Solving negative and mixed Toda hierarchies by Jacobi inversion problems

•Negative and mixed Toda hierarchies are presented.•They are proved to be integrable Hamiltonian systems.•The Jacobi inversion problems for the Hamiltonian systems are given.

The Jacobi inversion problems of negative and mixed Toda hierarchies are investigated through a symplectic map and some finite-dimensional Hamiltonian systems. Each negative equation is decomposed into the symplectic flow and a negative Hamiltonian flow, each mixed equation is decomposed into the symplectic flow and a mixed Hamiltonian flow. The separated variables are introduced to study these Hamiltonian systems. Based on the Hamilton–Jacobi theory, the relationship between the action-angle coordinates and the Jacobi-inversion problems is established.