# Higher time-derivative theories from space-time interchanged integrable
field theories

Andreas Fring∙, Takano Taira∘ and Bethan Turner∙
$\bullet$ Department of Mathematics, City, University of London, Northampton
Square,
London EC1V 0HB, UK
$\circ$ Research Fellow of Japan Society for Promotion of Science, Institute
of Industrial
Science, The University of Tokyo 5-1-5 Kashiwanoha, Kashiwa 277-8574, Japan
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract:

We compare a relativistic and a nonrelativistic version of Ostrogradsky’s
method for higher-time derivative theories extended to scalar field theories
and consider as an alternative a multi-field variant. We apply the schemes to
space-time rotated modified Korteweg-de Vries systems and, exploiting their
integrability, to Hamiltonian systems built from space-time rotated inverse
Legendre transformed higher-order charges of these systems. We derive the
equal-time Poisson bracket structures of these theories, establish the
integrability of the latter theories by means of the Painlevé test and
construct exact analytical period benign solutions in terms of Jacobi elliptic
functions to the classical equations of motion. The classical energies of
these partially complex solutions are real when they respect a certain
modified CPT-symmetry and complex when this symmetry is broken. The higher
order Cauchy and initial-boundary value problem are addressed analytically and
numerically. Finally, we provide the explicit quantization of the simplest
mKdV system, exhibiting the usual conundrum of having the choice between
either having to deal with a theory that includes non-normalizable states or
spectra that are unbounded from below. In our non-Hermitian system the choice
is dictated by the correct sign in the decay width.

††conference: Higher time-derivative theories from x-t interchanged integrable
field theories

##  1 Introduction

Standard quantum field theories only contain derivative terms in their kinetic
part up to second order. Ostrogradky’s fundamental theorem [1, 2] is the main
reason for this limit, in particular on the higher time-derivatives as their
presence will inevitably lead to linear momentum field terms that can not be
eliminated by partial integration and subsequent dropping of surface terms.
While the presence of these terms is highly unappealing, since at least
classically they lead to instabilities that can be reached in finite time,
higher time-derivative theories (HTDT) posses also many very attractive
features such as for instance being renormalizable [3, 4].

These latter properties have nurtured the hope that one might eventually
overcome the problematic issues and exploit the positive features. Several
proposals of dealing with the deficiencies have been made such as for instance
a Dirac-Pauli quantization scheme with an indefinite metric [5] or the complex
extensions of the models to a ${\cal PT}$-symmetric non-Hermitian system [6,
7]. A useful insight is gained by separating out theories that are already
very problematic on the classical level and making a distinction between ghost
states of malevolent and benign nature. On the classical level the malevolent
states are identified as solutions that reach singularities in finite time,
whereas benign solutions are oscillatory or might only diverge in infinite
time [8, 9, 10, 11].

Leaving the problematic features largely aside, HTDT appeared in a number of
different contexts, such as in attempts to quantise gravity when adding
curvature squared terms to the Einstein-Hilbert action [12] or the resolution
of the cosmological singularity problem [13]. Finite temperature physics may
be formulated in terms of HTDT [14] and also HTDT black hole solutions have
been constructed [15]. Their BRST symmetries have been identified [16, 17] in
some quantum field theories, they were used in a massless particle
descriptions of bosons and fermions [18, 19], used in the description of the
Higgs sector in the standard model and also some supersymmetric versions have
been investigated [20, 21]. Classical and quantum stability properties of HTDT
were investigated in [22, 23, 24, 25, 26, 27].

The theories studied in detail so far are often simply ad hoc examples or
tailored for the specific contexts they are considered in as mentioned above.
A systematic study of larger classes of HTDT remains an open issue. Recently
Smilga [9] pointed out that higher charges of integrable systems naturally
involve terms with higher derivatives and when interpreted as Hamiltonians
could be suitable candidates to identify classes of theories with benign
sectors in their parameter space. It turns out that even in their original
form these theories have not been studied systematically, but exhibit
interesting features, as seen in higher charge Hamiltonian systems of affine
Toda lattice theories type [28] and scattering theories for multi-particle
Calogero and Calogero-Moser systems [29]. The HTDT can be obtained from higher
charges of these theories by interchanging space and time. Some properties of
models obtained in this manner are trivially invariant when $x$ has been
interchanged with $t$. However, time is being kept as the flow parameter so
that for instance equal-time Poisson bracket change, the Cauchy and the
initial-boundary value problem are altered as they require more independent
boundary functions etc. As a specific class of integrable systems we
investigate here in more detail space-time rotated modified Korteweg-de Vries
(mKdV) systems as HTDT.

Our manuscript is organised as follows: In section 2 a comparison between
different variants of Ostrogradsky’s classical method adapted to scalar field
theories is carried out. We consider an alternative equivalent scheme in form
of a multi-field theory in which all higher time-derivatives are hidden and
set up the procedure on how to build HTDT Hamiltonian systems from higher
charges of integrable systems. In section 3 we apply the schemes to mKdV
systems initially for generic values of $n$ for which we derive the equal-time
Poisson bracket relations. Subsequently we derive separately for the KdV
($n=3$) and the standard mKdV ($n=4$) the higher charge Hamiltonians, convert
the entire theory into a multi-field theory that hides the higher derivatives
and establish their integrability by means of the Painlevé test. In section 4
we construct exact, partially complex, periodic solutions of benign nature to
the equations of motion in terms of Jacobi elliptic functions. We calculate
their classical energies, which turn out to be real when they are invariant
under certain ${\cal CPT}$-symmetries and complex when this symmetry is
broken. We demonstrate that the Cauchy and the initial-value boundary problem
is not solved with standard solutions for the $n=2$-theory, and identify some
sectors for the $n=3,4$-theories where this is possible. In section 5 we carry
out the quantization of the $n=2$-theory. Our conclusions are stated in
section 6.

##  2 Ostrogradsky’s method for scalar field theories

In this section we compare three versions of Ostrogradsky’s method for HTDT
extended to scalar field theories: a relativistic, a nonrelativistic, and a
multi-field variant. In each of them we recall and elaborate on the procedures
to derive the Hamiltonian densities from a given Lagrangian density involving
higher derivative terms in the time variable $t$ by employing Ostrogradsky’s
classical scheme [1, 2] adapted to scalar field theories.

