Quasiperiodic solutions of the negative-order Korteweg–de Vries hierarchy

We develop a complete algorithm for deriving quasiperiodic solutions of the negative-order KdV (nKdV) hierarchy using the backward Neumann systems. Starting with the nonlinearization of a Lax pair, the nKdV hierarchy reduces to a family of backward Neumann systems via separating temporal and spatial variables. We show that the backward Neumann systems are integrable in the Liouville sense and their involutive solutions yield finite-parameter solutions of the nKdV hierarchy. We present the negative-order Novikov equation, which specifies a finite-dimensional invariant subspace of nKdV flows. Using the Abel–Jacobi variable, we integrate the nKdV flows with Abel–Jacobi solutions on the Jacobian variety of a Riemann surface. Finally, we study the Riemann–Jacobi inversion of the Abel–Jacobi solutions, whence we obtain some quasiperiodic solutions of the nKdV hierarchy.