The nonlinear Klein-Gordon equation
is a good place to start our discussion, since the classical wave
equation and the linear Klein-Gordon equation can both be treated as
special cases of the function 
. Notice that the phase velocity is
missing from the equation; this corresponds to a choice of units for
which which v=1. Substituting eq:Der2E into eq:nonlinKGE,
setting 
, and rearranging terms, we obtain the
following equation for the 
row of the U matrix in terms of
the 
and 
rows:
Ablowitz, Kruskal and Ladik showed that this equation can be stabilized
by setting r=1 and using an average of the spatial coordinates for the
function, 
[1]
This choice of 
and 
not only reduces the number of
computations, it also separates the U matrix into two non-interacting
grids: 
and 
, i.e., n and j
have the same parity or opposite parity. This reduces the number of
calculations by a factor of two since only the even-even and
odd-odd grid points need to be calculated.
The numerical values for
the current wavefunction and the previous wavefunction are stored in two
global CUPS DVectors
named yVec
and yPrevVec
,
respectively. The corresponding x-values are stored in another
DVector, xVec
.