###  2.1 Lorentz invariant formulation

We start with a Lorentz invariant formulation as outlined in [30, 31, 32] for
relativistic field theories that generalise Ostrogradsky’s method further from
quantum mechanics to field theories. We consider a generic Lagrangian density
depending on a scalar field $\varphi$ and its higher derivatives with respect
to space and time

${\cal
L}(\varphi,\varphi_{\mu},\ldots,\varphi_{\mu_{1},\ldots,\mu_{m}}),\qquad\mu_{i}=x,t$
(1)

with $m$ denoting the highest order of the derivatives. Our metric is taken to
be flat. The generalised momenta derived from ${\cal L}$ fall into two
different types as

$\pi^{\mu_{1}\dots\mu_{m}}:=\frac{\partial{\cal
L}}{\partial\varphi_{\mu_{1}\dots\mu_{m}}},\qquad\pi^{\mu_{1}\dots\mu_{i}}:=\frac{\partial{\cal
L}}{\partial\varphi_{\mu_{1}\dots\mu_{i}}}-\partial_{\mu_{i+1}}\pi^{\mu_{1}\dots\mu_{i}\mu_{i+1}},\quad
i=1,\ldots,m-1.$ (2)

The Legendre transformation in terms of the canonical momenta then defines the
Hamiltonian density as

${\cal
H}=\pi^{\mu}\varphi_{\mu}+\dots+\pi^{\mu_{1}\ldots\mu_{m-1}}\varphi_{\mu_{1}\ldots\mu_{m-1}}+\pi^{\mu_{1}\ldots\mu_{m}}\hat{\varphi}_{\mu_{1}\ldots\mu_{m}}-{\cal
L}(\varphi,\varphi_{\mu},\ldots,\hat{\varphi}_{\mu_{1},\ldots,\mu_{m}}),$ (3)

where $\hat{\varphi}_{\mu_{1},\ldots,\mu_{m}}$ is to be understood as
replacing

$\varphi_{\mu_{1},\ldots,\mu_{m}}\rightarrow\hat{\varphi}_{\mu_{1},\ldots,\mu_{m}}(\varphi,\varphi_{\mu},\ldots,\varphi_{\mu_{1},\ldots,\mu_{m-1}};\pi^{\mu_{1},\ldots,\mu_{m}}),$
(4)

by solving (2). The Lorentz invariant Hamilton field equations then result to

$\displaystyle\partial_{\mu}\varphi$ $\displaystyle=$
$\displaystyle\frac{\partial{\cal
H}}{\partial\pi^{\mu}},\quad\,\,\,\,\,\partial_{\mu}\varphi_{\nu}=\frac{\partial{\cal
H}}{\partial\pi^{\mu\nu}},\quad\ldots\quad\,\partial_{\nu}\varphi_{\mu_{1}\dots\mu_{m-1}}=\frac{\partial{\cal
H}}{\partial\pi^{\nu\mu_{1}\ldots\mu_{m-1}}},$ (5)
$\displaystyle\partial_{\mu}\pi^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{\partial{\cal
H}}{\partial\varphi},\quad\partial_{\nu}\pi^{\mu\nu}=-\frac{\partial{\cal
H}}{\partial\varphi_{\mu}},\quad\ldots\,\,\partial_{\nu}\pi^{\mu_{1}\dots\mu_{m-1}\nu}=-\frac{\partial{\cal
H}}{\partial\varphi^{\mu_{1}\ldots\mu_{m-1}}}.$ (6)

The generalisation of the above to relativistic scalar field theories
involving multiple fields is straightforward. For our purposes here we reduce
the dependence on the $x$-derivatives, thus ending up with nonrelativistic
scalar field theory.

###  2.2 Nonrelativistic formulation

For a nonrelativistic theory we reduce the set of values $\mu_{i}$ can take
and consider Lagrangian densities that only involve a first order derivative
in $x$ and time-derivatives up to order $m$

${\cal L}(\varphi,\varphi_{x},\varphi_{t}\ldots,\varphi_{mt}).$ (7)

In the defining relations of the momenta we take $\mu_{i}=t$ for
$i=1,\ldots,m$ and do not associate any canonical momenta to derivatives with
respect to $x$. Thus (2) becomes

$\pi^{mt}:=\frac{\partial{\cal
L}}{\partial\varphi_{mt}},\qquad\pi^{nt}:=\frac{\partial{\cal
L}}{\partial\varphi_{nt}}-\partial_{t}\pi^{(n+1)t},\quad n=1,\ldots,m-1.$ (8)

Replacing iteratively the generalized momenta $\pi^{(n+1)t}$ in the second
relation in (8) we obtain

$\pi_{\ell}:=\pi^{\ell
t}=\sum_{k=\ell}^{m}(-1)^{k-\ell}\partial^{k-\ell}_{t}\left(\frac{\partial{\cal
L}}{\partial\varphi_{kt}}\right),\qquad\ell=1,\ldots,m,$ (9)

which corresponds to a formula that may be found in [33] after (2.11) in
there. The equal-time canonical Poisson bracket relations between the
canonical momentum fields $\pi_{i}$ and the canonical coordinate fields
$\Phi_{i}$ are

$\left\\{\Phi_{i}(x),\pi_{j}(x^{\prime})\right\\}=\delta_{ij}\delta(x-x^{\prime}),\qquad\text{with}\,\,\,\Phi_{i}:=\varphi_{(i-1)t},\quad
i,j=1,\ldots,m.$ (10)

The Legendre transformation leading to the Hamiltonian density is then
entirely expressed in terms of these fields

${\cal H}=\sum_{k=1}^{m-1}\pi_{k}\Phi_{k+1}+\pi_{m}\hat{\varphi}_{mt}-{\cal
L}(\Phi_{1},\ldots,\Phi_{m},\hat{\varphi}_{mt}),$ (11)

where $\hat{\varphi}_{mt}$ means that we replace

$\varphi_{mt}\rightarrow\hat{\varphi}_{mt}(\Phi_{1},\ldots,\Phi_{m};\pi_{m});$
(12)

by solving (8). Since $\varphi_{mt}$ is not a canonical coordinate field one
has achieved in this way that ${\cal H}$ only depends on canonical coordinate
and momentum fields. The time-evolution of the canonical fields is then given
by

$\left(\Phi_{i}\right)_{t}=\frac{\delta{\cal
H}}{\delta\pi_{i}}=\sum_{n=0}^{\infty}(-1)^{n}\frac{\partial{\cal
H}}{\partial\left[(\pi_{i})_{nx}\right]},\qquad\left(\pi_{i}\right)_{t}=-\frac{\delta{\cal
H}}{\delta\Phi_{i}}=\sum_{n=0}^{\infty}(-1)^{n}\frac{\partial{\cal
H}}{\partial\left[(\Phi_{i})_{nx}\right]}.$ (13)

Notice that ${\cal H}$ in (11) is not obtained as a direct special case from
(3), as a term of the form $\pi^{x}\varphi_{x}$ is not included, since we have
not identified $\pi^{x}$ as a canonical momentum.

