Dynamical systems with benign ghosts

Thibault Damour

Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France

<EMAIL_ADDRESS>

and

Andrei Smilga

SUBATECH, Université de Nantes,
4 rue Alfred Kastler, BP 20722, Nantes 44307, France;

<EMAIL_ADDRESS>

We consider finite and infinite-dimensional ghost-ridden dynamical systems
whose Hamiltonians involve non positive definite kinetic terms. We point out
the existence of three classes of such systems where the ghosts are benign,
i.e. systems whose evolution is unlimited in time: (i) systems obtained from
the variation of bounded-motion systems; (ii) systems describing motions over
certain Lorentzian manifolds and (iii) higher-derivative models related to
certain modified Korteweg–de Vries equations.

##  1 Introduction

A ghost-ridden dynamical quantum system is defined as a system whose spectrum
is not bounded neither from below, nor from above. This is in particular the
case for the quantum versions of the Ostrogradsky Hamiltonians [1] describing
the dynamics of higher-derivative Lagrangians (for a review, see, e.g., [2]).
Many ghost-ridden systems are sick: their evolution operator is not unitary.
Such systems involve classical trajectories that run into a singularity after
a finite time of evolution (a blow-up).

There are, however, ghost-ridden systems with benign ghosts, in the sense that
they exhibit no classical blow-up and have a unitary quantum evolution
operator. One of the simplest examples of such a benign-ghost system is the
Pais-Uhlenbeck oscillator [3] with the higher-derivative Lagrangian

$\displaystyle L\ =\
\frac{1}{2}\left[\ddot{x}^{2}-(\omega_{1}^{2}+\omega_{2}^{2})\dot{x}^{2}+\omega_{1}^{2}\omega_{2}^{2}x^{2}\right].$
(1.1)

The quantum version of the corresponding Ostrogradsky Hamiltonian does not
have a bottom. Indeed, there exists a canonical transformation [4] that brings
the Hamiltonian into the form111We assumed here that
$\omega_{1}\neq\omega_{2}$. In the case of equal frequencies, the situation is
somewhat more complicated because the canonical transformation mentioned above
is singular and the Hamiltonian is not reduced to the form (1.2). Still, the
Hamiltonian is well defined. It has a continuous spectrum [3, 5, 6].

$\displaystyle H=\
\frac{\hat{P}_{1}^{2}+\omega_{1}^{2}X_{1}^{2}}{2}-\frac{\hat{P}_{2}^{2}+\omega_{2}^{2}X_{2}^{2}}{2}\,.$
(1.2)

This Hamiltonian has a pure point spectrum with neither bottom, nor ceiling:

$\displaystyle E_{nm}\ =\
\left(n+\frac{1}{2}\right)\omega_{1}-\left(m+\frac{1}{2}\right)\omega_{2}\,.$
(1.3)

If the ratio $\omega_{1}/\omega_{2}$ is irrational, the spectrum is everywhere
dense. The evolution operator is unitary.

The ghosts generally strike back, however, when one leaves the realm of
quadratic Hamiltonians to include interactions. Consider for example the
Lagrangian

$\displaystyle L\ =\
\frac{1}{2}\left[\ddot{x}^{2}-2\omega^{2}\dot{x}^{2}+\omega^{4}x^{2}\right]-\frac{1}{4}\alpha
x^{4}\,,$ (1.4)

whose classical equation of motion reads

$\displaystyle\left(\frac{d^{2}}{dt^{2}}+\omega^{2}\right)^{2}x-\alpha x^{3}\
=\ 0\,.$ (1.5)

The classical trajectories depend on four initial conditions. There is an
obvious stationary point

$\displaystyle x(0)\ =\ \dot{x}(0)\ =\ \ddot{x}(0)\ =\ x^{(3)}(0)\ =\ 0\,.$
(1.6)

The behaviour of the system in the vicinity of this point depends on the sign
of $\alpha$. If $\alpha<0$, the Ostrogradsky Hamiltonian acquires an extra
negative contribution to the energy and all the trajectories other than
$x(t)=0$ run away to infinity in a finite time. The situation is somewhat
better for positive $\alpha$. The stationary point (1.6) lies at the center of
an “island of stability” — the trajectories with initial conditions
sufficiently close to (1.6) do not go astray, but exhibit an oscillatory
behavour. However, this island has a shore. When the deviations of initial
conditions from (1.6) are large enough, the trajectory blows up [7]. Numerical
studies show that most systems obtained by a nonlinear deformation of (1.1)
exhibit a similar behaviour. For instance, a similar island of stability
(surrounded by an infinite ocean of blow-up behavior) was observed in [8] for
a (cosmology-inspired) model obtained by adding to the Pais-Uhlenbeck
Hamiltonian (1.2) (with $\omega_{1}=0$ and $\omega_{2}\neq 0$) an interaction
term $\lambda X_{1}^{2}X_{2}^{2}$ (with $\lambda>0$).

When considering the quantum version of ghost-ridden Hamiltonians involving
non-trivial interactions it is difficult to reach definite answers applying to
general situations. A case by case study is required. For instance, it has
been argued that the quantum problem for the system (1.4) is also malignant
[2]. On the other hand, there are Hamiltonians which entail blowing up
classical motions but which lead to a well-defined, unitary quantum evolution.
A well-known example is the Hamiltonian describing the 3-dimensional motion of
a particle in an attractive $\frac{1}{r^{2}}$ potential:

$\displaystyle H\ =\ \frac{\vec{p}^{2}}{2m}-\frac{\kappa}{r^{2}}\,.$ (1.7)

Classically, for certain initial conditions, the particle falls to the center
in a finite time. The quantum dynamics of this system depends on the value of
${\kappa}$. If $m{\kappa}<1/8$, the ground state exists and unitarity is
preserved. If $m{\kappa}>1/8$, the spectrum is not bounded from below and,
what is worse, the quantum problem cannot be well posed until the singularity
at the origin is smoothed out [9]. One can say that for $m{\kappa}<1/8$
quantum fluctuations cope successfully with the attractive force of the
potential and prevent the system from collapsing.

The latter example suggests that quantum fluctuations can only make a ghost-
ridden system better, not worse. We therefore conjecture that, if the
classical dynamics of the system is benign, its quantum dynamics will also be
benign, irrespectively of whether the spectrum has, or does not have, a
bottom.

Some particular examples of benign ghost-ridden (nonlinearly interacting)
Hamiltonians have been presented in previous works [7, 10, 12, 11, 13, 14].
The aim of the present paper is to delineate new classes of benign
Hamiltonians, some of which have a large generality (in the sense that they
contain several arbitrary functions). More precisely, we shall present classes
of nontrivially interacting ghost-ridden systems such that all the classical
motions (and not only the motions restricted to a limited region of phase
space) admit an infinite-time evolution.222Note that we are not requiring that
the motions indefinitely stay within a compact region of phase space. We do
not exclude power-law, or exponential, runaway behaviors. We are simply
excluding finite-time blow-up. In view of the conjecture stated in the last
paragraph, we expect that the quantum dynamics of our benign ghost-ridden
systems will be well posed. We leave studies of this issue to future work.

##  2 Variational dynamics: a large class of benign ghosts

A general class of dynamical systems with benign ghosts is obtained by
considering the variational equations of motion of a bounded-motion333Here,
“bounded motion” means that the time evolution of the considered, unperturbed,
system stays within some compact domain of phase space. Hamiltonian system. An
example of such a system was found in [10] by studying a certain higher-
derivative supersymmetric mechanical system, though its variational nature was
not noticed.444The supersymmetric aspects of this problem are not relevant
here, and we shall forget here about fermions.

The general setting for defining such systems is the following. One starts
from a basic, unperturbed dynamical system (with $n$ degrees of freedom)
described, say, by a Lagrangian, $L_{0}(q^{i},{\dot{q}}^{i})$, $i=1,\cdots n$,
i.e. by the action

$\displaystyle S_{0}[q,{\dot{q}}]=\int dt\,L_{0}(q^{i},{\dot{q}}^{i})\,.$
(2.1)

Then, one considers the dynamics defined by varying the Lagrangian action
(2.1), i.e. by making in $L_{0}(q^{i},{\dot{q}}^{i})$ the replacement

$\displaystyle q^{i}(t)\to q^{i}(t)+\epsilon\,Q^{i}(t)\,,$
$\displaystyle{\dot{q}}^{i}(t)\to{\dot{q}}^{i}(t)+\epsilon\,{\dot{Q}}^{i}(t)\,,$
(2.2)

and by keeping only the term linear in $\epsilon$. In other words, one is
considering the new action (with $2n$ degrees of freedom)

$\displaystyle S_{1}[q,{\dot{q}};Q,{\dot{Q}}]=\int
dtL_{1}(q^{i},{\dot{q}}^{i};Q^{i},{\dot{Q}}^{i})\,,$ (2.3)

where

$\displaystyle
L_{1}(q^{i},{\dot{q}}^{i};Q^{i},{\dot{Q}}^{i})=Q^{j}\frac{\partial
L_{0}(q,{\dot{q}})}{\partial q^{j}}+{\dot{Q}}^{j}\frac{\partial
L_{0}(q,{\dot{q}})}{\partial{\dot{q}}^{j}}\,.$ (2.4)

The variational action (2.3) leads to the following equations of motion for
$q^{i}$ and $Q^{i}$ (with $\delta/\delta q$ denoting an Euler-Lagrange
variational derivative,
$\partial_{q}-\frac{d}{dt}\partial_{{\dot{q}}}+\frac{d^{2}}{dt^{2}}\partial_{\ddot{q}}+\cdots$):

$\displaystyle 0=\frac{\delta L_{1}}{\delta Q^{i}}=\frac{\partial
L_{0}(q,{\dot{q}})}{\partial q^{i}}-\frac{d}{dt}\frac{\partial
L_{0}(q,{\dot{q}})}{\partial{\dot{q}}^{i}}\equiv\frac{\delta L_{0}}{\delta
q^{i}}\,,$ $\displaystyle 0=\frac{\delta L_{1}}{\delta
q^{i}}=Q^{j}\frac{\partial L_{0}(q,{\dot{q}})}{\partial q^{j}\partial
q^{i}}-\frac{d}{dt}\left[{\dot{Q}}^{j}\frac{\partial
L_{0}(q,{\dot{q}})}{\partial{\dot{q}}^{j}\partial{\dot{q}}^{i}}\right]\,.$
(2.5)

We see that the first Euler-Lagrange equation $0={\delta L_{1}}/{\delta
Q^{i}}$ coincides with the unperturbed equation of motion of $q^{i}$, ${\delta
L_{0}}(q,{\dot{q}})/{\delta q^{i}}=0$. On the other hand, the second equation,
$0={\delta L_{1}}/{\delta q^{i}}$ coincides with the variation of the equation
${\delta L_{0}}(q,{\dot{q}})/{\delta q^{i}}=0$: it is obtained by replacing
there $q^{i}(t)$ and their first and second derivatives as in (2) and keeping
only the term linear in $\epsilon$.

