Modified Korteweg-de Vries Equation as a System with Benign Ghosts111A talk at
the AAMP conference, Prague, September 2021, to be published in Acta
Polytechnica.

Andrei Smilga

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

<EMAIL_ADDRESS>

###### Abstract

We consider the modified Korteweg-de Vries equation,
$u_{xxx}+6u^{2}u_{x}+u_{t}\ =\ 0$, and explore its dynamics in spatial
direction. Higher $x$ derivatives bring about the ghosts. We argue that these
ghosts are benign, i.e. the classical dynamics of this system does not involve
a blow-up. This probably means that also the associated quantum problem is
well defined.

##  1 Introduction

A system with ghosts is by definition a system where the quantum Hamiltonian
has no ground state so that its spectrum involves the states with arbitrarily
low and arbitrarily high energies. In particular, all non-degenerate theories
with higher derivatives in the Lagrangian (but not only them!) involve ghosts.
The ghosts show up there already at the classical level: the Ostrogradsky
Hamiltonians of higher derivative systems [1] include the linear in momenta
terms and are thus not positive definite [2]. This brings about the ghosts in
the quantum problem [3, 4].

In many cases, ghost-ridden systems are sick — the Schrödinger problem is not
well posed and unitarity is violated. Probably, the simplest example of such a
system is a system with 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)

Classically, for certain initial conditions, the particle falls to the center
in a finite time, as is shown in Fig. 1.

Figure 1: Falling on the center for the Hamiltonian (1) with $m=1$ and
$\kappa=.05$. The energy is slightly negative. The particles with positive
energies escape to infinity.

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 [5]. 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, i.e. the system does not run into
a singularity in finite time,222We still call a system benign if it runs into
a singularity at $t=\infty$. Such systems have well-defined quantum dynamics.
This refers e.g. to the problem of motion in a uniform electric field (see
e.g. [6], §24) and also to the inversed oscillator with the Hamiltonian
$H=(p^{2}-x^{2})/2$. In the latter problem, the classical trajectories $x(t)$
grow exponentially with time, but the quantum problem is still benign (see
e.g. [7], Ch. 3, corollary 13). The spectrum in this case is continuous, as it
is for the uniform field problem. its quantum dynamics will also be benign,
irrespectively of whether the spectrum has, or does not have, a bottom.

This all refers to ordinary mechanical or field theory systems, where energy
is conserved and the notion of Hamiltonian exists. The ghosts in gravity
(especially, in higher-derivative gravity) is a special issue that we are not
discussing here.

Besides malignant ghost-ridden systems, of which the system (1) with
$m{\kappa}>1/8$ represents an example, there also are many systems with ghosts
which are benign — unitarity is preserved and the quantum Hamiltonian is self-
adjoint with a well-defined real spectrum. To begin with, such is the famous
Pais-Uhlenbeck oscillator [8] — a higher derivative system with the Lagrangian

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

This system is free, its canonical Hamiltonian can be reduced to the
difference of the two oscillator Hamiltonians by a canonical transformation
[9]. The first example of a nontrivial benign ghost system involving nonlinear
interactions was built up in [10]. For other such examples, see Refs. [11, 12,
13, 14, 15].

In recent [16], we outlined two wide classes of benign ghost systems: (i) the
systems obtained by a variation of ordinary systems and involving, compared to
them, a double set of dynamic variables and (ii) the systems describing
geodesic motion over Lorenzian manifolds. In addition, we noticed that the
evolution of the modified Korteweg-de Vries (MKdV) system (9) in the spatial
direction also exhibits a benign ghost dynamics. This report is mostly based
on section 4 of Ref. [16] that deals with the MKdV dynamics.