###  2.3 Higher time-derivative theories in disguise as multi-field theories

It is well known, see e.g. [34], that the canonical Hamiltonian for the
standard KdV equation can be derived from a Legendre transformed Lagrangian
involving two scalar fields, instead of one, upon the implementation of Dirac
constraints. For the space-time rotated version this implies that the higher
time-derivatives are hidden in some extra fields, so that one can simply use
the standard variational scheme with no need to invoke the conceptually more
involved Ostrogradsky method at the cost of more algebraic complexity. In this
section we show in more generality how to systematically reformulate HTDT as
multi-field theories. Our starting point is a higher time-derivative
Lagrangian of the form (7) and the aim is to construct a transformation
$\Gamma$ that converts it into a multi-field Lagrangian ${\cal L}^{\prime}$
involving at most first order derivatives in time

${\cal
L}(\varphi,\varphi_{x},\varphi_{t}\ldots,\varphi_{mt})\xrightarrow{\Gamma}{\cal
L}^{\prime}\left(\phi_{1},(\phi_{1})_{x},(\phi_{1})_{t},\phi_{2},(\phi_{2})_{t},\ldots,\phi_{m^{\prime}},(\phi_{m^{\prime}})_{t}\right),$
(14)

where $m^{\prime}:=[m/2+1]$ with $[x]$, being the greatest integer function
returning the greatest integer less than or equal to $x$. We demand that
${\cal L}\equiv{\cal L}^{\prime}$ up to surface terms in the time integrations
so that the resulting Euler-Lagrange equations from both are identical. For
${\cal L}^{\prime}$ they are simply computed in the conventional manner as

$\sum_{n=0}^{m^{\prime}}\left\\{\frac{\partial{\cal
L}^{\prime}}{\partial\phi_{n}}-\partial_{t}\left[\frac{\partial{\cal
L}^{\prime}}{\partial(\phi_{n})_{t}}\right]\right\\}=\partial_{x}\left[\frac{\partial{\cal
L}^{\prime}}{\partial(\phi_{n})_{x}}\right].$ (15)

The conversion $\Gamma$ is achieved by relating the canonical momenta
$\pi_{\ell}$ obtained from ${\cal L}$ by means of (9) to $\pi^{\prime}_{\ell}$
computed from ${\cal L}^{\prime}$ as

$\pi_{\ell}=(-1)^{\ell}\frac{\partial^{\ell-1}\pi^{\prime}_{\ell}}{\partial
t^{\ell-1}},\qquad\text{with}\,\,\pi^{\prime}_{\ell}=\frac{\partial{\cal
L}^{\prime}}{\partial\phi_{2\ell-1}},\quad\ell=1,\ldots,m^{\prime}.$ (16)

Having computed all canonical momenta $\pi_{\ell}$, we solve these constraints
by integrating out these equations to determine ${\cal L}^{\prime}$

${\cal
L}^{\prime}=\\!\\!\int\left[(-1)^{\ell}\int\pi_{\ell}dt^{\ell-1}\right]d\varphi_{(2\ell-1)t}+f_{\ell}\left(\varphi,\varphi_{x},\varphi_{t}\ldots\check{\varphi}_{(2\ell-1)t}\ldots,\varphi_{(m+1)t}\right),\,\,\ell=1,\ldots,m^{\prime},$
(17)

where $\check{\varphi}_{(2\ell-1)t}$ indicates that the dependence on these
field is missing in $f_{\ell}$, as they result as integration functions from
the integrations with respect to these field. Subsequently the unknown
functions $f_{\ell}$ can be determined by comparing the $m^{\prime}$
equations. As we will see below the time integrations over the canonical
momenta $\pi_{\ell}$ are often trivially performed.

###  2.4 Higher time-derivative Hamiltonians from space-time rotated higher
charges

Our intention is to apply the above schemes not only to the standard space-
time rotated Hamiltonians, but also to rotated versions of higher charges
$Q_{n}$ interpreted as Hamiltonians. Here we briefly summarise how we
construct our models. We start with a Hamiltonian density ${\cal H}$
associated to an integrable system so that we can construct higher charge
densities ${\cal Q}_{n}$. Subsequently, we interpret these charge densities as
Hamiltonian densities, and use an inverse Legendre transform to construct
their associated Lagrangian densities ${\cal L}_{n}$. We then exchange space
and time leading to a “rotated” Lagrangian ${\cal L}^{r}_{n}$, which, given
the nature of the models we started with, will inevitably involve higher order
derivatives in time. Finally we employ Ostrogradsky’s generalised scheme to
derive the corresponding Hamiltonian densities ${\cal H}^{r}_{n}$. Our
construction scheme is summarised as

${\cal H}\xrightarrow{\text{integrability}}{\cal
Q}_{n}\xrightarrow{\text{inverse Legendre transform}}{\cal
L}_{n}\xrightarrow{x\leftrightarrow t}{\cal
L}^{r}_{n}\xrightarrow{\text{Ostrogradsky scheme}}{\cal H}^{r}_{n}$ (18)

The conserved charges $Q_{n}=\int{\cal Q}_{n}dx$ involving the densities
${\cal Q}_{n}$ obey conservation laws of the form

$\left({\cal
Q}_{n}\right)_{t}=\left(\chi_{n}\right)_{x},\qquad\Rightarrow\qquad\frac{dQ_{n}}{dt}=\int\left({\cal
Q}_{n}\right)_{t}dx=\int\left(\chi_{n}\right)_{x}dx=0,$ (19)

where we assume that the flux $\chi_{n}$ vanishes or exactly cancels at the
boundaries. Note that while the Euler-Lagrange equations maintain their form
under an exchange of $x$ and $t$, that is not the case for Hamilton’s field
equations. The equations resulting from ${\cal H}^{r}_{n}$ are therefore
different from those obtained from simply rotating ${\cal Q}_{n}$ and will
lead to a new set of equations of motion.

Next we apply the schemes from above to various integrable field theoretical
systems.

##  3 Canonical higher time-derivative Hamiltonians

In this section, we derive canonical Hamiltonians, their canonical variables
including their mutual Poisson brackets for a family of HTDT corresponding to
rotated versions of general mKdV Hamiltonians with a space-time exchange and
those resulting from higher charges as outlined in section 2.4. We compare the
different schemes from sections 2.1, 2.2 and 2.3, one presenting a Lorentz-
invariant version leading to Hamiltonians that are in general not preserved
over time, one leading to a nonrelativistic version with conserved
Hamiltonians and an equivalent version in terms of multiple fields with time-
derivatives at most of order one.

