The story of the first investigations by FPU –as well as the follow-on discoveries
by Kruskal, et. al has been told many times, so it will not be repeated here. More-
over, the subsequent literature on the both the FPU problem and KdV is too vast
and varied to summarize adequately in this space. For our purposes, the essential
elements are as follows: FPU considered their nonlinear lattice systems to be “toy”
models of a general, macroscopic, physical system in which nonlinear terms cause
interaction between the normal modes of the linear system (i.e., Fourier modes). In
accordance with the postulates that underly statistical derivations of macroscopic
thermodynamics, FPU expected to see a single-mode initial condition evolve to
a state in which the energy was evenly distributed across all the Fourier modes
(“thermalization”). Instead, numerical simulations yielded repeated approximate
recurrences of the single-mode initial state at evenly-spaced time intervals.
Given the discovery of recurrence by FPU and the derivation of KdV as an
asymptotic model of the FPU lattice (cf. e.g, [31]), Zabusky and Kruskal did not
identify recurrences observed in simulations of KdV [36] as a novelty. Instead, for
Zabusky and Kruskal, the novel surprising result observed in simulations of KdV
was the elastic interaction of solitons (in the approximate form of the classical
traveling-waves of KdV). The discovery of the solitons (more precisely, their elastic
interactions) inspired the development of the Inverse-Scattering Transform (IST)
method for the solution of KdV [16], which in turn led to the characterization
of KdV as a completely integrable Hamiltonian system (cf. e.g., [13]). Despite
the original suspicions of Kruskal [27], the theory of solitons and the IST is, of
course, not uniquely applicable to KdV, and has been extended to a broad array
of physically important and mathematically rich nonlinear evolution equations on
the line (cf. e.g., [3]), as well as systems with periodic and quasiperiodic boundary
conditions (cf. e.g., [11,23,29]).
The logical circle is completed by invoking the solitons of KdV as an expla-
nation for the recurrence observed by Zabusky and Kruskal (e.g., [30, 36]). By
extension, the recurrence observed in simulations of the FPU lattice is then ex-
plained on the grounds that FPU dynamics are governed by KdV (e.g., [8, 32]).
Even at first glance, the connections are non-trivial: the individual
and multisoliton solutions, are solutions of KdV with vanishing boundary condi-
tions on the whole line (u 0 as x ±∞), while solutions of KdV with periodic
boundary conditions are more complex (cf. e.g., [11, 23, 29]). Nevertheless, if an
individual soliton is narrow compared to the spatial period, the soliton does not
“see” the periodic boundary condition and exists as an approximate solution (the
limit of a periodic “cnoidal” wave). Indeed, the correspondence between periodic
and whole-line solutions can be developed even for a spatially periodic solution
whose support is not small compared the spatial period [30]. Even so, the periodic
boundary conditions are an important ingredient for the soliton view of recurrence:
to achieve (approximate) recurrence, the unidirectional solitons of whole-line KdV
“wrap around” in order to nearly recover their original arrangement as the system
The system integrated by Zabusky and Kruskal [36] was not, strictly speaking,
KdV (1.1) itself, but rather the lattice
un =
(un−1 + un + un+1)(un−1 un+1)
un−2 2un−1 + 2un+1 un+2
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