Quasiperiodic Solutions of the Negative-Order Korteweg-De Vries Hierarchy

Abstract

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.