###  3.1 Standard Hamiltonian for modified KdV systems, generic $n$

In order to provide a point of reference, we start with the standard
Lagrangian for the particle representation of the KdV system [35] in a
slightly generalised form that includes modified KdV systems. It is well known
[34], that the canonical Hamiltonian for the standard KdV equation can be
obtained through a Legendre transformation of a Lagrangian involving two
scalar fields, $\phi$ and $\psi$, along with their first-order derivatives,
after the implementation of Dirac constraints. The Lagrangian for the mKdV
system reads

${\cal
L}=\frac{1}{2}\phi_{t}\phi_{x}+\phi_{x}\psi_{x}+\phi_{x}^{n}+\frac{1}{2}\psi^{2},\qquad
n\in\mathbb{N},$ (20)

where $n=3$ and $n=4$ are special as they correspond to the integrable
standard KdV system and standard modified KdV system, respectively. For
simplicity we refer here by modified KdV systems to all cases with integer
values $n\neq 3$. The systems for $n>4$ are not integrable, see e.g. [36]. The
Euler-Lagrange equations derived from ${\cal L}$ become

$\displaystyle\frac{\partial{\cal
L}}{\partial\phi}-\partial_{\mu}\left(\frac{\partial{\cal
L}}{\partial_{\mu}\phi}\right)=0,\quad\Rightarrow\quad\phi_{xt}+n(n-1)\phi_{x}^{n-2}\phi_{xx}+\psi_{xx}=0,$
(21) $\displaystyle\frac{\partial{\cal
L}}{\partial\psi}-\partial_{\mu}\left(\frac{\partial{\cal
L}}{\partial_{\mu}\psi}\right)=0,\quad\Rightarrow\quad\psi=\phi_{xx}.$ (22)

Thus, one sees that the field $\psi$ was just concealing the higher $x$
derivatives in the $\phi$-field. When introducing the field $u:=\phi_{x}$,
equations (21) acquire the form of the modified KdV equations

$u_{t}+n(n-1)u^{n-2}u_{x}+u_{xxx}=0.$ (23)

As was pointed out in [34], the Lagrangian ${\cal L}$ is degenerate, since the
equations for the momentum fields $\pi_{\psi}=\partial{\cal
L}/\partial\psi_{t}=0$ and $\pi_{\phi}=\partial{\cal
L}/\partial\phi_{t}=\phi_{x}/2$ can not be solved for the velocity fields, and
therefore one can not convert from velocity phase space to the momentum phase
space in a straightforward manner. However, by properly including Dirac
constraints the canonical Hamiltonian with the correct Poisson bracket
structure can be constructed, see [34]. The unconstrained part of this
Hamiltonian density is obtained from a Legendre transform and reads

${\cal H}_{0}=\pi_{\phi}\psi_{t}+\pi_{\phi}\phi_{t}-{\cal
L}=-\phi_{x}^{3}-\phi_{x}\psi_{x}-\frac{1}{2}\psi^{2}.$ (24)

In the Hamiltonian we convert terms by integration by parts and a subsequent
dropping of surface terms in the usual way by assuming that all fields and
their $x$-derivatives vanish at the boundaries. In terms of the KdV-fields we
then obtain

$H_{0}=\int{\cal H}_{0}dx=\int\left(\frac{1}{2}u_{x}^{2}-u^{n}\right)dx.$ (25)

Here we are mainly interested in the energies of particular solution and since
the additional term $H_{1}$, required to derive the correct Poisson bracket
structure, does not contribute to them we will not report it here, but simply
refer to [34] for its explicit expression.

###  3.2 Rotated standard Hamiltonian for modified KdV systems, generic $n$

A space-time rotated, one scalar field, partially integrated surface term
adjusted version of the Lagrangian (20) is

${\cal
L}^{r}=\frac{1}{2}\varphi_{t}\varphi_{x}+\varphi_{t}^{n}-\frac{1}{2}\varphi_{tt}^{2},\qquad
n\in\mathbb{N}.$ (26)

At first we derive the Lorentz invariant version of Hamilton’s equations by
the scheme as outlined in section 2.1. Using the defining relations for the
generalised momenta (2) with $m=2$, we compute them to

$\pi^{x}=\frac{1}{2}\varphi_{t},\quad\pi^{t}=\frac{1}{2}\varphi_{x}+\varphi_{ttt}+n\varphi_{t}^{n-1},\quad\pi^{tt}=-\varphi_{tt},\quad\pi^{xx}=\pi^{xt}=\pi^{tx}=0.$
(27)

The Hamiltonian density according to (3) then results to

$\displaystyle{\cal H}^{r}$ $\displaystyle=$
$\displaystyle\pi^{x}\varphi_{x}+\pi^{t}\varphi_{t}+\pi^{tt}\hat{\varphi}_{tt}-{\cal
L}^{r}(\varphi,\varphi_{t},\varphi_{x},\hat{\varphi}_{tt})$ (28)
$\displaystyle=$
$\displaystyle\pi^{x}\varphi_{x}+\pi^{t}\varphi_{t}-\frac{1}{2}\left(\pi^{tt}\right)^{2}-\frac{1}{2}\varphi_{t}\varphi_{x}-\varphi_{t}^{3},$
(29)

where we identified $\hat{\varphi}_{tt}=-\pi^{tt}$. Hamilton’s field equations
are then consistently computed according (5) and (6)

$\displaystyle\partial_{\mu}\varphi$ $\displaystyle=$
$\displaystyle\frac{{{\cal
H}}^{r}}{\partial\pi^{\mu}}\quad\Leftrightarrow\quad\varphi_{t}=\varphi_{t},\,\,\varphi_{x}=\varphi_{x},$
(30) $\displaystyle\partial_{\mu}\varphi_{\nu}$ $\displaystyle=$
$\displaystyle\frac{{{\cal
H}}^{r}}{\partial\pi^{\mu\nu}}\quad\Leftrightarrow\quad\varphi_{tt}=-\pi^{tt},$
(31) $\displaystyle\partial_{\mu}\pi^{\mu}$ $\displaystyle=$
$\displaystyle-\frac{{{\cal
H}}^{r}}{\partial\varphi}\,\,\Leftrightarrow\,\,\partial_{t}\pi^{t}+\partial_{x}\pi^{x}=0,\,\,\Leftrightarrow\,\,n(n-1)\varphi_{t}^{n-2}\varphi_{tt}+\frac{1}{2}\varphi_{xt}+\varphi_{tttt}+\frac{1}{2}\varphi_{xt}=0,\qquad\,\,\,$
(32) $\displaystyle\partial_{\nu}\pi^{\mu\nu}$ $\displaystyle=$
$\displaystyle-\frac{{{\cal
H}}^{r}}{\partial\varphi_{\mu}}\quad\Leftrightarrow\quad\partial_{t}\pi^{tt}=-\frac{{{\cal
H}}^{r}}{\partial\varphi_{t}}\quad\Leftrightarrow\quad-\varphi_{ttt}=-\pi^{t}+\frac{1}{2}\varphi_{x}+n\varphi_{t}^{n-1}.$
(33)

While the Hamiltonian density ${{\cal H}}^{r}$ yields a consistent set of
Hamilton’s equations in the relativistic framework its corresponding
Hamiltonian ${H}^{r}=\int{{\cal H}}^{r}dx$ is in general not a conserved
quantity

$\frac{d{H}^{r}}{dt}=\int\frac{d{{\cal
H}}^{r}}{dt}dx=\int\frac{d(\pi^{x}\varphi_{x})}{dt}dx+\int\frac{d{\cal
H}^{\prime r}}{dt}dx=\int\frac{d(\pi^{x}\varphi_{x})}{dt}dx\neq 0.$ (34)

We will present the precise form of the conserved Hamiltonian ${\cal
H}^{\prime r}$ below, which is in fact the nonrelativistic version of $H^{r}$,
which we should be using as the modified KdV systems are not Lorentz
invariant.

