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Chains of KP, Semi-infinite 1-Toda Lattice Hierarchy and Kontsevich Integral
There are well-known constructions of integrable systems which are chains of
infinitely many copies of the equations of the KP hierarchy ``glued'' together
with some additional variables, e.g., the modified KP hierarchy. Another
interpretation of the latter, in terms of infinite matrices, is called the
1-Toda lattice hierarchy. One way infinite reduction of this hierarchy has all
solutions in the form of sequences of expanding Wronskians. We define another
chain of the KP equations, also with solutions of the Wronsksian type, which is
characterized by the property to stabilize with respect to a gradation. Under
some constraints imposed, the tau functions of the chain are the tau functions
associated with the Kontsevich integrals.Comment: LaTeX, 15 page
On the constrained KP hierarchy
An explanation for the so-called constrained hierarhies is presented by
linking them with the symmetries of the KP hierarchy. While the existence of
ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP
hierarchy to the KdV hierarchies, the existence of additional symmetries allows
to reduce KP to the constrained KP.Comment: 7pp, LaTe
Trajectory optimization by explicit numerical methods
Trajectory optimization by explicit numerical method
Why the general Zakharov-Shabat equations form a hierarchy?
The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature
equations with rational dependence on a spectral parameter, if properly
defined, can be considered as a hierarchy. The latter means a collection of
commuting vector fields in the same phase space. Further properties of the
hierarchy are discussed, such as additional symmetries, an analogue to the
string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian
definition of soliton solutions.Comment: 13p
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