Reductions to Korteweg-de Vries Soliton Hierarchy

Based on the nonlinearization of Lax pairs,the Korteweg-de Vries (KdV) soliton hierarchy is decomposedinto a family of finite-dimensional Hamiltonian systems,whose Liouville integrability is proved by means of the ellipticcoordinates.By applying the Abel-Jacobi coordinates on a Riemann surface of hyperelliptic curve,the resulting Hamil-tonian flows as well as the KdV soliton hierarchy are ultimately reduced into linear superpositions,expressed by theAbel-Jacobi variables.