Estimates, asymptotics and trace formulas for periodic vector NLS equations, II

TL;DR

This work analyzes the Lax operator for the periodic vector NLS (Manakov system) with a smooth 3×3 periodic potential, revealing a spectrum consisting of bands of multiplicity 3 separated by simple gaps. The authors develop a comprehensive spectral-analytic framework: high-energy asymptotics of multipliers and branch points on a 3-sheeted Riemann surface, a Lyapunov-discriminant theory, and trace formulas that link spectral data to the VNLS Hamiltonians. Central techniques include refined monodromy-matrix asymptotics via a conjugation that handles perturbations and a conformal-mapping approach to the averaged quasimomentum, yielding explicit relations between the Hamiltonians ${\mathcal H}_0,{\mathcal H}_1,{\mathcal H}_2$ and gap data. The results provide both structural insights into the periodic Manakov system and quantitative bounds for the Hamiltonian in terms of gap lengths, with special consideration given to the 2-sheeted reduction case and the ZS-operator correspondence. These developments extend the scalar and ZS theory to the matrix/vector setting and establish a robust bridge between spectral geometry and conserved quantities of periodic VNLS.

Abstract

We consider a first order operator with a smooth periodic 3x3 matrix potential on the real line. It is the Lax operator for the periodic vector NLS equation. Its spectrum covers the real line and it is union of the spectral bands of multiplicity 3, separated by intervals (gaps) of multiplicity 1. We prove and describe the following: \\ $\cdot$ The geometry of the Riemann surface and its branch points. \\ $\cdot$ The asymptotics of branch points are determined and they are real at high energy. \\ $\cdot$ Trace formulas for integral of motions, including the Hamiltonian of the NLS equation. \\ $\cdot$ Estimates of the Hamiltonian in terms of gap lengths. The proof is based on the analysis of averaged quasi-momentum as a conformal mapping of the upper half plane on the domain on the upper half plane and on the asymptotics of the monodromy matrix and multipliers at high energy.

Figures (3)

• Figure 1: The function $\mathfrak{D}$, the Lyapunov function $\Delta$ and the spectrum $\mathfrak{S}_3$
• Figure 2: $\Delta_1$, $\Delta_2$, and $\Delta_3$ are Lyapunov functions at high energy, $z_{1,n}$, $z_{2,n}$, and $z_{3,n}$ are periodic eigenvalues. The dotted curve is the case where there are no branching points . The solid curve is the case where there are 4 branching points
• Figure 3: The domain ${\mathbb K}$ and $k_n^\pm=k(\lambda_n^\pm)$