83 research outputs found
Factorization of a Matrix Differential Operator Using Functions in its Kernel
Just as knowing some roots of a polynomial allows one to factor it, a
well-known result provides a factorization of any scalar differential operator
given a set of linearly independent functions in its kernel. This note provides
a straight-forward generalization to the case of matrix coefficient
differential operators that applies even in the case that the leading
coefficient is singular
When is negativity not a problem for the ultra-discrete limit?
The `ultra-discrete limit' has provided a link between integrable difference
equations and cellular automata displaying soliton like solutions. In
particular, this procedure generally turns strictly positive solutions of
algebraic difference equations with positive coefficients into corresponding
solutions to equations involving the "Max" operator. Although it certainly is
the case that dropping these positivity conditions creates potential
difficulties, it is still possible for solutions to persist under the
ultra-discrete limit even in their absence. To recognize when this will occur,
one must consider whether a certain expression, involving a measure of the
rates of convergence of different terms in the difference equation and their
coefficients, is equal to zero. Applications discussed include the solution of
elementary ordinary difference equations, a discretization of the Hirota
Bilinear Difference Equation and the identification of integrals of motion for
ultra-discrete equations
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