When considering the first-order Hamiltonian version of the so-defined
variational dynamics, one encounters a slightly surprising feature. Namely, if
we denote the momenta conjugate to $q^{i}$ and $Q^{i}$ respectively as $p_{i}$
and $P_{i}$, the Hamiltonian $H_{1}(q^{i},Q^{i};p_{i},P_{i})$ describing the
varied dynamics (corresponding to
$L_{1}(q^{i},{\dot{q}}^{i};Q^{i},{\dot{Q}}^{i})$) is not obtained by varying
the unperturbed Hamiltonian $H_{0}(q^{i};p_{i})$ by means of the naively
expected variation $q^{i}\to q^{i}+\epsilon\,Q^{i};p_{i}\to
p_{i}+\epsilon\,P_{i}$. It is obtained by first doing the latter naive
variation, and then by swapping the momenta according to $p_{i}\leftrightarrow
P_{i}$.

The necessity to swap $p\leftrightarrow P$ in the naive variation of
$H_{0}(q^{i};p_{i})$ is easily seen by varying the Hamiltonian version of the
action (2.1),

$\displaystyle S_{0}^{H}[q,p]=\int
dt\left[p_{i}{\dot{q}}^{i}-H_{0}(q,p)\right]\,.$ (2.6)

Using the naively defined variations $q^{i}\to q^{i}+\epsilon\,Q^{i};p_{i}\to
p_{i}+\epsilon\,P_{i}$ we get

$\displaystyle\delta S_{0}^{H}[q,Q,p,P]=\int
dt\left[P_{i}{\dot{q}}^{i}+p_{i}{\dot{Q}}^{i}-\left(Q^{i}\frac{\partial
H_{0}(q,p)}{\partial q^{i}}+P_{i}\frac{\partial H_{0}(q,p)}{\partial
p_{i}}\right)\right]\,,$ (2.7)

in which the Hamiltonian kinetic term $P_{i}{\dot{q}}^{i}+p_{i}{\dot{Q}}^{i}$
shows that the conjugate momentum to $q^{i}$ is actually $P_{i}\sim\delta
p_{i}$, while the conjugate momentum to $Q^{i}$ is the unperturbed $p_{i}$.
After the swap $p\leftrightarrow P$, one finally gets the Hamiltonian for the
varied dynamics expressed in terms of the canonical pairs $(q,p);(Q,P)$:

$\displaystyle H_{1}(q,Q;p,P)=Q^{i}\frac{\partial H_{0}(q,P)}{\partial
q^{i}}+p_{i}\frac{\partial H_{0}(q,P)}{\partial P_{i}}\,.$ (2.8)

From our present perspective (namely, studying ghost-ridden dynamics), note
that all the varied Hamiltonians $H_{1}(q,Q;p,P)$ are necessarily unbounded
below (and above) because they have a linear dependence on the phase-space
variables $Q^{i}$ and $p_{i}$, see Eq. (2.8). This makes the quantum spectrum
also unbounded. Let us, however, see why this ghost feature leads generically
to a benign evolution.

The simplest type of varied dynamics is obtained by varying a simple
Hamiltonian (with one degree of freedom) of the form

$\displaystyle H_{0}(x,p)=\frac{1}{2}p^{2}+V(x)\,.$ (2.9)

In that case, the variation of the first term yields the symmetric structure
$pP$, invariant under the swap $p\leftrightarrow P$, which is thus
ineffective. One then gets (with the notation $D\equiv Q=\delta x$ used in
[10]) a varied-dynamics Hamiltonian involving two pairs of canonically
conjugated variables, $(x,p)$ and $(D,P)$:

$\displaystyle H_{1}(x,D;p,P)=pP+DV^{\prime}(x)\,.$ (2.10)

The classical equations of motion are

$\displaystyle\ddot{x}+V^{\prime}(x)\ =\ 0\,,\ \ \ \ \ \ \ \ \ \ \
\ddot{D}+V^{\prime\prime}(x)D=0\,.$ (2.11)

They admit two constants of motion: the varied Hamiltonian $H_{1}(x,D;p,P)$,
and the conserved energy $E_{0}=\frac{1}{2}\dot{x}^{2}+V(x)$ of the
unperturbed motion of $x$. Using the Hamilton equation $\dot{x}=\frac{\partial
H_{1}}{\partial p}=P$, the unperturbed energy $E_{0}(x,\dot{x})$ yields the
second integral of motion $N(x,P)$ with

$\displaystyle N(x,P)\ =\ \frac{P^{2}}{2}+V(x)\,.$ (2.12)

The model of Ref. [10] was of this type, with the simplest nontrivial
potential

$\displaystyle V(x)\ =\ \frac{\omega^{2}x^{2}}{2}+\frac{\lambda x^{4}}{4}\,,\
\ \ \ \ \ \lambda>0\,.$ (2.13)

If one introduces the variables

$\displaystyle X_{1,2}\ =\
\sqrt{\frac{\omega}{2}}x\pm\frac{1}{\sqrt{2\omega}}D\,,\ \ \ \ \ \ P_{1,2}\ =\
\frac{1}{\sqrt{2\omega}}p\pm\sqrt{\frac{\omega}{2}}P\,,$ (2.14)

the Hamiltonian (2.10) acquires the form [10]

$\displaystyle H_{1}\ =\
\frac{P_{1}^{2}+\omega^{2}X_{1}^{2}}{2}-\frac{P_{2}^{2}+\omega^{2}X_{2}^{2}}{2}+\frac{\lambda}{4\omega}(X_{1}-X_{2})(X_{1}+X_{2})^{3}\,.$
(2.15)

In other words, this is the Hamiltonian (1.2) with degenerate frequencies,
with an additional quartic interaction of a special form.

A nice distinguishing feature of the system (2.10) is its integrability.
Indeed, we have two degrees of freedom, and two commuting integrals of motion:
$H_{1}$ and $N$. (The vanishing of the Poisson bracket $\\{H_{1},N\\}$ is
easily checked.) This allows one to find, for the simple potential (2.13), the
solution analytically.

First, the unperturbed dynamics for $x$ describes oscillations in the quartic
potential (2.13). The solutions are elliptic functions whose parameters depend
on the integral of motion $N$ :

$\displaystyle x(t)\ =\ x_{0}\,{\rm cn}[\Omega(t-t_{0}),k]\,,$ (2.16)

where ${\rm cn}$ is a Jacobi elliptic function (with elliptic modulus
$k=\sqrt{m}$), and

$\displaystyle\alpha=\frac{\omega^{4}}{\lambda N},\ \ \ \Omega=[\lambda
N(4+\alpha)]^{1/4},$ $\displaystyle k^{2}\equiv
m=\frac{1}{2}\left[1-\sqrt{\frac{\alpha}{4+\alpha}}\right],\ \
x_{0}=\left(\frac{N}{\lambda}\right)^{1/4}\sqrt{\sqrt{4+\alpha}-\sqrt{\alpha}}\,.$
(2.17)

Bearing in mind (2.16), the equation for $D$ is a Hill equation describing an
oscillator with periodically varying frequency. The solutions of generic Hill
equations are obtained by Floquet theory and, depending on parameters, can be
either bounded for all times, or exponentially growing. However, we are here
in a special case. We know that the equation for $D$ describes the
infinitesimal variations of the general solution of the quartic oscillator
describing the $x$ dynamics. Therefore we can simply obtain the general
solution for $D$ by varying the integration constants entering the general
solution (2.16). It is enough to get two independent solutions. A first one
can be obtained by varying the integration constant $t_{0}$ in Eq. (2.16).
This yields

$\displaystyle D_{1}(t)\ =\ \frac{\partial[x(t)]}{\partial
t_{0}}=-\dot{x}(t)=+\Omega x_{0}\,{\rm sn}[\Omega t,k]{\rm dn}[\Omega t,k]\,,$
(2.18)

where we have set $t_{0}=0$ after variation. This solution is periodic in $t$.

A second solution is obtained by varying the value of $N$ in Eq. (2.16).
Varying $N$ implies that both the amplitude $x_{0}(N)$, and the frequency
$\Omega(N)$, of $x(t)$ must be varied. The variation of $\Omega(N)$ in $x(t)\
=\ x_{0}(N)\,{\rm cn}[\Omega(N)t,k]$ (setting again $t_{0}=0$, and noticing
that $k$ does not depend on $N$) brings about a prefactor $t$ multiplying the
derivative of ${\rm cn}(x)$ and generates a $D_{2}(t)$ of the form

$\displaystyle D_{2}(t)\ =\ t\frac{d}{dt}{\rm cn}[\Omega(N)t,k]+\beta\,{\rm
cn}[\Omega(N)t,k]\ =$ $\displaystyle-t\,\Omega(N)\,{\rm sn}[\Omega(N)t,k]{\rm
dn}[\Omega(N)t,k]+\beta\,{\rm cn}[\Omega(N)t,k],$ (2.19)

where $\beta$ is some constant. The contribution $-t\Omega\,{\rm sn}[\Omega
t,k]{\rm dn}[\Omega t,k]$ is the product of $t$ by a periodic function of $t$.
Therefore, while the first independent solution $D_{1}(t)$ is periodic, the
second independent solution $D_{2}(t)$ will exhibit an oscillatory behavior
with an amplitude rising linearly in time. This is illustrated in Fig. 1. What
is important for our present purpose is the benign nature of the general
solution, $D(t)=c_{1}D_{1}(t)+c_{2}D_{2}(t)$. Indeed, we consider as malignant
ghost only the cases leading to blow-up in a finite time. The linear growth in
time exhibited by the generic solution of the $D(t)$ equation is quite benign.

Figure 1: A typical behaviour of $D(t)$, as follows from solving Eq. (2.11).

