Symmetry reductions of the self-dual Yang–Mills equations for SL(2,C) bundles with the background metric ds2=2 du dv−dx2+f2(u)dy2 are considered. One of the field components in the reduced equations can be cast into Jordan normal form after gauge transformations. The reduced equations for the two possible normal forms are equivalent, respectively, to certain generalizations of the Korteweg–de Vries (KdV) equation and the nonlinear Schrödinger (NLS) equation. It is shown that the generalized KdV and NLS equations fail the Painlevé test except when the metric is flat. The generalized KdV equation is transformed to a simple form in the case when f(u)=ua and it is shown that one may obtain either the KdV equation or the cylindrical KdV equation by this method only when the metric is flat.
A class of spacetime metrics with coefficients depending on a single null
coordinate is introduced. The equation of the null cone at an arbitrary
point is obtained in explicit form. It is shown that every metric in this
class has a Weyl tensor of Petrov type N and is conformal to
a metric satisfying the Einstein vacuum equations. The metrics
belonging to a particular subclass are shown to be conformal
to certain plane-wave solutions in general relativity. The
properties of a generalization of these metrics to n + 1
dimensions are studied. The generalized metrics always have
2n-1 independent Killing vectors and 2n independent
conformal Killing vectors. The algebra of Killing vectors
turns out to be the Heisenberg algebra.
The solution of differential equations using the software package Mathematica is discussed in this paper. We focus on two functions, DSolve and NDSolve, and give various examples of how one can obtain symbolic or numerical results using these functions. An overview of the Wolfram Demonstrations Project is given, along with various novel user-contributed examples in the field of differential equations. The use of these Demonstrations in a classroom setting is elaborated upon to emphasize their significance for education.
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