##  2 Spatial dynamics of KdV and MKdV equations

Consider first the ordinary KdV equation,

$\displaystyle u_{xxx}+6uu_{x}+u_{t}=0\,,$ (3)

where $u_{x}=\partial u/\partial x,u_{t}=\partial u/\partial t$ etc.) It has
an infinite number of integrals of motion and is exactly soluble.333Exact
solvability always makes the behaviour of a system more handy. In particular,
many mechanical models including benign ghosts, which were mentioned above,
are exactly solvable. The KdV equation is derived from the field Lagrangian

$\displaystyle L[\psi(t,x)]\ =\
\frac{1}{2}\psi_{xx}^{2}-\psi_{x}^{3}-\frac{1}{2}\psi_{t}\psi_{x}\,$ (4)

if one denotes $u(t,x)\equiv\psi_{x}$ after having varied over $\psi(t,x)$.
This Lagrangian involves higher spatial derivatives, but not higher time
derivatives and does not involve ghosts in the ordinary sense. We can,
however, simply rename

$\displaystyle t\to X,\qquad x\to T\,,$ (5)

in which case the equation acquires the form

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

and higher time derivatives appear. According to our conjecture, to study the
question of whether the quantum Hamiltonian corresponding to the thus rotated
Lagrangian (4) is Hermitian and unitarity is preserved, it is sufficient to
study its classical dynamics: if it does not involve a blow-up and all
classical trajectories exist at all times $T$, one can be sure that the
quantum system is also benign.

Note that the question whether or not blowing up trajectories are present is
far from being trivial. The ordinary Cauchy problem for the equation (4)
consists in setting the initial value of $u(t_{0},x)$ at a given time moment,
say, $t_{0}=0$. And we are interested now [staying with Eq. (4) and not
changing the name of the variables according to (5)] in the Cauchy problem in
$x$ direction. The presence of third spatial derivatives in (4) makes it
necessary to define at the line $x=x_{0}$ three different functions:
$u(t,x_{0}),u_{x}(t,x_{0})$ and $u_{xx}(t,x_{0})$. The presence of three
arbitrary functions makes the space of solutions to the spatial Cauchy problem
much larger than for the ordinary Cauchy problem. The solutions to the latter
represent a subset of measure zero in the set of the solutions in the former,
and the fact that the solutions to the ordinary Cauchy problem are all benign
does not mean that it is also the case for the rotated $x$-directed problem.

And, indeed, for the ordinary KdV equation (4) the problem is not benign. It
is best seen if we choose a $t$-independent Ansatz $u(t,x)\to u(x)$ and plug
it into (3). The equation is reduced to

$\displaystyle\partial_{x}(u_{xx}+3u^{2})=0\qquad\Longrightarrow\qquad
u_{xx}+3u^{2}=C\,.$ (7)

This equation describes the motion in the cubic potential $V(u)=u^{3}-Cu$. It
has blow-up solutions. If $C=0$, they read

$\displaystyle u(x)\ =\ -\frac{2}{(x-x_{0})^{2}}\,.$ (8)

However, the situation is completely different for the modified KdV
equation,444In Ref. [16], we wrote this equation as  $u_{xxx}+12\kappa
u^{2}u_{x}+u_{t}=0\,$ and kept $\kappa$ in all subsequent formulas. But here
we have chosen for simplicity to fix $\kappa=1/2$.

$\displaystyle u_{xxx}+6u^{2}u_{x}+u_{t}=0\,.$ (9)

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

$\displaystyle\partial_{t}u=-\partial_{x}(u_{xx}+2u^{3})\,,$ (10)
$\displaystyle\partial_{t}u^{2}=-2\partial_{x}\left[\frac{3}{2}u^{4}+uu_{xx}-\frac{1}{2}u_{x}^{2}\right]\,,$
(11)

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

For the time-independent Ansatz, we obtain instead of (7)

$\displaystyle\partial_{x}(u_{xx}+2u^{3})=0\qquad\Longrightarrow\qquad
u_{xx}+2u^{3}=C\,.$ (13)

This describes the motion in a quartic potential $V(u)=u^{4}/2-Cu$. This
motion is bounded, the solutions being elliptic functions.

This observation presents an argument that the rotated Cauchy problem for the
equation (9) with arbitrary initial conditions on the line $x=const$ might be
benign.