We refer to Ref. [10] for a study of the quantum version of the Hamiltonian
(2.10) (due to integrability, the eigenvalues and eigenfunctions of the
quantum Hamiltonian can be found explicitly in this case).

We have used here the specific quartic potential (2.13) to be able to exhibit
explicit solutions for the one-degree-of-freedom variational dynamics (2.10).
However, the conclusions we reached about the benign dynamics of the
$(x,D;p,P)$ system hold for a general class of potentials. Indeed, if we take
for $V(x)$ a smooth confining potential growing as $x\to\pm\infty$, the
solutions for $x(t)$ will represent a nonlinear oscillation of a certain type
— periodic functions of time with a frequency $\Omega$ depending on the
integral of motion

$\displaystyle
E_{0}(x,\dot{x})\equiv\frac{1}{2}\dot{x}^{2}+V(x)=\frac{P^{2}}{2}+V(x)\equiv
N\,.$ (2.20)

In other words,

$\displaystyle x(t)=X\left[\Omega(N)(t-t_{0}),N\right]\,,$ (2.21)

where $X(\theta,N)$ is a periodic function of $\theta$, with period $2\pi$. It
can be expanded into a Fourier series,

$\displaystyle X(\theta,N)={\rm
Re}\left\\{\sum_{k\in\mathbb{Z}}A_{k}(N)e^{{\rm i}k\theta}\right\\}\,.$ (2.22)

The existence of the two commuting integrals of motion $H_{1}$ and $N$ allows
one to solve the motion of $x(t)$ by a quadrature, giving some type of
hyperelliptic function.

The general solution for $D(t)$ can then be simply obtained by varying the
general solution for $x(t)$ with respect to the two integration constants it
contains, namely $t_{0}$ and $N$. Again, it is clear that the first
independent solution

$\displaystyle D_{1}(t)=\partial x(t)/\partial t_{0}=-\dot{x}(t)=\Omega(N){\rm
Im}\left\\{\sum_{k}kA_{k}(N)e^{{\rm i}k\theta(t)}\right\\}\,,$ (2.23)

will be periodic in time (with the same period as $x(t)$), while the second
solution

$\displaystyle D_{2}(t)=\frac{\partial}{\partial
N}X[\Omega(N)(t-t_{0}),N]={\rm Re}\frac{\partial}{\partial
N}\left\\{\sum_{k}A_{k}(N)e^{{\rm i}k\Omega(N)(t-t_{0})}\right\\}\,,$ (2.24)

will be the sum of a periodic function, and of the function

$\displaystyle\frac{d\ln\Omega(N)}{dN}(t-t_{0})\dot{x}(t)\,.$ (2.25)

The latter function is a product of $t-t_{0}$ and of a periodic function of
$t$. As in the case illustrated in Fig. 1, we have again an oscillatory
behavior with an amplitude rising linearly in time. A benign ghost again.

Let us consider the generalization of these results to $n$ degrees of freedom,
i.e. to the system with the varied Lagrangian (2.4) and the corresponding
Hamiltonian (2.8). In that case, the results will strongly depend on the
integrable or non-integrable character of the unperturbed dynamics described
by the action (2.1), or equivalently, (2.6).

Let us first assume that the unperturbed dynamics (2.6) is integrable, and
that we are considering bound motions. In that case, the general solution of
the dynamics (2.6) corresponds to a quasi-periodic motion where the phase-
space coordinates $q^{i},p_{i}$ admit representations of the form

$\displaystyle q^{i}(t)$ $\displaystyle=$ $\displaystyle q^{i}[{\bf
I},{\bm{\theta}}(t)]=\sum_{\bf k}a^{i}_{k_{1}\cdots
k_{n}}(I_{1},\cdots,I_{n})e^{{\rm
i}(k_{1}\theta_{1}(t)+\cdots+k_{n}\theta_{n}(t))}\,,$ $\displaystyle p_{i}(t)$
$\displaystyle=$ $\displaystyle p_{i}[{\bf I},{\bm{\theta}}(t)]=\sum_{\bf
k}b^{i}_{k_{1}\cdots k_{n}}(I_{1},\cdots,I_{n})e^{{\rm
i}(k_{1}\theta_{1}(t)+\cdots+k_{n}\theta_{n}(t))}\,.$ (2.26)

Here $({\bf I},{\bm{\theta}})=(I_{i},\theta_{i})$, $i=1,\cdots,n$ are action-
angle variables, ${\bf k}=(k_{i}),i=1,\cdots,n$ are multiplets of (relative)
integers (summed over $\mathbb{Z}^{n}$), and the time evolution of the angles
is of the form

$\displaystyle\theta_{i}(t)=\omega_{i}({\bf I})\,t+\theta_{i}^{0}\,.$ (2.27)

We assume that the $2n$ integration constants entering this solution are
$I_{i}$, and $\theta_{i}^{0}$. The general solution for the variation
$Q^{i}=\delta q^{i}$ can then be obtained as a linear superposition of the
$2n$ particular varied solutions defined by varying the $2n$ integration
constants, i.e.

$\displaystyle Q^{i}(t)=\sum_{j}\left[C_{I_{j}}\frac{\partial}{\partial
I_{j}}q^{i}[{\bf
I},{\bm{\theta}}(t)]+C_{\theta_{j}}\frac{\partial}{\partial\theta^{0}_{j}}q^{i}[{\bf
I},{\bm{\theta}}(t)]\right]\,.$ (2.28)

This expression is the sum of some quasi-periodic functions (of the type (2))
and of the functions coming from varying the frequencies $\omega_{i}(I_{j})$,
namely

$\displaystyle t\,\sum_{j,\ell}C_{I_{j}}\frac{\partial\omega_{\ell}}{\partial
I_{j}}\frac{\partial q^{i}[{\bf I},{\bm{\theta}}]}{\partial\theta_{\ell}}\,.$
(2.29)

The latter functions are the products of $t$ and quasi-periodic functions of
time.555Here we assume a fast enough decay for the coefficients
$a^{i}_{k_{1}\cdots k_{n}}$ as the $k_{i}$ tend to infinity to ensure the
quasi-periodic nature of the right factor in Eq. (2.29). This is again a
benign behavior of linearly-growing oscillatory form, which is essentially a
quasi-periodic version of Fig. 1.

Let us now consider the generic case where the unperturbed system (2.1) is not
integrable, and exhibits a chaotic behavior. [We are still assuming that the
(unperturbed) evolution stays within a compact domain of phase-space.] In that
case, the behavior of the varied dynamical system can be much worse than in
the integrable case assumed above. First, on mid-term time scales, the
variations $Q^{i}=\delta q^{i}$ may grow exponentially with time (Lyapunov
instability). On longer time scales, such a Lyapunov exponential instability
might (via an Arnold-type diffusion) evolve into a more chaotic behavior.
Anyway, $Q^{i}(t)$ satisfies an homogeneous linear ordinary differential
equation (ODE) with time-dependent coefficients, say (in the simple, equal-
mass, potential case)

$\displaystyle\ddot{Q}^{i}+\frac{\partial^{2}V(q)}{\partial q^{i}\partial
q^{j}}Q^{j}=0\,,$ (2.30)

in which $q^{i}$ must be replaced by a solution of
$\ddot{q}^{i}+\frac{\partial V(q)}{\partial q^{i}}=0$. General theorems about
linear ODE’s then guarantee that $Q^{i}(t)$ can only have singularities at a
finite time, if the coefficients of the ODE become singular. As we assume here
that the unperturbed solution $q^{i}(t)$ is regular for all times, we are
guaranteed that the behavior of $Q^{i}(t)$ will be benign in our general sense
(i.e. no finite-time blow-up).

Let us finally mention that our variational approach can be straightforwardly
generalized to field theory systems (i.e. to an infinite number of dynamical
variables). For instance, if we start with the unperturbed Lagrangian (using
here the signature $(x_{\mu})^{2}\equiv t^{2}-{\bf x}^{2}$),

$\displaystyle{\cal L}_{0}\ =\ \frac{1}{2}(\partial_{\mu}\phi)^{2}-V(\phi)\,,$
(2.31)

we are led to the following variational Lagrangian for two real fields
$\phi(t,{\bf x})$ and $D(t,{\bf x})\equiv\delta\phi(t,{\bf x})$

$\displaystyle{\cal L}\ =\
\partial_{\mu}\phi\partial_{\mu}D-DV^{\prime}(\phi).$ (2.32)

If we use the same potential as in (2.10), the equations of motion for the
system (2.32) are

$\displaystyle\Box\phi+\omega^{2}\phi+\lambda\phi^{3}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle\Box D+(\omega^{2}+3\lambda\phi^{2})\,D$
$\displaystyle=$ $\displaystyle 0\,.$ (2.33)

The particular case of a (1+1)-dimensional spacetime has been considered (and
numerically investigated) in Ref. [13].

General mathematical results on the nonlinear Klein-Gordon equation (with
potential $\lambda\phi^{p}/p$, with $\lambda>0$ and $p<2+\frac{4}{d-2}$ in $d$
spatial dimensions) [15] guarantee the global existence of solutions of Eq.
(2) for $\phi(t,{\bf x})$, with suitable (finite-energy) Cauchy data at $t=0$.
These generic solutions have been shown to exhibit asymptotic decay in time.
We then expect that these exact results on the nonlinear Klein-Gordon equation
for $\phi$ imply (by varying the Cauchy data) a benign behavior (with
asymptotic decay) for generic solutions of the linear, varied Klein-Gordon
equation satisfied by $D=\delta\phi$. Ref. [13] numerically investigated the
case of a one-dimensional space ($d=1$) compactified on a circle. The space
compactification suppresses the asymptotic time decay of $\phi$ and seemingly
induced a mild (linear in time) growth for $D$ [13].

Evidently our variational construction can be set up basically for any system.
One can vary the Yang-Mills Lagrangian,

$\displaystyle{\cal L}^{\rm YM}_{0}\ =\ -\frac{1}{2}{\rm
Tr}\\{F_{\mu\nu}F^{\mu\nu}\\}\,,$ (2.34)

setting $A_{\mu}\to A_{\mu}+\epsilon B_{\mu}$ and keeping in (2.34) the terms
linear in $B_{\mu}$, namely

$\displaystyle{\cal L}^{\rm YM}_{1}=B_{\mu}\frac{\delta{\cal L}^{\rm
YM}[A]}{\delta A_{\mu}}\,.$ (2.35)

At the linearized level (in $A$), ${\cal L}^{\rm YM}_{1}$ describes two
massless spin-1 fields, one of which is a ghost. At the nonlinear level, the
number of degrees of freedom is preserved because of the invariance of ${\cal
L}^{\rm YM}_{1}$ under two distinct gauge transformations: the usual one
acting on $A$, and a separate one (involving $A$-covariant derivatives) acting
on $B$. This system is again a benign ghost-ridden system.

