Volume 635, 2015
Recurrence in the Korteweg-de Vries equation?
Ben Herbst, Garrett Nieddu, and A. David Trubatch
Abstract. The Zabusky-Kruskal lattice (ZK) was derived as a finite-
difference approximation of the Korteweg-de Vries equation for the purpose
of numerical simulation of that PDE. Like the Fermi-Pasta-Ulam lattice from
which it was ultimately derived, ZK was also observed to exhibit near-re-
currence of its initial state at regular time intervals. The recurrence has not
been completely explained, though it has been attributed to the solitons or,
less specifically, the integrability of the KdV continuum limit The attribution
of recurrence to the integrablity of the continuum limit (i.e., KdV) leads nat-
urally to the hypotheses that simulations (i) on smaller grid separations, (ii)
with discretizations that are integrable as spatially discrete systems or (iii)
with higher-order discretizations should all exhibit stronger recurrences than
observed in the original simulations of ZK. However, systematic simulations
over a range of grid scales with ZK, as well as an integrable discretization intro-
duced here and a spectral discretization of KdV are not consistent with these
hypotheses. On the contrary, for the ZK and an integrable finite-difference
discretization of KdV, recurrence of a low-mode initial state is observed to be
strongest and most persistent at an intermediate scale. We conclude that the
observed recurrences is a lattice property and not a reflection of the integrable
dynamics of KdV.
When Zabusky and Kruskal  performed numerical simulations of the
Korteweg-de Vries equation (KdV),
(1.1) ut + uux +
with periodic boundary conditions (u(0,t) = u(2,t)), they were looking for the
recurrence of the single-mode initial condition
(1.2) u(x, 0) = sin(πx) , 0 ≤ x ≤ 2 ,
at regular time intervals. The recurrence of the initial condition was expected
because, in this instance, KdV (1.1) was considered an asymptotic limit of the
nonlinear oscillator lattice first investigated by Fermi, Pasta and Ulam (FPU) .
In addition to observing repeated (partial) recurrences, Zabusky and Kruskal also
famously observed the elastic nonlinear interaction of localized waves and coined
the term soliton to denote their particle-like coherence .
2010 Mathematics Subject Classification. Primary 34C60, 37J35.
The second and third authors gratefully acknowledge support for this work by NSF DMS
c 2015 American Mathematical Society