Note that this behaviour is specific for the equation (9) with the positive
sign of the middle term (the so-called focusing case). Plugging the time-
independent Ansatz in the defocusing MKdV equation,555The coefficient 6 is a
convention. It can be changed by rescaling $t$ and $x$. But the sign stays
invariant under rescaling.

$\displaystyle u_{xxx}-6u^{2}u_{x}+u_{t}=0\,,$ (14)

the problem would be reduced to the motion in the potential $V(u)=-u^{4}/2-Cu$
characterized by a blow-up. This conforms to the well-known fact that any
solution $u(t,x)$ of the ordinary KdV equation is related to a solution
$v(t,x)$ of the defocusing MKdV equation by the Miura transformation,

$\displaystyle u\ =\ -(v^{2}+v_{x})\,.$ (15)

A different (though related) analytic argument indicating the absence of real
blow-up solutions for the focusing MKdV equation comes from the analysis of
its scaling properties. It is easily seen that Eq. (9) is invariant under the
rescalings $u=\lambda_{u}\bar{u}$, $x=\lambda_{x}\bar{x}$,
$t=\lambda_{t}\bar{t}$ if

$\displaystyle\lambda_{t}=\lambda_{x}^{3},\qquad\lambda_{u}=\lambda_{x}^{-1}\,.$
(16)

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

$\displaystyle u(t,x)=\frac{1}{[3(t-t_{0})]^{1/3}}w(z)\,,$ (17)

where

$\displaystyle z\ =\ \frac{x-x_{0}}{[3(t-t_{0})]^{1/3}}\,.$ (18)

Inserting the ansatz (17) in Eq. (9), one easily verifies that the function
$w(z)$ satisfies the equation

$\displaystyle
0=w^{\prime\prime\prime}+(6w^{2}-z)w^{\prime}-w=\frac{d}{dz}\\!\left[w^{\prime\prime}+2w^{3}-zw\right]$
(19)

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,

$\displaystyle w^{\prime\prime}=-2w^{3}+zw+C\,.$ (20)

For the equation (14), the same analysis would give the equation

$\displaystyle w^{\prime\prime}=2w^{3}+zw+C\,.$ (21)

These are Painlevé II equations [17]. In general, Painlevé equations have pole
singularities. And indeed, a local analysis of Eq. (21) (keeping the leading-
order terms $w^{\prime\prime}\approx 2w^{3}$) shows that (21) admits simple
poles, $w(z)\approx\pm 1/(z-z_{0})$. 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 $u(t,x)\propto\left[x-x_{0}-z_{0}[3(t-t_{0})]^{1/3}\right]^{-1}$.
But for the equation (20) [and hence for (9)], the singularities are absent.

The third argument in favor of the conjecture that the $x$ evolution of
sufficiently smooth Cauchy data on the line $x=const$ for the MKdV equation
(9) does not bring about singularities in $u(t,x)$ comes from numerical
simulations. To simplify the numerical analysis, we considered the problem on
the band $0\leq t\leq 2\pi$, where we imposed [as is allowed by Eq. (9)]
periodic boundary conditions:

$\displaystyle u(t+2\pi,x)\ =\ u(t,x)\,.$ (22)

We have chosen the Cauchy data

$\displaystyle u(t,0)\ =\ \sin t,\ u_{x}(t,0)\ =\ u_{xx}(t,0)\ =\ 0\,.$ (23)

We first checked that the use of such Cauchy data for the defocusing MKdV
equation (14) was leading to a blow-up rather fast (at $x=2.2630\ldots$). This
is illustrated in Fig. 2, where the function $u(t,x)$ is plotted just before
the blow-up, at $x=2.26$.

Figure 2: $u(t,x=2.26)$ for the defocusing MKdV.

By contrast, our numerical simulations of the $x$ evolution of the focusing
MKdV equation showed that $u(t,x)$ stayed bounded for all the values of $x$
that we explored. We met, however, another problem associated with the
instability of Eq. (9) under high-frequency (HF) perturbations.