Mutatis mutandis, one can also consider the ghost-ridden action obtained by
varying the Einstein action [we use here the signature $(-+++)$ and $16\pi
G=1$],

$\displaystyle{\cal L}^{\rm E}_{0}=\sqrt{-g}g^{\mu\nu}R_{\mu\nu}[g]\,.$ (2.36)

Varying $g_{\mu\nu}\to g_{\mu\nu}+\epsilon h_{\mu\nu}$ and keeping the term
linear in $\epsilon$ yields

$\displaystyle{\cal L}^{\rm
E}_{1}=-\sqrt{-g}\left(R^{\mu\nu}[g]-\frac{1}{2}g^{\mu\nu}R[g]\right)h_{\mu\nu}\,.$
(2.37)

At the linearized level around flat spacetime
($g_{\mu\nu}=\eta_{\mu\nu}+\epsilon f_{\mu\nu}$), ${\cal L}^{\rm E}_{1}$
describes two massless spin-2 fields, one of which is a ghost. At the
nonlinear level, the number of degrees of freedom is preserved because of the
invariance of ${\cal L}^{\rm E}_{1}$ under two distinct gauge transformations:
the usual coordinate transformations, $x^{\prime\mu}=f^{\mu}(x^{\nu})$, acting
both on $g_{\mu\nu}$ and on the tensor $h_{\mu\nu}$, and a separate one
(involving $g$-covariant derivatives) acting linearly on $h$, namely
$h_{\mu\nu}\to
h_{\mu\nu}+\nabla^{g}_{\mu}\xi_{\nu}+\nabla^{g}_{\nu}\xi_{\mu}$. Note that the
Euler-Lagrange equations derived from ${\cal L}^{\rm E}_{1}$ imply that
$g_{\mu\nu}$ must satisfy the (vacuum) Einstein equations,

$\displaystyle{\cal
E}^{\mu\nu}[g_{\alpha\beta}]\equiv\sqrt{-g}\left(R^{\mu\nu}[g]-\frac{1}{2}g^{\mu\nu}R[g]\right)=0\,,$
(2.38)

while $h_{\mu\nu}$ must satisfy the linearized Einstein equations

$\displaystyle\frac{\delta{\cal E}^{\mu\nu}}{\delta
g_{\alpha\beta}}\,h_{\alpha\beta}=0\,.$ (2.39)

This system is again a benign ghost-ridden system. For instance, the
mathematical results on the global nonlinear stability of the Minkowski
spacetime [16, 17, 18] have shown the global existence of solutions of
Einstein vacuum equations for $g_{\mu\nu}(t,{\bf x})$, with small Cauchy data
at $t=0$. These generic solutions have been shown to exhibit asymptotic decay
in time. We believe that the good control (in all spacetime directions) of the
geometric properties of these nonlinear, but small perturbations of Minkowski
space [16] suffice to prove the global existence of the Green’s function
needed to solve the linearized Einstein equation satisfied by $h_{\mu\nu}$
(after a suitable gauge fixing).

The basic reason why the ghost-ridden systems considered in this section were
benign is that the equations of motion of the variables $(Q,P)$ were linear
[though influenced by the nonlinear dynamics of the unperturbed variables
$(q,p)$]. In the following, we are going to introduce more interesting ghost-
ridden systems where the dynamics of the ghost degrees of freedom is
nonlinear. It is then more delicate to delineate ghost-ridden systems that
stay benign in spite of such nonlinear interactions.

##  3 Geodesics on Lorentzian manifolds

A general class of nonlinear ghost-ridden Hamiltonians with benign solutions
are the (quadratic) Hamiltonians describing geodesic motion on geodesically
complete Lorentzian manifolds. Namely, to any given $D$-dimensional Lorentzian
manifold $M_{D}$ with metric tensor $g_{\mu\nu}(x)$ one can associate the
Hamiltonian

$\displaystyle
H(x^{\lambda},p_{\lambda})=\frac{1}{2}g^{\mu\nu}(x)p_{\mu}p_{\nu}\,.$ (3.1)

Because of the $-+\cdots+$ signature that we choose to use here, this
Hamiltonian contains $D-1$ positive terms and a ghost-like negative term. The
Hamiltonian (3.1) describes geodesic motions on $M_{D}$. More precisely, the
Hamilton equations of motion

$\displaystyle\frac{dx^{\mu}}{d\tau}$ $\displaystyle=$
$\displaystyle\frac{\partial H}{\partial p_{\mu}}=g^{\mu\nu}(x)p_{\nu}\,,$
$\displaystyle\frac{dp_{\mu}}{d\tau}$ $\displaystyle=$
$\displaystyle-\frac{\partial H}{\partial
x^{\mu}}=-\frac{1}{2}\partial_{\mu}g^{\nu\lambda}(x)p_{\nu}p_{\lambda}\,,$
(3.2)

describe a motion in the $2D$-dimensional phase-space
$(x^{\lambda},p_{\lambda})$ [the cotangent space of $M_{D}$] with respect to
an Hamiltonian “time variable” $\tau$, which is an affine parameter along the
considered geodesic. For a general curved spacetime, the only conserved
quantity of the dynamics (3) is the “energy” (the factor 2 being introduced
for convenience),

$\displaystyle E=2\,H(x,p)\equiv g^{\mu\nu}(x)p_{\mu}p_{\nu}\,.$ (3.3)

A positive value of $E$ describes spacelike geodesics, a negative value
describes timelike ones, while $E=0$ describes null geodesics. Actually, as
the affine parametrization of geodesics is defined modulo an arbitrary affine
transformation $\tau\to a\tau+b$, one can, without loss of generality only
consider the three cases $E=+1$, $E=-1$ and $E=0$. The values $E=\pm 1$ mean
that $\tau$ is equal to the proper length $\sqrt{\pm ds^{2}}$ along the
geodesic, with

$\displaystyle ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}\,.$ (3.4)

We are interested in the systems for which the Hamiltonian evolution (3) can
be continued indefinitely with respect to the Hamiltonian time variable
$\tau$. Then, as was argued in the Introduction, the ghosts are benign. In the
context of the geodesic Hamiltonian (3.1), our condition boils down to saying
that the Lorentzian manifold $(M_{D},g)$ is geodesically complete. We can
therefore conclude that the ghost-ridden Hamiltonian (3.1) defines a benign
dynamics (for all values of the energy $E$) on any geodesically complete
Lorentzian manifold.

Mathematical investigations have given large classes of geodesically complete
Lorentzian manifolds. Of particular physical significance is the fact, proven
in Refs. [19, 20], that the vacuum Einstein spacetimes close to Minkowski that
were constructed in Refs. [16, 17, 18] are geodesically complete for all
values of the energy. Evidently, this property will not extend if one
considers spacetimes containing black holes.

As a very particular type of geodesically-complete spacetimes, we can also
mention the de Sitter, as well as anti-de Sitter (AdS), spacetimes (of any
dimension) [21]. For concreteness, let us consider the AdS spacetime, with the
metric (in global coordinates)

$\displaystyle
ds^{2}=-\left(1+\frac{r^{2}}{\ell^{2}}\right)dt^{2}+\frac{dr^{2}}{1+\frac{r^{2}}{\ell^{2}}}+r^{2}d\Omega^{2}_{D-2}\,.$
(3.5)

The associated geodesic Hamitonian reads

$\displaystyle
H=-\frac{1}{2}\frac{p_{t}^{2}}{1+\frac{r^{2}}{\ell^{2}}}+\frac{1}{2}\left(1+\frac{r^{2}}{\ell^{2}}\right)p_{r}^{2}+\frac{1}{2}\frac{J^{2}}{r^{2}}\,.$
(3.6)

Here, $\ell$ is the AdS length scale giving the constant negative curvature
$K=-1/\ell^{2}$ and $J^{2}$ is the squared angular momentum linked to the
motion on the sphere $S_{D-2}$.

The AdS spacetime is a homogeneous, symmetric space, equivalent to the coset
$O(2,D-1)/O(1,D-1)$. Because of its homogeneity we can reduce the study of
geodesics to the geodesics starting from any given point, say the origin
$t=0,r=0$ in the global coordinates of (3.5). In addition, as the isotropy
group of the coset $O(2,D-1)/O(1,D-1)$, namely $O(1,D-1)$ is the local Lorentz
group, we can use this group to reduce the study of the three types of
geodesics to a particular timelike geodesic (e.g. the geodesic $r=0$), a
particular spacelike one (e.g. the radial geodesic $t=0,\Omega={\rm cst}$),
and a particular null geodesic (a radial one directed along the light cone).

In these cases, the solutions to the equations (3) are very simple:666To
simplify them still further, we have set $\ell=1$. the spacelike geodesic with
$E=1$ is $r(\tau)=\sinh\tau,t=0,\Omega={\rm cst}$, the timelike geodesic with
$E=-1$ is $r=0,t=\tau,\Omega={\rm cst}$, while the null geodesic is $r=\tau$,
$t={\rm arctan}\,\tau$, $\Omega={\rm cst}$. We see that all three types of
geodesics have an infinite affine length.

The class of Ricci-flat, near-Minkowski geodesically complete Lorentzian
manifolds mentioned above [19, 20] might naively suggest that any Lorentzian
manifold with small enough curvature will be geodesically complete (as is the
case for Riemannian manifolds). In other words, one might think that the
incompleteness of geodesics must be linked to the presence of curvature
singularities, or at least high-curvature regions. This is incorrect. The
(toy-Taub-NUT) Misner manifold [22] (see also section 5.8 in [21]) yields a
simple example of a smooth Lorentzian manifold, which is locally flat, but
geodesically incomplete (because of its nontrivial topology). The Misner
manifold is two-dimensional and has the topology ${\mathbb{R}}\times S_{1}$.
In global coordinates $(x,\phi)$, where $x\in{\mathbb{R}}$ and where
$\phi\in[0,2\pi]$ is an angle describing the circle $S_{1}$, the metric reads
777See [21] for the local coordinate transformation needed to exhibit the
flatness of the Misner metric (3.7).

