Quasi-Periodic Solutions to the Nonlocal Nonlinear Schrödinger Equations

A hierarchy of nonlocal nonlinear Schrodinger equations is derived by using the Lenard gradients and the zero-curvature equation. According to the Lax matrix of the nonlocal nonlinear Schrodinger equations, we introduce a hyperelliptic Riemann surface Kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {K}}_{n}$$\end{document} of genus n, from which Dubrovin-type equations, meromorphic function, and Baker-Akhiezer function are established. By the theory of algebraic curves, the corresponding flows are straightened by resorting to the Abel-Jacobi coordinates. Finally, we obtain the explicit Riemann theta function representations of the Baker-Akhiezer function, specifically, that of solutions for the hierarchy of nonlocal nonlinear Schrodinger equations in regard to the asymptotic properties of the Baker-Akhiezer function.