We will demonstrate this scheme for the equivalent Lagrangian ${\cal
L}^{\prime r}$ obtained from ${\cal L}^{r}$ by initially hiding the higher
derivative fields as explained in general terms in section 2.3. With the
canonical momenta computed as in (27) the integrated constraints (16) yield
with (17)

$\displaystyle{\cal L}^{\prime r}$ $\displaystyle=$
$\displaystyle\int\pi^{t}d\varphi_{t}=\frac{1}{2}\varphi_{x}\varphi_{t}+\varphi_{t}^{n}+\varphi_{t}\varphi_{ttt}+f_{1}(\varphi,\varphi_{tt},\varphi_{ttt}),$
(35) $\displaystyle{\cal L}^{\prime r}$ $\displaystyle=$
$\displaystyle-\int\left[\int\pi^{tt}dt\right]d\varphi_{ttt}=\varphi_{tt}\varphi_{ttt}+f_{2}(\varphi,\varphi_{t}\varphi_{tt}).$
(36)

The comparison of (35) and (36) then gives

${\cal L}^{\prime
r}=\frac{1}{2}\varphi_{x}\varphi_{t}+\varphi_{t}^{n}+\varphi_{t}\varphi_{ttt}+g(\varphi,\varphi_{tt}),$
(37)

where $g(\varphi,\varphi_{tt})$ is the last unknown function. Equating ${\cal
L}^{r}$ and ${\cal L}^{\prime r}$ we identify at first
$g(\varphi,\varphi_{tt})=-\varphi_{t}\varphi_{ttt}-1/2\varphi_{tt}^{2}$, but
since $g$ does not depend on $\varphi_{ttt}$ we must convert the first term
with an integration by parts, so that
$g(\varphi,\varphi_{tt})=1/2\varphi_{tt}^{2}$ when surface terms are dropped.
Finally, we obtain

${\cal L}^{\prime
r}=\frac{1}{2}\varphi_{x}\varphi_{t}+\varphi_{t}^{n}+\varphi_{t}\varphi_{ttt}+\frac{1}{2}\varphi_{tt}^{2},$
(38)

which differs from ${\cal L}^{r}$ just by surface terms as anticipated.
Introducing now the new fields $\phi:=\varphi$ and $\psi:=\varphi_{tt}$, the
Lagrangian ${\cal L}^{\prime r}$ acquires the form

${\cal L}^{\prime
r}=\frac{1}{2}\phi_{t}\phi_{x}+\phi_{t}\psi_{t}+\phi_{t}^{n}+\frac{1}{2}\psi^{2},\qquad
n\in\mathbb{N},$ (39)

so that indeed all higher order time-derivatives have been hidden. When
rotating back by swapping $x$ and $t$ we recover the version (20).

The Euler-Lagrange equations are now obtained from ${\cal L}^{\prime r}$ by
using (15) to

$\phi_{xt}+n(n-1)\phi_{t}^{n-2}\phi_{tt}+\psi_{tt}=0,\qquad\text{and}\qquad\psi=\phi_{tt}.$
(40)

With $u:=\phi_{t}$ the first equations converts into the rotated mKdV
equations

$u_{x}+n(n-1)u^{n-2}u_{t}+u_{ttt}=0.$ (41)

Unlike the original unrotated Lagrangian, the rotated Lagrangians ${\cal
L}^{r}$ and ${\cal L}^{\prime r}$ are not degenerate. The canonical momenta
for ${\cal L}^{\prime r}$ are computed directly to

$\pi_{\psi}=\frac{\partial{\cal L}^{\prime
r}}{\partial\psi_{t}}=\phi_{t},\qquad\pi_{\phi}=\frac{\partial{\cal L}^{\prime
r}}{\partial\phi_{t}}=\frac{1}{2}\phi_{x}+n\phi_{t}^{n-1}+\psi_{t},$ (42)

which can be inverted for the velocity fields

$\phi_{t}=\pi_{\psi},\qquad\psi_{t}=\pi_{\phi}-\frac{1}{2}\phi_{x}-n\pi_{\psi}^{n-1}.$
(43)

The Legendre transformed Lagrangian ${\cal L}^{\prime r}$ then yields the
Hamiltonian density

${\cal H}^{\prime r}=\pi_{\psi}\psi_{t}+\pi_{\phi}\phi_{t}-{\cal L}^{\prime
r}=\pi_{\phi}\pi_{\psi}-\frac{1}{2}\pi_{\psi}\phi_{x}-\pi_{\psi}^{n}-\frac{1}{2}\psi^{2}.$
(44)

This is indeed the canonical Hamiltonian with the correct underlying canonical
Poisson bracket structure. As a crucial difference compared to the nonrotated
version we did not require any Dirac constraints in its derivation. We assume
the standard equal-time canonical Poisson bracket relations for the canonical
fields

$\left\\{\psi(x),\pi_{\psi}(x^{\prime})\right\\}=\delta\left(x-x^{\prime}\right),\qquad\left\\{\phi(x),\pi_{\phi}(x^{\prime})\right\\}=\delta\left(x-x^{\prime}\right),$
(45)

with all other combinations of $\psi,\phi,\pi_{\psi},\pi_{\phi}$ vanishing.
When translating the vanishing relations to the $\phi$-fields and their time-
derivatives we obtain

$\left\\{\phi_{tt}(x),\phi_{ttt}(x^{\prime})\right\\}=-n(n-1)\phi_{t}^{n-2}\partial_{x}\delta\left(x-x^{\prime}\right),\quad\left\\{\phi_{ttt}(x),\phi_{ttt}(x^{\prime})\right\\}=-\partial_{x}\delta\left(x-x^{\prime}\right),$
(46)

and

$\left\\{\phi(x),\phi(x^{\prime})\right\\}=\left\\{\phi(x),\phi_{t}(x^{\prime})\right\\}=\left\\{\phi(x),\phi_{tt}(x^{\prime})\right\\}=\left\\{\phi_{t}(x),\phi_{t}(x^{\prime})\right\\}=\left\\{\phi_{t}(x),\phi_{ttt}(x^{\prime})\right\\}=0.$
(47)

