RECURRENCE IN THE KORTEWEG-DE VRIES EQUATION? 3

which, by inspection, is an

O(h2)

discretization of KdV. The system of equations

for the lattice can be written in vector-matrix form as

d

dt

u = − (A3u) × (Dcu) −

δ2D3cu

where: u is an N-component vector of the values un, the matrices A3, Dc and

D3c represent the differences in (1.3) and × represents a component-by-component

vector product. (Of course, a scheme is also required to discretize the time steps.

However, consideration of the time-stepping method used in [36] is not essential,

as the results described there, and here, can be obtained with any of a number of

suﬃciently accurate ODE integration schemes.) Further simulations [17] with the

same lattice (1.3), but with a two-mode intial condition and a different value of

δ2,

also showed repeated near-recurrences of the initial state.

The averaging coeﬃcient, A, in the nonlinear term of the lattice (1.3) leads to

the exact preservation of the quantities

m1 =

n

un m2 =

n

un,2

with the imposition of periodic or vanishing (on a bi-infinite lattice) boundary

conditions. These conserved quantities correspond to the preserved quantities

m1 = u dx m2 =

u2

dx

for KdV under the same boundary conditions. In particular, the conservation of

m2 on the lattice precludes runaway growth of the solution as it evolves.

An alternative explanation for the observed near-recurrence of the low-mode

initial condition is to assert that the evolution of the ZK lattice (1.3) is governed

by the properties of KdV (1.1) as a completely integrable (infinite-dimensional)

Hamiltonian system, but without reference to the solitons of KdV (1.1) on the line.

In rough summary, for KdV with periodic boundary conditions, the “finite gap”

quasiperiodic (in time) solutions are dense in the space of solutions [25, 26], and

these quasiperiodic solutions are almost periodic (in time).

While a generic perturbation of an integrable system destroys the integrability,

the well-known KAM theorem describes the persistence of quasiperiodic orbits of

finite-dimensional integrable systems under Hamiltonian perturbations (cf. [9] and

references therein). In this regard, we note that, even though the ZK lattice (1.3) is

not integrable as a system of ODEs [24], it can be understood as a perturbation of

KdV, which, as noted, is a completely integrable Hamiltonian PDE. In this regard,

there has been significant work to extend KAM to infinite-dimensional systems,

including especially KdV with peridoc boundary conditions [20,22].

The Nekhoroshev theorem [28] complements the KAM theorem by providing

a bound, valid for exponentially-long times, on the divergence of orbits of the

perturbed system from the quasiperiodic orbits of the unperturbed (integrable)

system. (See [10] for a unified treatment of KAM and Nekhoroshev theorems.) In

the standard Nekhoroshev theorem, both (i) the bound on the divergence of the

orbit of the perturbed system from the orbit of the integrable system and (ii) the

expression for the maximum time of the validity for that bound include the a term

in which the perturbation parameter is raised to a power of

1

N

, where N is the

dimension of the system. If we consider the ZK lattice (1.3) as a perturbation of

KdV with spatial period L, we have L = Nh, where N is the number of grid points