57 research outputs found
Functional representation of the Ablowitz-Ladik hierarchy. II
In this paper I continue studies of the functional representation of the
Ablowitz-Ladik hierarchy (ALH). Using formal series solutions of the
zero-curvature condition I rederive the functional equations for the
tau-functions of the ALH and obtain some new equations which provide more
straightforward description of the ALH and which were absent in the previous
paper. These results are used to establish relations between the ALH and the
discrete-time nonlinear Schrodinger equations, to deduce the superposition
formulae (Fay's identities) for the tau-functions of the hierarchy and to
obtain some new results related to the Lax representation of the ALH and its
conservation laws. Using the previously found connections between the ALH and
other integrable systems I derive functional equations which are equivalent to
the AKNS, derivative nonlinear Schrodinger and Davey-Stewartson hierarchies.Comment: arxiv version is already officia
Functional representation of the negative AKNS hierarchy
This paper is devoted to the negative flows of the AKNS hierarchy. The main
result of this work is the functional representation of the extended AKNS
hierarchy, composed of both positive (classical) and negative flows. We derive
a finite set of functional equations, constructed by means of the Miwa's
shifts, which contains all equations of the hierarchy. Using the obtained
functional representation we convert the nonlocal equations of the negative
subhierarchy into local systems of higher order, derive the generating function
of the conservation laws and the N-dark-soliton solutions for the extended AKNS
hierarchy. As an additional result we obtain the functional representation of
the Landau-Lifshitz hierarchy.Comment: 20 page
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