Suppressing the nonlinear term in the KdV or MKdV equations, we obtain

$\displaystyle u_{xxx}+u_{t}=0\,,$ (24)

This equation describes the fluctuations around the solution $u(t,x)=0$. Its
analysis gives us an idea on the behaviour of fluctuations around other
solutions. Decomposing $u(t,x)$ as a Fourier integral, in plane waves $e^{{\rm
i}(\omega t+kx)}$, we obtain the dispersion law

$\displaystyle\omega=k^{3}\,.$ (25)

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

$\displaystyle u(0,x)=v(x)\equiv\int\frac{dk}{2\pi}\,v(k)e^{{\rm i}kx}\,,$
(26)

the time evolution of the initial data $v(x)$ yields the solution

$\displaystyle u(t,x)=\int\frac{dk}{2\pi}\,v(k)e^{{\rm i}(k^{3}t+kx)}\,.$ (27)

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.,
[18]). This allows one to take the initial data in low-$s$ Sobolev spaces
$H^{s}$ (describing pretty rough initial data).

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 $u_{xx}(t,0)=u_{2}(t)$, as in
(23). Assuming that the three Cauchy data $u_{a}(t)$, $a=0,1,2$, are Fourier-
transformable, we can represent them as

$\displaystyle u_{a}(t)\ =\ \int\frac{d\omega}{2\pi}\,u_{a}(\omega)e^{{\rm
i}\omega t}\,.$ (28)

The three Cauchy data determine a unique solution which, when decomposed in
plane waves, satisfies the same dispersion law (25) as before. However, the
dispersion law (25) must now be solved for $k$ in terms of $\omega$. As it is
a cubic equation in $k$, it has three different roots:

$\displaystyle k_{a}(\omega)\ =\ \omega^{\frac{1}{3}}e^{2\pi ia/3}\,\qquad
a=0,1,2\,.$ (29)

This yields a solution for $u(t,x)$ of the form

$\displaystyle
u(t,x)=\sum_{a=0,1,2}\int\frac{d\omega}{2\pi}\,v_{a}(\omega)e^{{\rm i}(\omega
t+k_{a}x)}\,,$ (30)

where the three coefficients $v_{a}(\omega)$ are uniquely determined by the
three initial conditions at $x=0$. 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 imaginary parts of
$k_{1}(\omega)$ and $k_{2}(\omega)$.

Figure 3: $u(t,x=3)$ for the focusing MKdV. Figure 4: $u_{x}(t,x=3)$ for the
focusing MKdV.

This can be avoided if the initial data are sufficiently smooth, not involving
HF modes. As a minimum condition for a local existence theorem, one should
require the Fourier transforms $v_{a}(\omega)$ to decrease like
$e^{-\alpha|\omega|^{\frac{1}{3}}}$ for some positive constant $\alpha$.666See
Ref.[16] for more detailed discussion.

However, it is difficult to respect these essential smoothness constraints on
the behavour of $u(t,x)$ in the numerical calculations. The standard
Mathematica algorithms do not do so, and that is why we, starting from some
values of $x$, observe the HF noise in our results.

In Figs. 3,4, we present the results of numerical calculations of $u(t,x)$ and
$u_{x}(t,x)$ for $x=3$. There is no trace of blow-up. For the plot of
$u(t,x)$, one also does not see a HF noise, but it is seen in the plot for
$u_{x}(t,x)$. For larger values of $x$, the noise also shows up in the plot of
$u(t,x)$. At $x\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle>}}{{\sim}}\;$}3.8$, the
noise overwhelms the signal.

The observed noise is a numerical effect associated with a finite computer
accuracy. To confirm this, we performed a different calculation choosing the
initial conditions which correspond to the exact solitonic solution to Eq.
(9).