With the help of these relations we compute Hamilton’s equations of motion

$\displaystyle\phi_{t}$ $\displaystyle=$
$\displaystyle\frac{\partial{H}^{\prime
r}}{\partial\pi_{\phi}}=\left\\{\phi,{H}^{\prime r}\right\\}=\pi_{\psi},$ (48)
$\displaystyle\psi_{t}$ $\displaystyle=$
$\displaystyle\frac{\partial{H}^{\prime
r}}{\partial\pi_{\psi}}=\left\\{\psi,{H}^{\prime
r}\right\\}=\pi_{\phi}-\frac{1}{2}\phi_{x}-n\pi_{\psi}^{n-1},$ (49)
$\displaystyle\left(\pi_{\psi}\right)_{t}$ $\displaystyle=$
$\displaystyle-\frac{\partial{H}^{\prime
r}}{\partial\psi}=\left\\{\pi_{\psi},{H}^{\prime r}\right\\}=\psi=\phi_{tt},$
(50) $\displaystyle\left(\pi_{\phi}\right)_{t}$ $\displaystyle=$
$\displaystyle-\frac{\partial{H}^{\prime
r}}{\partial\phi}=\left\\{\pi_{\phi},{H}^{\prime
r}\right\\}=-\frac{1}{2}\left(\pi_{\psi}\right)_{x},$ (51)

where (51) is the first equation of (40). We can express the Hamiltonian
density in terms of the KdV-field $u(x,t)$ as

${\cal H}^{\prime r}=(n-1)u^{n}+uu_{tt}-\frac{1}{2}u_{t}^{2},$ (52)

which is a conserved quantity as is easily verified when using the equation of
motion (41)

$\frac{dH^{\prime r}}{dt}=\int\frac{d{\cal H}^{\prime r}}{dt}dx=\int
u\left[u_{ttt}+n(n-1)u^{(n-2)}u_{t}\right]dx=-\frac{1}{2}\int(u^{2})_{x}dx=0,$
(53)

and when dropping surface terms. The nonvanishing equal-time Poisson brackets
for the KdV fields are obtained by direct convertion of the above

$\displaystyle\left\\{u(x),u_{t}(x^{\prime})\right\\}$ $\displaystyle=$
$\displaystyle-\delta\left(x-x^{\prime}\right),$ (54)
$\displaystyle\left\\{u_{tt}(x),u_{tt}(x^{\prime})\right\\}$ $\displaystyle=$
$\displaystyle-\partial_{x}\delta\left(x-x^{\prime}\right),$ (55)
$\displaystyle\left\\{u_{t}(x),u_{tt}(x^{\prime})\right\\}$ $\displaystyle=$
$\displaystyle-n(n-1)u^{n-2}\partial_{x}\delta\left(x-x^{\prime}\right),$ (56)

such that

$u_{t}=\left\\{u,H^{\prime r}\right\\}.$ (57)

Thus, $H^{\prime}_{r}$ is a conserved quantity interpreted in the usual
fashion as energy and also as a Hamiltonian since it generates the evolution
in time. Notice that $H_{r}$ does not admit such an interpretation.

Solutions $u_{4}$ and $u_{3}$ to the equation of motion (41) for $n=4$ and
$n=3$, respectively, are related by means of the rotated Miura transformation

$u_{3}=2u_{4}^{2}+i\sqrt{2}(u_{4})_{t},$ (58)

offering the possibility to construct a complex solution $u_{3}$ from a real
solution $u_{4}$ or possibly a real solution $u_{3}$ from a complex solution
$u_{4}$.

###  3.3 xt-rotated first higher charge Hamiltonians for the KdV system, $n=3$

Next we discuss a HTDT obtained from a rotated higher charge following the
scheme outlined in (18). Starting with the KdV system, the next highest charge
density satisfying (19) beyond the Hamiltonian reads

${\cal Q}_{3}=\frac{5}{3}uu_{x}^{2}-\frac{5}{6}u^{4}-\frac{1}{6}u_{xx}^{2},$
(59)

with corresponding flux

$\chi_{3}=\frac{1}{6}\left[24u^{5}-u_{xxx}^{2}+10\left(2u^{3}+u_{x}^{2}\right)u_{xx}-90u^{2}u_{x}^{2}+2u_{4x}u_{xx}-20uu_{x}u_{xxx}+16uu_{xx}^{2}\right].$
(60)

Identifying $u:=\varphi_{x}$, we obtain the associated Lagrangian by an
inverse Legendre transform as

${\cal
L}_{3}=\frac{1}{2}\varphi_{t}\varphi_{x}-\frac{5}{3}\varphi_{x}\varphi_{xx}^{2}+\frac{5}{6}\varphi_{x}^{4}+\frac{\varphi_{xxx}^{2}}{6},$
(61)

Following the scheme in (18), the rotated version

${\cal
L}_{3}^{r}=\frac{1}{2}\varphi_{t}\varphi_{x}-\frac{5}{3}\varphi_{t}\varphi_{tt}^{2}+\frac{5}{6}\varphi_{t}^{4}+\frac{\varphi_{ttt}^{2}}{6},$
(62)

is used next as the starting point of the Ostrogradsky procedure. The
generalised momentum and coordinate fields are computed from (8) and (10) to

$\displaystyle\pi_{1}$ $\displaystyle=$
$\displaystyle\frac{10}{3}\left(\varphi_{t}\varphi_{ttt}+\varphi_{t}^{3}\right)+\frac{5\varphi_{tt}^{2}}{3}+\frac{\varphi_{5t}}{3}+\frac{\varphi_{x}}{2},\,\,\,\pi_{2}=-\frac{10}{3}\varphi_{t}\varphi_{tt}-\frac{\varphi_{4t}}{3},\,\,\,\pi_{3}=\frac{\varphi_{ttt}}{3},\,\,\,\,$
(63) $\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\varphi,\,\,\,\Phi_{2}=\varphi_{t},\,\,\,\Phi_{3}=\varphi_{tt}.$
(64)

Also this system is not degenerate so that the Hamiltonian density is obtained
directly from (11)

$\displaystyle{\cal H}^{\prime r}_{3}$ $\displaystyle=$
$\displaystyle\pi_{1}\Phi_{2}+\pi_{2}\Phi_{3}+\frac{3}{2}\pi_{3}^{2}-\frac{1}{2}\Phi_{2}\left(\Phi_{3}\right)^{2}+\frac{5}{3}\Phi_{2}\Phi_{3}^{2}-\frac{5}{6}\Phi_{2}^{4},$
(65) $\displaystyle=$
$\displaystyle-\frac{u_{t}u_{ttt}}{3}+\frac{1}{3}uu_{4t}+\frac{10u^{2}u_{tt}}{3}+\frac{u_{tt}^{2}}{6}+\frac{5u^{4}}{2},$
(66)

with $u:=\varphi_{t}$.