$\displaystyle ds^{2}=2dxd\phi+xd\phi^{2}\,.$ (3.7)

This corresponds to the geodesic Hamiltonian multiplied by a factor two
compared to Eq. (3.1))

$\displaystyle H(x,\phi;p_{x},p_{\phi})=2p_{x}p_{\phi}-xp_{x}^{2}\,.$ (3.8)

Note that the metric and the Hamiltonian stay regular as one crosses the line
$x=0$ (which is a Killing horizon). The determinant of the metric is
everywhere equal to $-1$, and the signature is globally $-+$, so that the
Hamiltonian is not positive definite, representing locally a difference
between two squares. In other words, it involves an ordinary and a ghost
degree of freedom.

The equations of motion read

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle 2p_{\phi}-2xp_{x}\,,$
$\displaystyle\dot{\phi}$ $\displaystyle=$ $\displaystyle 2p_{x}\,,$
$\displaystyle\dot{p}_{x}$ $\displaystyle=$ $\displaystyle p_{x}^{2}\,,$
$\displaystyle\dot{p}_{\phi}$ $\displaystyle=$ $\displaystyle 0\,.$ (3.9)

Denoting here the Hamiltonian time variable as $t$ (i.e.
$\dot{x}=\frac{dx}{dt}$), it is simple to solve the equations of motion. We
see that $p_{\phi}$ is an integral of motion. Assuming that the initial value
of $p_{x}$ is not zero, say $p_{x}(0)=1/c$, the equation of motion for $p_{x}$
is easily integrated, giving

$\displaystyle p_{x}(t)=\frac{1}{c-t}\,.$ (3.10)

This leads to a blow-up for $p_{x}(t)$ at the finite time $t=c$, which can be
positive or negative. Correspondingly, $\phi(t)$ blows up logarithmically as
$t\to c$ according to

$\displaystyle\phi(t)=\phi(0)-2\ln\left(1-\frac{t}{c}\right)\,,$ (3.11)

assuming $t/c<1$. While $p_{\phi}$ stays constant, the time-evolution of the
remaining phase space variable $x(t)$ is conveniently obtained by using the
constancy of the energy: $E=H(x,\phi;p_{x},p_{\phi})$. This yields

$\displaystyle x=\frac{2p_{x}p_{\phi}-E}{p_{x}^{2}}\,.$ (3.12)

When one approaches the blow-up (where $p_{x}\to\infty$), $x$ tends to zero.

One can visualize the Misner manifold as a cylinder (with $\phi$ being the
angular coordinate, and $x$ the coordinate along its axis). The geodesic
incompleteness of the Misner manifold means that most geodesics [apart from
the ones with $p_{x}(0)=0$] spiral [either for $t<0$ or $t>0$, depending on
the sign of $p_{x}(0)$] towards the horizon circle $x=0$ by making infinitely
many turns within a finite (affine) time. This exemplifies how the
nonlinearity of the ghost-ridden Hamiltonian (3.8) leads to a finite-time
blow-up in phase space, i.e. is a system with a malignant ghost.

##  4 Modified Korteweg-de Vries equation as a benign higher-derivative model.

We started our discussion with the Pais-Uhlenbeck oscillator (1.1), a higher-
derivative model with benign ghosts. We noted in the Introduction that the
ghosts generically cease to be benign if a nonlinear interaction term is added
to the Lagrangian (1.1). The ghost models considered in Sec. 2 were benign
essentially because the ghost degrees of freedom satisfied linear equations of
motion. The ghost models considered in Sec. 3 involved nonlinear evolution
equations, and could be benign (or not) depending on the global geometric
properties of the considered Lorentzian manifold. However, they were not
models linked to higher-derivative Lagrangians.

A natural question at this stage is: Are there nonlinear higher-derivative
models with benign ghosts? In Ref.[25], one of us suggested that the usual
2-dimensional $(t,x)$ Korteweg-de Vries (KdV) system, with the roles of
temporal and spatial variables interchanged might be benign because of the
existence of infinitely many local conservation laws. Indeed, let us rename

$\displaystyle t\to X,\qquad x\to T\,.$ (4.1)

Then the local flux conservations,
$0=\partial_{t}J^{t}_{n}+\partial_{x}J^{x}_{n}\,{\equiv}\,\partial_{T}J^{x}_{n}+\partial_{X}J^{t}_{n}$,
imply the $T$-conservation of the fluxes

$\displaystyle{\cal F}_{n}=\int
dX\left[J^{x}_{n}\right]_{T=cst}\,{\equiv}\,\int
dt\left[J^{x}_{n}\right]_{x=cst}\,.$ (4.2)

Let us recall that the ordinary KdV equation for $u(t,x)$, namely888As usual,
we denote the partial derivatives of $u(t,x)$ by subscripts; e.g.
$u_{x}\equiv\partial u/\partial x$.

$\displaystyle u_{xxx}+6uu_{x}+u_{t}=0\,,$ (4.3)

derives from the action $\int dtdx\,L[\psi(t,x)]$ with the 2-dimensional
Lagrangian density

$\displaystyle L[\psi(t,x)]\ =\
\frac{1}{2}\psi_{xx}^{2}-\psi_{x}^{3}-\frac{1}{2}\psi_{t}\psi_{x}\,$ (4.4)

if one denotes $u(t,x)\equiv\psi_{x}$ after having varied over $\psi(t,x)$.

When changing the names of spacetime variables according to Eq.(4.1) and
denoting $\psi(t,x)=\psi(X,T)\to\Psi(T,X)$, the action reads $\int
dTdXL[\Psi(T,X)]$ with a Lagrangian density

$\displaystyle L[\Psi(T,X)]\ =\
\frac{1}{2}\Psi_{TT}^{2}-\Psi_{T}^{3}-\frac{1}{2}\Psi_{T}\Psi_{X}\,,$ (4.5)

which contains higher-order $T$-time derivatives. The corresponding equation
of motion

$\displaystyle u_{TTT}+6uu_{T}+u_{X}=0\,,$ (4.6)

is of third order in the $T$-time derivative of $u=\Psi_{T}$. As we will see
below, higher-order $T$-derivatives in Eq. (4.6) bring about exponential
instabilities when considering the evolution in the $T$ (i.e. $x$) direction.

In order to express our results on the KdV and modified KdV equations in a
more transparent way we willl not henceforth use the notation $T$ for $x$ and
$X$ for $t$. We will simply consider a non-standard Cauchy problem for the
equations (4.3), (4.10). Namely, instead of setting the initial value of
$u(t,x)$ at999As the KdV equation is invariant under $t$ translations (and $x$
translations), one can fix the initial-$t$ Cauchy hypersurface at $t=0$ (and
the initial-$x$ Cauchy hypersurface at $x=0$). $t=0$ [i.e. giving one function
of $x$, $v(x)$, with the condition $u(0,x)=v(x)$], we shall now pose a
“rotated” Cauchy problem on the $x=0$ axis. However, as Eq. (4.3) features the
third $x$-derivative of $u$, we must now give as Cauchy data three independent
functions of $t$, $u_{0}(t)$, $u_{1}(t)$ and $u_{2}(t)$, with the three
conditions

$\displaystyle u(t,0)=u_{0}(t)\ ,\ u_{x}(t,0)=u_{1}(t)\ ,\
u_{xx}(t,0)=u_{2}(t).$ (4.7)

Note that this means that we are now considering more general solutions of the
KdV equation. Indeed, any solution of the ordinary Cauchy problem [determined
by one function $v(x)=u(0,x)$] will evolve in $t$ (in both directions, $t>0$
and $t<0$) and will induce on the $x=0$ axis some values for the three
functions $u(t,0)$, $u_{x}(t,0)$ and $u_{xx}(t,0)$, which are not functionally
independent because they are all determined by the single function $v(x)$.

The ordinary ($t$-direction) Cauchy problem for the KdV equation (4.3) has
been shown to be globally well-posed when the single Cauchy datum $v(x)$
belongs to suitable functional spaces, namely $H^{s}$ Sobolev spaces with
$s\geq 1$ (see [23] and references therein). However, the fact that the KdV
equation has good dynamic behaviour when evolved in the $t$ direction does not
mean that the same is true for its evolution in the $x$ direction.

Indeed, we will explicitly see below that the $x$ evolution gives rise [when
considering the linearized approximation of (4.3)] to exponentially growing
modes (which are absent when considering the $t$ evolution). These
exponentially growing modes amplify any small-scale structure present in the
$x$-evolution Cauchy data $u_{0}(t)$, $u_{1}(t)$ and $u_{2}(t)$ and are not
tamed by the existence of the infinitely many conserved fluxes ${\cal F}_{n}$
defined in Eq. (4.2), because their integrands $J^{x}_{n}$ are not positive
definite.

Worse, Eq. (4.3) admits approximate (as well as exact) solutions which blow up
along a line located at some finite distance in the $x$ direction. Indeed, if
one looks for power-law singularities of $u(t,x)$ of the general form
$u(t,x)\approx C(t)[x-x_{0}(t)]^{\alpha}$, it is easily seen that they must
necessarily be of the form

$\displaystyle u(t,x)^{\rm KdV}_{\rm blowup}\ \approx\
-2\left[x-x_{0}(t)\right]^{-2}\,,$ (4.8)

with an arbitrary possible blow-up line $x=x_{0}(t)$. A specific example of
such a blow-up solution is the following exact $t$-independent solution of the
KdV equation:

$\displaystyle u(t,x)\ =\ -\frac{2}{(x-c)^{2}}\,,$ (4.9)

where $c$ is any constant. The function (4.9) has smooth Cauchy data on the
$x=0$ axis, and blows up on the line $x=c$. We also performed some numerical
simulations (with periodic Cauchy data given at $x=0$) and found that many
such initial data develop a singularity of the form (4.8) during their
$x$-direction evolution. This seems to confirm the standard “Ostrogradsky-
ghost” lore that the evolution of a higher-derivative model, such as (4.5)
becomes singular during their evolution. And one cannot expect that the
quantum problem for the system (4.5) would be benign.

