Solitons, numerical experiments, and that mysterious lady
August 2008, page 10
The interesting article by Thierry Dauxois on "Fermi,
Pasta, Ulam, and a Mysterious Lady" (PHYSICS TODAY, January 2008, page 55) relates the subject
of solitons to that of the Fermi-Pasta-Ulam (FPU) problem. The term "soliton" was introduced by
Norman Zabusky and Martin Kruskal1 in 1965 because the nonlinear waves studied did
not lose their identity after colliding. In a sense, they resembled particles. The study by Zabusky
and Kruskal was a numerical one of the Korteweg–de Vries equation, but the motivation was
to study the propagation of waves in a collisionless plasma containing a magnetic field. Fifty
years ago John Adlam and I studied that problem2 and found an analytical solution for
strong, collision-free hydromagnetic solitary waves for Alfvén Mach numbers less than
2. The solution was not valid for faster, stronger waves. Further work in 1960 dealt with the excitation
of a train of such waves;3 that time the equations were solved numerically. The work
with Adlam seems to have been largely overlooked until recently,4 presumably because
it predated the term "soliton."
Rediscovering Mary
Tsingou's role in the Fermi-Pasta-Ulam problem is laudable. However, Thierry Dauxois is
incorrect in calling the FPU problem "the first-ever numerical experiment" that marked the beginning
of "computer simulations of scientific problems."
Lewis F. Richardson's
landmark 1922 work on numerical weather prediction predated the FPU problem by more than three
decades and far surpasses it in complexity.1 The first successful numerical weather
forecast was performed on the ENIAC computer in 1950 by a team of scientists that included John von
Neumann.2 Both of those numerical experiments were highly nonlinear in character
and involved approximations of the Navier–Stokes equation. Dauxois's oversight confirms
the statement that "meteorologists . . . are the Rodney Dangerfields of science.
They get no respect from . . . physics and chemistry."3
References
1. L. F. Richardson, Weather Prediction by Numerical Process, Cambridge U. Press, Cambridge, UK (1922).
2. J. G. Charney, R. Fjörtoft, J. von Neumann, Tellus2, 237 (1950).
3. J. Fishman, R. Kalish, The Weather Revolution: Innovations and Imminent Breakthroughs in Accurate Forecasting, Plenum Press, New York (1994), p. 29.
Dauxois replies:
I did not attempt to present a complete history of the soliton concept, so all possibly relevant
papers were not cited. However, I think the paper by Norman Zabusky and Martin Kruskal (J. E. Allen's
reference 1) ought to be emphasized for several reasons. First, it dealt directly with the understanding
of the puzzling observation made by Enrico Fermi, John Pasta, Stanislaw Ulam, and Mary Tsingou.
Second, it highlighted the soliton, a concept of general interest1 that goes beyond
the observation of "collision free" wave interactions. Third, the suffix"-on" in the name emphasizes
that those waves have properties of particles.
I know that using a computer
to solve an equation was done before FPU-Tsingou. (Working in physical oceanography and having
a wife in fluid mechanics, I do respect meteorologists!) Solving equations, with or without approximations,
is different from conducting a numerical experiment, which asks the computer a physical question.
One studies a system simpler than the real one in order to use the computer to test theories that could
not have been tested with real experiments, affected as they are by uncontrollable effects and
noise (see the epistemological paper in reference 2). I am not aware of any previous use of computers
in that way, nor, apparently, was Ulam.3
References
1. T. Dauxois, M. Peyrard, Physics of Solitons, Cambridge U. Press, New York (2006).
2. R. Livi, S. Ruffo, M. Pettini, A. Vulpiani, Giornale de Fisica26(4), 285 (1985).
3. S. M. Ulam, Adventures of a Mathematician, Scribner, New York (1976).
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