# Quasi-periodic solutions to the negative-order KdV hierarchy

## TL;DR

This paper develops a comprehensive method to derive quasi-periodic solutions for the negative-order KdV hierarchy using backward Neumann systems, integrability, and algebraic geometry techniques involving Riemann surfaces.

## Contribution

It introduces a novel algorithm linking backward Neumann systems with the nKdV hierarchy and employs Abel-Jacobi variables for explicit solution construction.

## Key findings

- Derived finite parametric solutions for nKdV hierarchy.
- Established integrability of backward Neumann systems in Liouville sense.
- Obtained explicit quasi-periodic solutions via Riemann surface methods.

## Abstract

A complete algorithm is developed to deduce quasi-periodic solutions for the negative-order KdV (nKdV) hierarchy by using the backward Neumann systems. From the nonlinearization of Lax pair, the nKdV hierarchy is reduced to a family of backward Neumann systems via separating temporal and spatial variables. The backward Neumann systems are shown to be integrable in the Liouville sense, whose involutive solutions yield the finite parametric solutions of nKdV hierarchy. The negative-order Novikov equation is given, which specifies a finite-dimensional invariant subspace of nKdV flows. By the Abel-Jacobi variable, the nKdV flows are integrated with Abel-Jacobi solutions on the Jacobi variety of a Riemann surface. Finally, the Riemann-Jacobi inversion of Abel--Jacobi solutions is studied, from which some quasi-periodic solutions to the nKdV hierarchy are obtained.