However, the situation appears to be much better for the
modified101010Modified KdV equations are generally defined by replacing the
nonlinear KdV term $6uu_{x}$ by $n(n-1){\kappa}u^{(n-2)}u_{x}$. The usual KdV
equation is the case $n=3$. In this case, a rescaling of the variables can set
the coefficient ${\kappa}$ to $1$. Here we consider the next modified KdV
equation for $n=4$. In that case, a rescaling of the variables can set the
coefficient ${\kappa}$ either to $1$ or $-1$; these two cases having different
physical properties — the so-called focusing and defocusing cases. KdV
equation,

$\displaystyle u_{xxx}+12{\kappa}u^{2}u_{x}+u_{t}=0\,.$ (4.10)

This equation admits an infinite number of integrals of motion, as the
ordinary KdV equation does. The first three local conservation laws are

$\displaystyle\partial_{t}u=-\partial_{x}(u_{xx}+4{\kappa}u^{3})\,,$ (4.11)
$\displaystyle\partial_{t}u^{2}=-2\partial_{x}\left[3{\kappa}u^{4}+uu_{xx}-\frac{1}{2}u_{x}^{2}\right]\,,$
(4.12)

$\displaystyle\partial_{t}\left({\kappa}u^{4}-\frac{1}{2}u_{x}^{2}\right)\ =\
\partial_{x}\left[{\kappa}u_{x}(12u^{2}u_{x}+u_{xxx})-\frac{1}{2}u_{xx}^{2}-4{\kappa}u^{3}u_{xx}-8{\kappa}^{2}u^{6}\right]\,.$
(4.13)

We will argue in the following that, in constrast to Eq. (4.3), the equation
(4.10) does not involve a blow-up.

Let us first briefly discuss the linearized KdV equation,

$\displaystyle u_{xxx}+u_{t}=0\,,$ (4.14)

which describes the fluctuations around the trivial solution $u(t,x)=0$ of the
usual KdV equation (4.3), as well as of all its modified versions with the
nonlinear term $n(n-1)\kappa u^{(n-2)}u_{x}$, $n\geq 3$.

Let us emphasize the drastic difference between the $t$-evolution Cauchy
problem, and the $x$-evolution Cauchy problem of the linearized equation
(4.14). Equation (4.14) can be solved by decomposing the solution $u(t,x)$ in
plane waves $e^{{\rm i}(\omega t+kx)}$. This yields the dispersion law

$\displaystyle\omega=k^{3}\,.$ (4.15)

If one poses the usual Cauchy problem with some Fourier-transformable initial
data

$\displaystyle u(0,x)=v(x)\equiv\int\frac{dk}{2\pi}\,v(k)e^{{\rm i}kx}\,,$
(4.16)

the $t$-evolution of the initial data $v(x)$ yields the solution (valid for
both signs of $t$)

$\displaystyle u(t,x)=\int\frac{dk}{2\pi}\,v(k)e^{{\rm i}(k^{3}t+kx)}\,.$
(4.17)

The important point here is that $u(t,x)$ is obtained from $v(k)$ by a purely
oscillatory complex kernel $e^{{\rm i}(k^{3}t+kx)}$ of unit modulus. It has
been shown that this oscillatory kernel has smoothing properties (see, e.g.,
[23]). This allows one to take the initial data in low-$s$ Sobolev spaces
$H^{s}$ (describing pretty rough initial data) [23].

On the other hand, if one considers the $x$-evolution Cauchy problem, one
starts from three independent functions of $t$ along the $x=0$ axis:
$u(t,0)=u_{0}(t)$, $u_{x}(t,0)=u_{1}(t)$ and $u_{xx}(t,0)=u_{2}(t)$, see Eq.
(4.7). Assuming that the three Cauchy data $u_{a}(t)$, $a=0,1,2$, are Fourier-
transformable, we can represent them as

$\displaystyle u_{a}(t)\equiv\int\frac{d\omega}{2\pi}\,u_{a}(\omega)e^{{\rm
i}\omega t}\,.$ (4.18)

The three Cauchy data determine a unique solution which, when decomposed in
plane waves, satisfies the same dispersion law (4.15) as before. However, the
dispersion law (4.15) must now be solved for $k$ in terms of $\omega$. As it
is a cubic equation in $k$, it has three different roots, namely

$\displaystyle k_{a}(\omega)=\omega^{\frac{1}{3}}j^{a}\,,$ (4.19)

where $\omega^{\frac{1}{3}}$ denotes the unique real cubic root of $\omega$
and where $j^{a}=1,j,j^{2}$ (with $j\equiv e^{\frac{2\pi\rm
i}{3}}=-\frac{1}{2}+{\rm i}\frac{\sqrt{3}}{2}$) are the three complex roots of
unity. This yields a solution for $u(t,x)$ of the form

$\displaystyle
u(t,x)=\sum_{a=0,1,2}\int\frac{d\omega}{2\pi}\,v_{a}(\omega)e^{{\rm i}(\omega
t+k_{a}x)}\,,$ (4.20)

where the three coefficients $v_{a}(\omega)$ are (uniquely) determined by the
three initial conditions at $x=0$, namely by the following system of three
linear equations111111The determinant of this system is $-3\sqrt{3}\omega$.

$\displaystyle u_{0}(\omega)$ $\displaystyle=$
$\displaystyle\sum_{a=0,1,2}v_{a}(\omega)\,,$ $\displaystyle u_{1}(\omega)$
$\displaystyle=$ $\displaystyle{\rm
i}\sum_{a=0,1,2}k_{a}(\omega)v_{a}(\omega)\,,$ $\displaystyle u_{2}(\omega)$
$\displaystyle=$
$\displaystyle-\sum_{a=0,1,2}k_{a}^{2}(\omega)v_{a}(\omega)\,.$ (4.21)

The point of this exercise was to exhibit the fact that, when considering the
$x$ evolution with arbitrary Cauchy data $u_{0}(t)$, $u_{1}(t)$, $u_{2}(t)$,
the solution involves exponentially growing modes in the $x$ direction, linked
to the fact that ${\rm i}k_{1}={\rm i}\omega^{\frac{1}{3}}j$ and ${\rm
i}k_{2}={\rm i}\omega^{\frac{1}{3}}j^{2}$ have real parts
$\pm\frac{\sqrt{3}}{2}\omega^{\frac{1}{3}}$ (this holds for both signs of
$x$). This instability is associated with the presence of higher time
derivatives.121212We hasten to comment, however, that higher derivatives do
not necessarily entail such an instability. For example, it does not show up
in the equations of motion for the Pais-Uhlenbeck system (1.1).

From the mathematical point of view, this means that high-frequency (HF)
wiggles, in the sense of high $\omega$, in the initial Cauchy data (at $x=0$)
will be amplified, on both sides of the $x$ axis, by the exponentially growing
factor $e^{\frac{\sqrt{3}}{2}|\omega|^{\frac{1}{3}}|x|}$. This indicates that
the Cauchy problem will be well-posed only if one takes initial data whose
Fourier transforms $v_{a}(\omega)$ decrease sufficiently fast as
$|\omega|\to+\infty$. As a minimum condition for a local existence theorem,
one should require the Fourier transforms $v_{a}(\omega)$ to decrease like
$e^{-c|\omega|^{\frac{1}{3}}}$ for some positive constant $c$. This
essentially defines the $s=3$ Gevrey class of functions on ${\mathbb{R}}$
(see, e.g.,[24]). Note that such a regularity condition is stronger than
infinite differentiability, but weaker than analyticity (which corresponds to
a decrease of the Fourier transforms of the type $e^{-c|\omega|}$, i.e. to the
$s=1$ Gevrey class).

We therefore expect that it will be mathematically possible to prove (at least
for sufficiently weak Cauchy data) a local existence theorem for solutions of
the $x$-evolution of the modified KdV equation (4.10) in suitable Gevrey
classes (say, with $1<s\leq 3$).131313 Nader Masmoudi confirmed (private
communication to TD) that it was likely that taking Cauchy data in the $s=3$
Gevrey class would suffice for local existence of the solution of the $x$
evolution. This would still leave open the issue of finding adequate function
spaces for global existence.

We wish to further argue that the special nonlinearity of Eq. (4.10), when
taking ${\kappa}>0$, is likely to allow sufficiently smooth $x=0$ Cauchy data
to define global solutions, extending to arbitrary large values of $|x|$. Our
main argument for believing that the local $x$-evolution of Eq. (4.10) can be
extended to large values of $|x|$ is the absence of blow-up solutions. Indeed,
let us look for power-law singularities of $u(t,x)$ of the general form

$\displaystyle u(t,x)\approx\beta(t){[}x-x_{0}(t){]}^{\alpha}\,,$ (4.22)

with $\alpha<0$. Inserting such an asymptotic behavior in Eq. (4.10), it is
easily seen that the $t$-derivative term $u_{t}$ is necessarily subdominant
with respect to the $x$-derivative terms. Therefore, blow-up solutions must,
in lowest approximation, solve the truncated equation

$\displaystyle 0=u_{xxx}+12{\kappa}u^{2}u_{x}\ =\ \frac{\partial}{\partial
x}(u_{xx}+4{\kappa}u^{3})\,.$ (4.23)

This equation is a third-order dynamical equation for the $x$ evolution which
admits the “constant of motion”

$\displaystyle u_{xx}+4{\kappa}u^{3}\ =\ C\,.$ (4.24)

In turn, the latter equation can be rewritten as $u_{xx}=-\frac{dV(u)}{du}$,
i.e. as the Newtonian equation of motion in $x$, playing the role of time, for
a particle with position $u$ in the following potential:

$\displaystyle V(u)={\kappa}u^{4}-Cu\,.$ (4.25)

When ${\kappa}>0$ (the focusing case), this potential grows for large values
of $|u|$, and therefore prevents the existence of blow-up solutions.141414By
contrast, for the ordinary KdV, one obtains a non-confining cubic potential,
and the corresponding equations of motion admit the singular run-away
solutions (4.9). General modified KdV equations with the nonlinearity
$n(n-1){\kappa}u^{(n-2)}u_{x}$ give rise to a potential
$V(u)={\kappa}u^{n}-Cu$, which is confining if $n$ is even and ${\kappa}>0$.
Technically, if we look for blow-up solutions of the type (4.22), one finds
that the blow-up exponent $\alpha$ must be equal to $\alpha=-1$ and that the
coefficient $\beta$ must satisfy the cubic equation
$\beta(2{\kappa}\beta^{2}+1)=0$. When ${\kappa}<0$ (the defocusing case),
there exist possible non-trivial (real) blow-up solutions with
$\beta=\pm(-2{\kappa})^{-\frac{1}{2}}$. [These solutions correspond to the
“fall” of the $u$ particle down the unstable quartic potential
${-|\kappa|}u^{4}-Cu$. ] By contrast, when ${\kappa}>0$, the only real
solution of the cubic equation of $\beta$ is $\beta=0$. In other words, there
exists no real blow-up solutions of the type indicated in Eq. (4.22).151515We
are aware that the ansatz (4.22) might be too restrictive. It was shown in
Ref. [27] that, in the formally confining case $n=6$ with ${\kappa}>0$, a
blow-up can occur in the $t$-evolution for data close to the corresponding
soliton.

