Method of Characteristics, Navier-Stokes Equation and Turbulent Map

I'm hoping to use the Method of Characteristics, from Krall &
Trivelpiece, Hazeltine's Plasma Text and Kreischer's Dissertation, to
combine x & t into a single variable. This may allow me to add another
layer of complexity to my (still incomplete) solution to the the Navier-
Stokes Equation. As it is, my solution is still lacking the most
important ingredient, which is the Navier-Stokes Equation itself.

Once again, I'm using the tried and true method of working backwards
from a known answer, or in this case partial answer, to the first
principle. Using references to Landau & Lifschitz and Tajima's Text,
I've managed to show that a bifurcation tree will exist, with many of
the same features as the well known Logistic Map, although an similar
number of equally important equations are still unsolved. Thus on a
microscopic level, I expect the Navier-Stokes Map to be identical to the
Logistic Map, while on a macroscopic level, I expect it to be completely
different.

What visual similarities in Feigenbaum's Diagram will be produced by the
common traits and what will be different are of course unknown, but it
is hoped that the pictures will be striking. Of course, with bad luck,
the accumulation of microscopic similarities may be so severe that the
two pix could be completely dissimilar, not allowing any visual
analogies to be made. Of course, it can't be known where this
comparison will lead, if anywhere, until the problem is solved.

John Schutkeker wrote in
. 102:


I'm hoping to use the Method of Characteristics, from Krall &
Trivelpiece, Hazeltine's Plasma Text and Kreischer's Dissertation, to
combine x & t into a single variable. This may allow me to add
another layer of complexity to my (still incomplete) solution to the
the Navier- Stokes Equation. As it is, my solution is still lacking
the most important ingredient, which is the Navier-Stokes Equation
itself.

Once again, I'm using the tried and true method of working backwards
from a known answer, or in this case partial answer, to the first
principle. Using references to Landau & Lifschitz and Tajima's Text,
I've managed to show that a bifurcation tree will exist, with many of
the same features as the well known Logistic Map, although an similar
number of equally important equations are still unsolved. Thus on a
microscopic level, I expect the Navier-Stokes Map to be identical to
the Logistic Map, while on a macroscopic level, I expect it to be
completely different.

What visual similarities in Feigenbaum's Diagram will be produced by
the common traits and what will be different are of course unknown,
but it is hoped that the pictures will be striking. Of course, with
bad luck, the accumulation of microscopic similarities may be so
severe that the two pix could be completely dissimilar, not allowing
any visual analogies to be made. Of course, it can't be known where
this comparison will lead, if anywhere, until the problem is solved.

Well, this idea died quickly. After checking Hazeltine's Book [I
couldn't find it in Krall & Trivelpiece, and I couldn't even find my
folder with Kreischer's thesis] ? to the best of my understanding, the
Method of Characteristics is not used to solve the equation of motion,
but to solve the evolution of the distribution function.

I also looked at the MOC to see if it could be adapted for the equation
of motion, but it didn't look like it could. From a bout of insomnia,
my head is quite foggy this week, so I'll come back to this last issue
when I'm feeling better, to make sure I didn't miss something.