which, by inspection, is an
discretization of KdV. The system of equations
for the lattice can be written in vector-matrix form as
u = (A3u) × (Dcu)
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
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 =
un m2 =
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 =
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
, 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
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