Hamilton’s equation of motion lead again to many trivially satisfied
identities with the equation of motion resulting to

$\displaystyle\left(\pi_{1}\right)_{t}=-\frac{{\cal H}^{\prime
r}_{3}}{\partial\varphi}\,\,$ $\displaystyle\Leftrightarrow$
$\displaystyle\varphi_{xt}+10\varphi_{t}^{2}\varphi_{tt}+\frac{10}{3}\varphi_{t}\varphi_{4t}+\frac{1}{3}\varphi_{6t}+\frac{20}{3}\varphi_{tt}\varphi_{ttt},$
(67) $\displaystyle\Leftrightarrow$ $\displaystyle
u_{x}+10u^{2}u_{t}+\frac{10}{3}uu_{ttt}+\frac{1}{3}u_{5t}+\frac{20}{3}u_{t}u_{tt},$
(68)

We observe that ${\cal H}^{\prime r}_{3}$ is a conserved quantity

$\displaystyle\frac{dH^{\prime r}_{3}}{dt}$ $\displaystyle=$
$\displaystyle\int\frac{d{\cal H}^{\prime r}_{3}}{dt}dx=\int
u\left[10u^{2}u_{t}+\frac{10}{3}uu_{ttt}+\frac{1}{3}u_{5t}+\frac{20}{3}u_{t}u_{tt}\right]dx$
(69) $\displaystyle=$ $\displaystyle-\frac{1}{2}\int(u^{2})_{x}dx=0,$ (70)

up to the validity of the equation of motion (68) and the vanishing of surface
terms.

####  3.3.1 Multi-field theory

Next we demonstrate that one may also re-express the higher charge Lagrangian
${\cal L}^{\prime r}_{3}$ and Hamiltonian ${\cal H}^{\prime r}_{3}$ densities
in terms of new fields that hide the higher time-derivatives. Solving the
constraints (16) using the canonical momenta (63) we obtain the equivalent
multi-field Lagrangian

$\displaystyle\mathcal{L}^{\prime r}$ $\displaystyle=$
$\displaystyle\frac{5}{6}\varphi_{t}^{4}+\frac{5}{3}\varphi_{2t}^{2}\varphi_{t}+\frac{1}{2}\varphi_{x}\varphi_{t}+\frac{5}{3}\varphi_{t}^{2}\varphi_{3t}+\frac{1}{3}\varphi_{5t}\varphi_{t}+\frac{1}{6}\varphi_{3t}^{2}+\frac{1}{3}\varphi_{2t}\varphi_{4t},$
$\displaystyle=$
$\displaystyle\frac{5}{6}\left(\phi_{1}\right)_{t}^{4}+\frac{5}{3}\phi_{2}^{2}\left(\phi_{1}\right)_{t}+\frac{1}{2}\left(\phi_{1}\right)_{x}\left(\phi_{1}\right)_{t}+\frac{5}{3}\left(\phi_{1}\right)_{t}^{2}\left(\phi_{2}\right)_{t}+\frac{1}{3}\left(\phi_{3}\right)_{t}\left(\phi_{1}\right)_{t}+\frac{1}{6}\left(\phi_{1}\right)_{t}^{2}+\frac{1}{3}\phi_{1}\phi_{2}.$

where $\phi_{1}:=\varphi$, $\phi_{2}:=\varphi_{2t}$ and
$\phi_{3}:=\varphi_{4t}$. Computing the corresponding canonical momenta gives

$\displaystyle\pi^{\prime}_{1}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}^{\prime
r}}{\partial\left(\phi_{1}\right)_{t}}=\frac{10}{3}\left(\phi_{1}\right)_{t}^{3}+\frac{5}{3}\phi_{2}^{2}+\frac{1}{2}\left(\phi_{1}\right)_{x}+\frac{10}{3}\left(\phi_{1}\right)_{t}\left(\phi_{2}\right)_{t}+\frac{1}{3}\left(\phi_{3}\right)_{t},$
(72) $\displaystyle\pi^{\prime}_{2}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}^{\prime
r}}{\partial\left(\phi_{2}\right)_{t}}=\frac{5}{3}\left(\phi_{1}\right)_{t}^{2}+\frac{1}{3}\left(\phi_{2}\right)_{t},$
(73) $\displaystyle\pi^{\prime}_{3}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathcal{L}^{\prime
r}}{\partial\left(\phi_{3}\right)_{t}}=\frac{1}{3}\left(\phi_{1}\right)_{t},$
(74)

so that the Legendre transformed $\mathcal{L}^{\prime r}$ yields the
Hamiltonian

$\displaystyle\mathcal{H}^{\prime r}$ $\displaystyle=$ $\displaystyle
3\pi^{\prime}_{3}\pi^{\prime}_{1}+270{\pi^{\prime}}_{\\!\\!3}^{4}-5\left(\phi_{2}\right)^{2}{\pi^{\prime}}_{\\!\\!3}-\frac{3}{2}{\pi^{\prime}}_{\\!\\!3}\left(\phi_{1}\right)_{x}-45{\pi^{\prime}}_{\\!\\!3}^{2}{\pi^{\prime}}_{\\!\\!2}+\frac{3}{2}{\pi^{\prime}}_{\\!\\!2}^{2}-\frac{1}{3}\phi_{2}\phi_{3},$
(75) $\displaystyle=$
$\displaystyle\frac{5}{2}\left(\phi_{1}\right)_{t}^{4}+\frac{10}{3}\left(\phi_{1}\right)_{t}^{2}\left(\phi_{2}\right)_{t}+\frac{1}{6}\left(\phi_{2}\right)_{t}^{2}+\frac{1}{3}\left(\phi_{1}\right)_{t}\left(\phi_{3}\right)_{t}-\frac{1}{3}\phi_{2}\phi_{3}.$
(76)

With the identification $\varphi_{t}=u$ the expression (76) is identical to
(66) previously derived. The version (75) is tailored to generate the
equations of motion. We find

