Generalized Miura Transformations, Two-Boson KP Hierarchies and their Reduction to KDV Hierarchies

Bracket preserving gauge equivalence is established between several two-boson generated KP type of hierarchies. These KP hierarchies reduce under symplectic reduction (via Dirac constraints) to KdV, mKdV and Schwarzian KdV hierarchies. Under this reduction the gauge equivalence is taking form of the conventional Miura maps between the above KdV type of hierarchies.Comment: 12 pgs., LaTeX, IFT-P/011/93, UICHEP-TH/93-

Completion of the Ablowitz-Kaup-Newell-Segur integrable coupling

Integrable couplings are associated with non-semisimple Lie algebras. In this paper, we propose a new method to generate new integrable systems through making perturbation in matrix spectral problems for integrable couplings, which is called the `completion process of integrable couplings'. As an example, the idea of construction is applied to the Ablowitz-Kaup-Newell-Segur integrable coupling. Each equation in the resulting hierarchy has a bi-Hamiltonian structure furnished by the component-trace identity

Generalized r-matrix structure and algebro-geometric solution for integrable systems

The purpose of this paper is to construct a generalized r-matrix structure of finite dimensional systems and an approach to obtain the algebro-geometric solutions of integrable nonlinear evolution equations (NLEEs). Our starting point is a generalized Lax matrix instead of usual Lax pair. The generalized r-matrix structure and Hamiltonian functions are presented on the basis of fundamental Poisson bracket. It can be clearly seen that various nonlinear constrained (c-) and restricted (r-) systems, such as the c-AKNS, c-MKdV, c-Toda, r-Toda, c-Levi, etc, are derived from the reduction of this structure. All these nonlinear systems have {\it r}-matrices, and are completely integrable in Liouville's sense. Furthermore, our generalized structure is developed to become an approach to obtain the algebro-geometric solutions of integrable NLEEs. Finally, the two typical examples are considered to illustrate this approach: the infinite or periodic Toda lattice equation and the AKNS equation with the condition of decay at infinity or periodic boundary.Comment: 41 pages, 0 figure

Meromorphic Lax representations of (1+1)-dimensional multi-Hamiltonian dispersionless systems

Rational Lax hierarchies introduced by Krichever are generalized. A systematic construction of infinite multi-Hamiltonian hierarchies and related conserved quantities is presented. The method is based on the classical R-matrix approach applied to Poisson algebras. A proof, that Poisson operators constructed near different points of Laurent expansion of Lax functions are equal, is given. All results are illustrated by several examples.Comment: 28 page