The soliton is a travelling wave, $u(t,x)=u(x-ct)\equiv u(\bar{x})$. Plugging
this Ansatz into (9), we obtain an ordinary differential equation

$\displaystyle\frac{\partial}{\partial\bar{x}}\left[u_{\bar{x}\bar{x}}+2u^{3}-cu\right]=0\,.$
(31)

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

$\displaystyle u_{\bar{x}\bar{x}}=-\frac{d}{du}{\cal V}(u)\,,$ (32)

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

$\displaystyle{\cal V}(u)\ =\ \frac{u^{4}-cu^{2}}{2}-C^{\prime}u\,.$ (33)

As was also the case for the time-independent Ansatz, the problem is reduced
to the dynamics of a particle moving in the confining quartic potential ${\cal
V}(u)$. The trajectory of the particle depends on three parameters: the
celerity $c$, the constant $C^{\prime}$, and the particle energy,

$\displaystyle E=\frac{1}{2}u_{\bar{x}}^{2}+{\cal V}(u)\,.$ (34)

The usually considered solitonic solutions (such that $u(\bar{x})$ tends to
zero when $\bar{x}\to\pm\infty$) are obtained by taking $c>0$, $C^{\prime}=0$
(so that the potential represents a symmetric double-well potential) and
$E=0$. The zero-energy trajectory describes a particle starting at “time”
$\bar{x}=-\infty$, at $u=0$ with zero “velocity” $u_{\bar{x}}$, gliding down,
say, to the right, reflecting on the right wall of the double well and then
turning back to end up again at $u=0$ when $\bar{x}=+\infty$. The explicit
form of the corresponding solution defined on the infinite $(t,x)$ plane is

$\displaystyle u(t,x)\ =\ \frac{\sqrt{c}}{\cosh[\sqrt{c}\,(x-ct)]}\,.$ (35)

To make contact with our numerical calculations, we need, however, a periodic
soliton solution. Such solutions can be easily constructed by considering
bound mechanical motions in the potential ${\cal V}(u)$ having a non-zero
energy. Periodic solutions exist both for positive and negative $c$. The
trajectories are the elliptic functions. It was more convenient for us to
assume $c=-|c|$, in which case we could make contact with Ref. [10], where the
expressions for the trajectories of motion in the same quartic potential were
explicitly written, one only had to rename the parameters. Choosing $E=1$ and
$c=-1$, we obtain the following solution:

$\displaystyle u(t,x)\ =\ {\rm cn}\left[\sqrt{3}(x+t),m\right]\,,$ (36)

where cn$(z)$ is the Jacobi elliptic cosine function with the elliptic modulus
$m=1/3$. The function (36) is periodic both in $t$ and $x$ with the period

$\displaystyle T\ =\ L\ =\frac{4}{\sqrt{3}}K\left(\frac{1}{3}\right)\ \approx
4\,.$ (37)

We fixed the initial conditions for $x=0$ and periodic conditions in time as
is dictated by (36) and then numerically solved (9). The numerical solution
should reproduce the exact one, and it does for $x\lower
3.01385pt\hbox{$\;\stackrel{{\scriptstyle\textstyle<}}{{\sim}}\;$}4$. However,
at larger values of $x$, the HF noise appears. The result of the calculation
for $x=4.5$ is given on Fig. 5.

Figure 5: HF noise for the periodic soliton evolution. $x=4.5$.

One can suppress the HF noise by increasing the step size, but then the form
of the soliton is distorted. To find a numerical procedure that suppresses the
noise and gives correct results for large values of $x$ remains a challenge
for future studies.

##  3 Discrete models with benign ghosts

One of the possible solutions to this numerical problem could consist in
discretizing the model in time direction and assuming that the variable $t$
takes only the discrete values $t=h,2h,\cdots,Nh$, for some integer $N\geq 3$
and by replacing the continuous time derivative $\psi_{t}$ by a discrete
(symmetric) time derivative $[\psi(t+h,x)-\psi(t-h,x)]/(2h)$. Then the
Lagrangian777It is quite analogous to (4). After variation with respect to
$\psi(t,x)$, one gets Eq. (9) after posing $u(t,x)=\psi_{x}(t,x)$.

