An alternative approach to the quasi-Periodic solutions of the Hunter-Saxton hierarchy

TL;DR

This paper develops algebro-geometric solutions for the entire Hunter-Saxton hierarchy using advanced mathematical tools like the zero curvature method, generalized Jacobian varieties, and inverse spectral problems.

Contribution

It introduces a new approach to construct global algebro-geometric solutions for all equations in the Hunter-Saxton hierarchy, expanding understanding of their integrability.

Findings

Explicit algebro-geometric solutions for the HS hierarchy

Application of inverse Sturm-Liouville problems

Use of generalized Jacobian and Riemann theta functions

Abstract

This paper is dedicated to provide the global solutions of algebro-geometric type for all the equations of a new commuting hierarchy containing the Hunter-Saxton (HS) equation. Our main tools include the zero curvature method to derive the HS hierarchy, the generalized Jacobian variety, the generalized Riemann theta function, the Weyl $m$-fucntions $m_\pm(x,t,z)$, and the pole motion obtained by solving an inverse problem for the Sturm-Liouville equation $L(\psi_1)=-\psi_1^{\prime\prime}=zy\psi_1$. Based on these tools and the theory of nonautonomous differential systems, topological dynamics and ergodic theory, the algebro-geometric solutions are obtained for the entire HS hierarchy.