A different (though related) analytic argument indicating the absence of real
blow-up solutions comes from the analysis of the scaling properties of the
modified KdV equation. It is easily seen that Eq. (4.10) is invariant under
the rescalings $u=\lambda_{u}\bar{u}$, $x=\lambda_{x}\bar{x}$,
$t=\lambda_{t}\bar{t}$ if

$\displaystyle\lambda_{t}=\lambda_{x}^{3}\ ;\ \lambda_{u}=\lambda_{x}^{-1}\,.$
(4.26)

The quantities $xu$ and $\frac{x}{t^{1/3}}$ are invariant under these
rescalings. Using also the space and time translational invariance of the
modified KdV equation, we can look for scaling solutions of the type

$\displaystyle u(t,x)=\frac{\alpha}{[3(t-t_{0})]^{1/3}}w(z)\,,$ (4.27)

where

$\displaystyle z\equiv\frac{x-x_{0}}{[3(t-t_{0})]^{1/3}}\,.$ (4.28)

Inserting the ansatz (4.27) in Eq. (4.10) and choosing the normalization
constant $\alpha=\frac{1}{\sqrt{2}}$, it is easily checked that the function
$w(z)$ must satisfy the equation

$\displaystyle
0=w^{\prime\prime\prime}+(6{\kappa}w^{2}-z)w^{\prime}-w=\frac{d}{dz}\left[w^{\prime\prime}+2{\kappa}w^{3}-zw\right]\,.$
(4.29)

Denoting as $C$ the constant value of the bracket in the last right-hand side,
we conclude that $w(z)$ satisfies a second-order equation of the type

$\displaystyle w^{\prime\prime}=-2{\kappa}w^{3}+zw+C\,.$ (4.30)

This is a Painlevé II equation [26]. In general, Painlevé equations have
(moving) pole singularities. And indeed, a local analysis of Eq. (4.30)
(keeping the leading-order terms161616Note in passing that this leading-order
equation describe the dynamics of a particle in the potential
$V(w)=\frac{1}{2}{\kappa}w^{4}$ $w^{\prime\prime}\approx-2{\kappa}w^{3}$)
shows that (4.30) admits simple poles, $w(z)=R/(z-z_{0})$, as local
singularities if the residue $R$ satisfies the equation ${\kappa}R^{2}=-1$.
When ${\kappa}<0$, the residue $R$ will be real, so that real solutions of
Painlevé II can have (and generally do have) poles on the real $z$ axis. The
existence of a real simple pole at $z=z_{0}$ would then correspond to a
singular (blow-up) behavior of $u(t,x)$ of the form
$u(t,x)\propto\left[x-x_{0}-z_{0}[3(t-t_{0})]^{1/3}\right]^{-1}$. However,
when ${\kappa}>0$, real solutions cannot have real poles. This excludes the
existence of (real) singular scaling solutions.

Another way to understand why the negative-${\kappa}$ modified KdV equation
has blow-up solutions is to use its relation with the usual KdV equation (4.3)
(which admits real blow-up solutions, as we emphasized above). Indeed, it is
well-known that the Miura transformation,

$\displaystyle u\ =\ -(2v^{2}+\sqrt{2}v_{x})\,,$ (4.31)

transforms the ordinary KdV equation (4.3) for $u(t,x)$ into the equation

$\displaystyle v_{t}-12v^{2}v_{x}+v_{xxx}\ =\ 0\,,$ (4.32)

which coincides with the modified KdV equation (4.10) for $v(t,x)$, with
${\kappa}=-1$.

To confirm our conjecture that the $x$-evolution of sufficiently smooth Cauchy
data (4.7) stays bounded when evolved with the modified KdV equation with
${\kappa}>0$, we have performed some numerical simulations (done with
Mathematica) for the case ${\kappa}=+1$. To simplify the numerical analysis we
imposed [as is allowed by Eq. (4.10)] periodicity along the $t$ direction
Using scaling invariance, we can assume $2\pi$ periodicity:

$\displaystyle u(t+2\pi,x)\ =\ u(t,x)\,.$ (4.33)

In order to study the effect of the nonlinear term $12u^{2}u_{x}$, we used
$t$-periodic Cauchy data for which the nonlinear term is initially (i.e. at
$x=0$) larger than the linear term $u_{t}$. In particular, we used the Cauchy
data

$\displaystyle u(t,0)\ =\ \cos t,\ u_{x}(t,0)=\cos t,\ u_{xx}(t,0)\ =\ 0\,.$
(4.34)

We first checked that the use of such Cauchy data for the modified KdV
equation with ${\kappa}=-1$ was leading quite fast (namely at $x=0.887717$) to
a singularity. By contrast, our numerical simulations of the $x$ evolution of
the ${\kappa}=+1$ modified KdV equation showed that $u(t,x)$ stayed bounded
for all the values of $x$ that we explored. This is illustrated in Fig. 2
which displays the solution $u(t,x)$ generated by the Cauchy data (4.34) in
the domain $0\leq t\leq 2\pi$, $0\leq x\leq 15$.

Figure 2: Solution $u(t,x)$ of the $\kappa=+1$ modified KdV equation (4.10)
corresponding to the $t$ periodicity (4.33) and the $x=0$ Cauchy data (4.34).

Fig. 2 illustrates the benign nature of the ghostful $x$-dynamics of the
modified KdV equation in the positive-$\kappa$ case. The presence (when
neglecting the term $u_{t}$) of an approximate $x$ dynamics governed by the
confining potential (4.25) reflects itself in the oscillations in the $x$
evolution of $u(t,x)$, i.e. in the “stormy-sea” aspect of $u(t,x)$ in the
$x>5$ part of Fig. 2. Note that we have taken here analytic ($C^{\omega}$)
data, which generate (at least locally) an analytic solution. We leave to
future work to clarify how less regular Cauchy data (e.g. taken in Gevrey
classes, or suitable Sobolev-type spaces) would evolve under the $x$
evolution.

We expect generic Cauchy data (4.7) for the $x$ evolution to evolve into
stormy-sea solutions similar to the one illustrated in Fig. 2. However, there
will also exist special (measure-zero) Cauchy data that will evolve into much
tamer solutions. Indeed, if we start with a smooth function of $x$, say
$v(x)$, and use it as unique Cauchy datum at $t=0$ [namely, $u(t=0,x)=v(x)$],
its $t$-evolution will define a smooth solution $u^{v\,\rm sol}(t,x)$. The
restriction to the $x=0$ axis of $u^{v\,\rm sol}$, $u^{v\,\rm sol}_{x}$, and
$u^{v\,\rm sol}_{xx}$ will then define Cauchy data for the $x$-evolution that
generate the smooth solution $u^{v\,\rm sol}(t,x)$. We have numerically
checked this fact by using solitonic solutions of the modified KdV equation.

Let us recall that, like the ordinary KdV equation, the modified equation
(4.10) admits solitonic solutions. Indeed, one can look for travelling-wave
solutions $u(t,x)=u(x+ct)$, moving with some celerity $c$, by inserting the
ansatz $u(t,x)=u(\bar{x})$, with $\bar{x}\equiv x+ct$, in Eq. (4.10). It is
easily seen that this yields the equation

$\displaystyle\frac{\partial}{\partial\bar{x}}\left[cu+4{\kappa}u^{3}+u_{\bar{x}\bar{x}}\right]=0\,.$
(4.35)

Denoting as $C_{0}$ the constant quantity within the bracket, we then get the
following second-order equation for the function $u(\bar{x})$:

$\displaystyle u_{\bar{x}\bar{x}}=-\frac{d}{du}{\cal V}(u)\,,$ (4.36)

with a potential function ${\cal V}(u)$ now given by

$\displaystyle{\cal V}(u)={\kappa}u^{4}+\frac{1}{2}cu^{2}-C_{0}u\,.$ (4.37)

We are again reduced to the dynamics of a particle moving in the confining
quartic potential ${\cal V}(u)$, considered for some fixed ${\kappa}>0$ (say,
${\kappa}=+1$) . The general solutions of this problem then depend on three
parameters: the celerity $c$, the constant $C_{0}$, and the constant energy of
the $\bar{x}$ dynamics:

$\displaystyle E=\frac{1}{2}u_{\bar{x}}^{2}+{\cal V}(u)\,.$ (4.38)

The usually considered solitonic solutions (such that $u(\bar{x})$ tend to
zero when $\bar{x}\to\pm\infty$) are obtained by taking $c<0$, $C_{0}=0$ and
$E=0$. This gives a symmetric double-well potential:
${\kappa}\,u^{4}-\frac{1}{2}|c|\,u^{2}$. The zero-energy solution then
describes a $u$-motion which starts, at “time”$\bar{x}=-\infty$, at $u=0$ with
zero “velocity” $u_{\bar{x}}$, glides down (say) to the right, reflects on the
right wall of the double well and then turns back to end up again at $u=0$
when $\bar{x}=+\infty$. The explicit form of the corresponding solution is

$\displaystyle{u({\bar{x}})\ =\
\sqrt{\frac{|c|}{2{\kappa}}}\,\frac{1}{\cosh(\sqrt{|c|}\,\bar{x})}\,.}$ (4.39)

Here, we are interested in constructing periodic travelling-wave solutions
satisfying $u(\bar{x}+\bar{T})=u(\bar{x})$ for some $\bar{T}$. Such solutions
can be easily constructed by considering bound mechanical motions in the
potential ${\cal V}(u)$ having a non-zero energy. We have already given such
oscillatory solutions in a (symmetric) quartic potential (i.e. for $C_{0}=0$)
in Eq. (2.16) above. The corresponding periodic travelling-wave soliton is
then of the form

$\displaystyle u(\bar{x})\ =\ x_{0}\,{\rm
cn}[\Omega(\bar{x}-{\bar{x}}_{0}),k]\,,$ (4.40)

as simply obtained by using in Eqs. (2.13), (2.16), (2) the replacements

$\displaystyle x\ \to\ u,\;t\to\bar{x},\;\omega^{2}\ \to\ c,\;\lambda\ \to\
4{\kappa},\;N\ \to E\,.$ (4.41)

We have used the $t$-periodic Cauchy data defined by restricting $u(x+c\,t)$,
and its first two $x$ derivatives, to $x=0$ to check the accuracy of our
numerical simulations. The numerical solution generated from these Cauchy data
agreed well with the analytical solution for values of $x$ of order $\bar{T}$.
For larger values they exhibited some numerical noise. The specific form of
the numerical noise depended on the numerical scheme used. When using the same
scheme as the one used to produce Fig. 2, the noise stayed at low frequencies.
This gives us confidence that Fig. 2 yields a reasonably accurate picture of
the benign nature of the $x$-evolution of smooth Cauchy data.

