori estimates of sequences $g=(|g_n|)_{n\in \Z},\m^\pm=(\m_n^\pm)_{n\in \Z}, h=(h_n)_{n\in \Z}$ in terms of weighted $\ell^p-$norms at $p\ge 1$. The proof is based on the analysis of the quasimomentum as the conformal mapping.

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.