$\displaystyle L[\psi(t,x)]\ =\
\frac{\psi_{xx}^{2}-\psi_{x}^{4}-\psi_{x}\psi_{t}}{2}\,.$ (38)

acquires the form

$\displaystyle
L_{N}=\sum_{k=1}^{N}\left\\{\frac{[\psi_{xx}(kh,x)]^{2}-[\psi_{x}(kh,x)]^{4}}{2}-\right.$
$\displaystyle\left.\frac{1}{2}\psi_{x}(kh,x)\frac{\psi[(k+1)h,x]-\psi[(k-1)h,x]}{2h}\right\\}\,,$
(39)

where we impose the periodicity: $\psi(0,x)\equiv\psi(Nh,x)$ and
$\psi[(N+1)h,x]\equiv\psi(h,x)$.888It is also possible to impose the
Dirichlet-type boundary conditions, $\psi(0h,x)=\psi[(N+1)h,x]=0$. For $N=2$,
periodicity cannot be imposed and Dirichlet conditions is the only option.

The Lagrangian (3) includes a finite number of degrees of freedom and
represents a mechanical system. This system involves higher derivatives in $x$
(playing the role of time) and hence involves ghosts. Defining the new
dynamical variables $a^{k}(x)=\psi_{x}(kh,x)$, the equations of motion derived
from the Lagrangian (3) read

$\displaystyle a^{k}_{xxx}+6(a^{k})^{2}a^{k}_{x}\ +\
\frac{a^{k+1}-a^{k-1}}{2h}\ =\ 0\,.$ (40)

There are two integrals of motion: the energy

$\displaystyle E\ =\
\sum_{k=1}^{N}\left[\frac{(a^{k}_{x})^{2}-3(a^{k})^{4}}{2}-a^{k}a^{k}_{xx}\right]\,.$
(41)

and

$\displaystyle Q\ =\ \sum_{k=1}^{N}\left[a^{k}_{xx}+2(a^{k})^{3}\right]\,.$
(42)

The expression (41) is the discretized version of the integral

$\displaystyle\int dt\left[\frac{(u_{x})^{2}-3u^{4}}{2}-uu_{xx}\right]$ (43)

in the continuous MKdV system, which is conserved during the $x$ evolution, as
follows from the local conservation law (11). The second integral of motion is
related to the conservation law (10). By contrast, the currents in the higher
conservation laws of the MKdV equation, starting with Eq.(2), do not translate
into integrals of motion of the discrete systems. We have only two integrals
of motion and many variables, which means that the equation system (40) is not
integrable and exhibits a chaotic behaviour.

We fed these equations to Mathematica and found out that their solution stays
bounded up to $x=10000$ and more — the ghosts are benign! This represents a
further argument in favour of the conjecture that also in the continuous
theory the evolution in spatial direction is benign. Indeed, one may 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 $x$).

Anyway, we tried the solitonic initial conditions and found out that the
discrete system for large $N=350$ (the limit of Mathematica skills) behaves
better than the PDE. As is seen from Fig. 6, the discrete solution stays close
to the exact soliton solution up to $x\approx 10$, to be compared to $x\approx
4.5$, which was the horizon of the numerical procedure of the previous
section. Hopefully, a clever mathematician, an expert in numerical
calculations, would be able to increase the horizon still more…

Figure 6: The solution of the system (40) for $a^{N}(x)$ ($N=350,\ h=T/N)$.

Lastly, we note that, irrespectively to the relationship of the systems (3) to
the MKdV equation, these systems represent an interest by their own because
they provide a set of nontrivial interacting higher derivative systems with
benign ghosts. Such systems were not known before.

## Acknowledgements

I am grateful to the organizers of the AAMP conference for the invitation to
make a talk there. Working on our paper [16], we benefited a lot from
illuminating discussions with Piotr Chrusciel, Alberto De Sole, Victor Kac,
Nader Masmoudi, Frank Merle and Laure Saint-Raymond.

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