##  5 Discrete nonlinear systems with benign ghosts.

In this final section, we define some higher-derivative dynamical models
having only a finite number of degrees of freedom and exhibiting a benign
behavior in their evolution. These models are constructed by discretizing the
modified KdV equation in the $t$ direction, keeping continuous the $x$
evolution. [We recall that $x$ is our timelike evolution variable.] This will
replace the partial differential equation (4.10) by a system of coupled ODEs
with respect to $x$.

Similarly to the derivation of the ordinary KdV equation from the Lagrangian
(4.4), the modified KdV equation (4.10) follows from the two-dimensional
action $S=\int dtdxL[\psi(t,x)]$, with the field-theory Lagrangian density

$\displaystyle L[\psi(t,x)]\ =\
\frac{1}{2}\psi_{xx}^{2}-{\kappa}\psi_{x}^{4}-\frac{1}{2}\psi_{x}\psi_{t}\,.$
(5.1)

After varying the action with respect to $\psi(t,x)$, one gets Eq. (4.10) by
substituting $\psi_{x}(t,x)\to u(t,x)$.

We can then define a discretized version of the modified KdV dynamics by
assuming that the variable $t$ takes only the discrete values
$t=h,2h,\cdots,Nh$, for some integer $N\geq 2$, and by replacing the
continuous time derivative $\psi_{t}$ by a discrete (symmetric) time
derivative $\frac{\psi(t+h,x)-\psi(t-h,x)}{2h}$. This yields an action of the
form $S_{N}=\int dx\,L_{N}$, where the Lagrangian $L_{N}$ is given by a sum of
$N$ terms:

$\displaystyle
L_{N}=\sum_{t=h}^{t=Nh}\left[\frac{1}{2}[\psi_{xx}(t,x)]^{2}-{\kappa}[\psi_{x}(t,x)]^{4}-\frac{1}{2}\psi_{x}(t,x)\frac{\psi(t+h,x)-\psi(t-h,x)}{2h}\right]\,.$
(5.2)

To define the model we further need to specify boundary conditions. Namely, we
need to define $\psi(0h,x)$ and $\psi[(N+1)h,x]$, which enter the discrete $t$
derivative for $t=h$ and $t=Nh$, respectively. This can be done in two
different ways: (i) we can use Dirichlet-type boundary conditions, namely
$\psi(0h,x)=0$ and $\psi[(N+1)h,x]=0$ or (ii) periodic boundary conditions,
$\psi(0h,x)=\psi(Nh,x)$, and $\psi[(N+1)h,x]=\psi(h,x)$. The periodicity
condition can only be imposed when $N\geq 3$.

We expect that taking larger and larger values of $N$ would allow one to
simulate better and better the continuous theory (though the presence of chaos
might make such a convergence non uniform in $x$).

The simplest discretized model is obtained by choosing $N=2$ and taking
Dirichlet-type boundary conditions. This model has only two dynamical
variables: $\psi(x)\equiv\psi(h,x)$ and $\chi(x)\equiv\psi(2h,x)$. We recall
that $x$ is playing the role of time. The Lagrangian $L_{2}\equiv L_{N=2}$
reads

$\displaystyle L_{2}\ =\
\frac{1}{2}\psi_{xx}^{2}+\frac{1}{2}\chi_{xx}^{2}-{\kappa}\psi_{x}^{4}-{\kappa}\chi_{x}^{4}-\frac{1}{4h}\psi_{x}\chi+\frac{1}{4h}\chi_{x}\psi\,.$
(5.3)

Adding a total $x$-derivative, the last two terms can be traded for
$+\frac{1}{2h}\chi_{x}\psi$. Using suitable rescalings, we can set
${\kappa}=1$ and $h=\frac{1}{2}$. For simplicity, we will use these values in
the following.

Defining the two new dynamical variables
$a(x)\equiv\psi_{x},b(x)\equiv\chi_{x}$, the equations of motion derived from
the Lagrangian (5.3) read

$\displaystyle a_{xxx}+12a^{2}a_{x}+b$ $\displaystyle=$ $\displaystyle 0\,,$
$\displaystyle b_{xxx}+12b^{2}b_{x}-a$ $\displaystyle=$ $\displaystyle 0\,.$
(5.4)

This is a system of two coupled higher-order evolution equations. The
important point is that, while we had technical difficulties in numerically
simulating in a stable manner the $x$-evolution of the (periodic) modified KdV
equation (4.10), we could easily perform long-term integrations of the coupled
system (5) up to $x=10000$ and more in a numerically stable manner. And though
the conserved energy of the system, namely

$\displaystyle E\ $ $\displaystyle=$ $\displaystyle\
\frac{1}{2}(\psi_{xx}^{2}+\chi_{xx}^{2})-3(\psi_{x}^{4}+\chi_{x}^{4})-\psi_{x}\psi_{xxx}-\chi_{x}\chi_{xxx}$
$\displaystyle=$ $\displaystyle\
\frac{1}{2}(a_{x}^{2}+b_{x}^{2})-3(a^{4}+b^{4})-aa_{xx}-bb_{xx}$ (5.5)

[which is a discrete version of the conserved current on the right-hand side
of Eq. (4.12)], is not positive-definite and can take arbitrarily positive or
negative values, our numerical simulations indicate that the classical motions
have a benign behavior, without any blow-up. In other words, the simple
discrete model $L_{2}$, Eq. (5.3), provides a nontrivial example of an
interacting higher-derivative system with benign ghosts.

One can also write the corresponding Ostrogradsky Hamiltonian. It reads

$\displaystyle H\ =\
\frac{1}{2}(P_{\psi}^{2}+P_{\chi}^{2})+p_{\psi}\psi_{x}+p_{\chi}\chi_{x}+\psi_{x}^{4}+\chi_{x}^{4}-\psi\chi_{x}\,,$
(5.6)

where $\psi,\psi_{x},\chi,\chi_{x}$ should all be treated as independent
variables with corresponding canonical momenta $p_{\psi},P_{\psi},p_{\chi}$
and $P_{\chi}$, respectively. The Hamilton equations of motion following from
(5.6) coincide with (5).

For completeness, let us also write the periodic, discrete dynamics defined by
$L_{N}$, Eq. (5.2), with $N\geq 3$. Setting as before ${\kappa}=1$ and
$h=\frac{1}{2}$ and defining the $N$ $x$-evolving discrete variables
$a^{k}(x)\equiv\psi(kh,x)_{x}$, with $k=1,\cdots,N$, and the periodicity
conditions, $a^{0}\equiv a^{N}$ and $a^{N+1}\equiv a^{1}$, the equations of
motion following from Eq. (5.2) read

$\displaystyle a^{k}_{xxx}+12(a^{k})^{2}a^{k}_{x}\ =\ a^{k+1}-a^{k-1}\,.$
(5.7)

The conserved energy is

$\displaystyle E\ =\
\sum_{k=1}^{N}\left[\frac{1}{2}(a^{k}_{x})^{2}-3(a^{k})^{4}-a^{k}a^{k}_{xx}\right]\,.$
(5.8)

The periodic systems with $N\geq 3$ enjoy also a second integral of motion
(linked to the periodicity in $t_{k}=kh$):

$\displaystyle Q\ =\ \sum_{k=1}^{N}\left[a^{k}_{xx}+4(a^{k})^{3}\right]\,.$
(5.9)

In the continuous theory, this follows from integrating over the periodic
variable $t$ the current in the right-hand side of Eq.(4.11). By contrast, the
currents in the higher conservation laws of the modified KdV equation,
starting with Eq. (4.13), do not translate into integrals of motion of the
discrete systems.

The $N$-th periodic discrete system has only two integrals of motion, Eqs.
(5.8), (5.9), and $2N$ pairs of phase space variables. [For $N=2$ one had only
one integral of motion for four pairs of phase space variables.] The systems
(5.2) are thus not integrable and their trajectories are expected to exhibit a
chaotic behavior. Our numerical simulations did confirm this expectation.

The numerical confirmations of the benign nature of the discrete systems up to
rather large values of $x$ can be considered as a further argument in favour
of the conjecture that the continous modified KdV system (4.10) is also
benign.

##  6 Conclusions

We presented several nontrivial examples of higher-derivative systems
including ghosts, but where the ghosts are of benign nature, i.e. they do not
lead to a blow-up in the classical case (and hence will not give rise to
unitarity violation in the quantum case). The most interesting example is the
two-dimensional modified KdV system (5.1) (with ${\kappa}>0$), viewed as a
higher-derivative evolution in the $x$ variable. We presented several
arguments strongly indicating that this system does not involve any blow-up
during its $x$ evolution. Mathematically proving that this is the case (for
smooth enough data) is a challenge for the future.

In the last section, we presented mechanical systems, with a finite number of
degrees of freedom, which are $t$-discretized avatars of the modified KdV
system (5.1). These systems are of interest on their own, notably because they
provide a set of nontrivial interacting higher derivative systems with benign
ghosts. Such systems were not known before.

## Acknowledgements

We are indebted to Piotr Chrusciel, Alberto De Sole, Victor Kac, Nader
Masmoudi, Frank Merle, and Laure Saint-Raymond for informative discussions.

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