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Gr\"obner Bases and Generation of Difference Schemes for Partial Differential Equations
In this paper we present an algorithmic approach to the generation of fully
conservative difference schemes for linear partial differential equations. The
approach is based on enlargement of the equations in their integral
conservation law form by extra integral relations between unknown functions and
their derivatives, and on discretization of the obtained system. The structure
of the discrete system depends on numerical approximation methods for the
integrals occurring in the enlarged system. As a result of the discretization,
a system of linear polynomial difference equations is derived for the unknown
functions and their partial derivatives. A difference scheme is constructed by
elimination of all the partial derivatives. The elimination can be achieved by
selecting a proper elimination ranking and by computing a Gr\"obner basis of
the linear difference ideal generated by the polynomials in the discrete
system. For these purposes we use the difference form of Janet-like Gr\"obner
bases and their implementation in Maple. As illustration of the described
methods and algorithms, we construct a number of difference schemes for Burgers
and Falkowich-Karman equations and discuss their numerical properties.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
-analogue of modified KP hierarchy and its quasi-classical limit
A -analogue of the tau function of the modified KP hierarchy is defined by
a change of independent variables. This tau function satisfies a system of
bilinear -difference equations. These bilinear equations are translated to
the language of wave functions, which turn out to satisfy a system of linear
-difference equations. These linear -difference equations are used to
formulate the Lax formalism and the description of quasi-classical limit. These
results can be generalized to a -analogue of the Toda hierarchy. The results
on the -analogue of the Toda hierarchy might have an application to the
random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are
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