$\displaystyle\left(\phi_{1}\right)_{t}$ $\displaystyle=$
$\displaystyle\int\\{\phi(x),\mathcal{H}(y)\\}dy=3{\pi^{\prime}}_{\\!\\!3}$
(77) $\displaystyle\left(\phi_{2}\right)_{t}$ $\displaystyle=$
$\displaystyle\int\\{\psi(x),\mathcal{H}(y)\\}dy=-45{\pi^{\prime}}_{\\!\\!1}^{2}+3{\pi^{\prime}}_{\\!\\!2}$
(78) $\displaystyle\left(\phi_{3}\right)_{t}$ $\displaystyle=$
$\displaystyle\int\\{\theta(x),\mathcal{H}(y)\\}dy=3{\pi^{\prime}}_{\\!\\!1}+1080{\pi^{\prime}}_{\\!\\!3}^{3}-5\phi_{1}^{2}-\frac{3}{2}\left(\phi_{0}\right)_{x}-90{\pi^{\prime}}_{\\!\\!2}{\pi^{\prime}}_{\\!\\!3}$
(79) $\displaystyle\left({\pi^{\prime}}_{\\!\\!1}\right)_{t}$ $\displaystyle=$
$\displaystyle\int\\{\pi_{\phi}(x),\mathcal{H}(y)\\}dy=-\frac{3}{2}({\pi^{\prime}}_{\\!\\!3})_{x}$
(80) $\displaystyle\left({\pi^{\prime}}_{\\!\\!2}\right)_{t}$ $\displaystyle=$
$\displaystyle\int\\{\pi_{1}(x),\mathcal{H}(y)\\}dy=10\phi_{1}{\pi^{\prime}}_{\\!\\!3}+\frac{1}{3}\phi_{2}$
(81) $\displaystyle\left({\pi^{\prime}}_{\\!\\!3}\right)_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\phi_{1}$ (82)

Using the explicit expression for the canonical momenta (72)-(74) we see that
all equations are trivially satisfied, except (80) which corresponds to the
equation of motion. Thus one can eliminate all time-derivatives beyond order
one by introducing new fields in the described manner and simply use the
standard variational principle, avoiding the use Ostrogradsky’s method
altogether. However, the conversion (17) becomes increasingly complicated as
we have checked for the next highest charge not reported here.

####  3.3.2 Integrability from Painlevé test

Having obtained a new version of a conserved Hamiltonian it is unclear whether
the system is integrable or not. In order to settle this question we carry out
a Painlevé test [37, 38, 39, 40, 41, 42] for the equation of motion in the
form (68). Our starting point is a Painlevé expansion of the form

$u(x,t)=\sum_{k=0}^{\infty}\lambda_{k}(x,t)w(x,t)^{k+\alpha},$ (83)

where the functions $\lambda_{k}(x,t)$ are analytic and $-\alpha\in\mathbb{N}$
characterises the leading order singularity when $w(x,t)\rightarrow 0$.
Substituting the expansion into the PDE under investigation the function
$\lambda_{k}(x,t)$ may in principle be computed recursively. The Painlevé test
consists of determining how many free parametric functions $\lambda_{k}(x,t)$,
referred to as resonances, occur in this recursive process or whether the
procedure would impose constraints on the functions $w(x,t)$. In case the
number of resonances is greater or equal to the order of the differential
equation the system is said to have passed the Painlevé test and is considered
to be integrable. In reverse, if that is not the case the test is failed and
one can deduce that the system is not integrable. We will not establish here
the stronger Painlevé property, which consists of proving in addition that the
series (83) is convergent.

For our system we need to establish at first the order of the leading
singularity. Substituting the first term from the expansion (83) into (68) we
can identify the leading order contributions from each of the terms $u_{x}\sim
w^{\alpha-1}$, $u^{2}u_{t}\sim w^{3\alpha-1}$, $uu_{ttt}\sim w^{\alpha-3}$,
$u_{5t}\sim w^{\alpha-5}$, $u_{t}u_{tt}\sim w^{\alpha-2}$. Matching the powers
of the second and fourth term, $3\alpha-1=\alpha-5$, we find that $\alpha=-2$.
Using this value in the expansion (83), we read off the coefficients in the
powers of $w$ and set them to zero

$w^{-7}:\quad\lambda_{0}w_{t}\left(\lambda_{0}+2w_{t}^{2}\right)\left(\lambda_{0}+6w_{t}^{2}\right)=0.$
(84)

For the solution $\lambda_{0}=-2w_{t}^{2}$ to (84) we find at the next orders

$\displaystyle w^{-6}:$
$\displaystyle\,\,w_{t}^{5}\left(\lambda_{1}-2w_{tt}\right)=0\quad\Rightarrow\quad\lambda_{1}=2w_{tt},$
(85) $\displaystyle w^{-5}:$
$\displaystyle\,\,w_{t}^{3}\left(\lambda_{1}-6w_{tt}\right)\left(1-2w_{t}\partial_{t}\right)\left(\lambda_{1}-2w_{tt}\right)=0,\quad\Rightarrow\quad\lambda_{2}\,\text{arbitrary},$
$\displaystyle w^{-4}:$
$\displaystyle\,\,\lambda_{3}=\frac{w_{t}\left[w_{t}\left(2w_{tt}w_{ttt}-2w_{t}\left(\lambda_{2}\right)_{t}+w_{4t}\right)\right]-w_{tt}^{3}}{2w_{t}^{4}},$
$\displaystyle w^{-3}:$
$\displaystyle\,\,\lambda_{4}=\frac{40w_{t}w_{tt}^{2}w_{ttt}+10w_{t}^{4}\left(\lambda_{2}\right)_{tt}-25w_{4t}w_{t}^{2}w_{tt}-20w_{tt}^{4}+30\lambda_{2}^{2}w_{t}^{4}}{20w_{t}^{6}}$
$\displaystyle\qquad\qquad+\frac{10\lambda_{2}w_{t}^{2}\left(4w_{t}w_{ttt}-3w_{tt}^{2}\right)+w_{t}^{3}\left(6w_{5t}+3w_{x}-10\left(\lambda_{2}\right)_{t}w_{tt}\right)}{20w_{t}^{6}},$
$\displaystyle w^{-2}:$ $\displaystyle\,\,\lambda_{5}\,\text{arbitrary},$
$\displaystyle w^{-1}:$ $\displaystyle\,\,\lambda_{6}\,\text{arbitrary},$
$\displaystyle w^{0}:$
$\displaystyle\,\,\lambda_{7}=\frac{1}{80w_{t}^{5}}\left\\{180\lambda_{4}w_{tt}w_{ttt}-40\lambda_{3}^{2}w_{t}w_{tt}-30\lambda_{5}w_{t}w_{tt}^{2}-80\lambda_{3}\lambda_{4}w_{t}^{3}-10\lambda_{4}w_{4t}w_{t}\right.$
$\displaystyle+21\lambda_{3}w_{5t}+30\lambda_{2}^{2}\left[\left(\lambda_{2}\right)_{t}+\lambda_{3}w_{t}\right]+45w_{4t}\left(\lambda_{3}\right)_{t}-160\lambda_{6}w_{t}^{3}w_{tt}-80\lambda_{5}w_{t}^{2}w_{ttt}+3\lambda_{3}w_{x}$
$\displaystyle+150w_{tt}^{2}\left(\lambda_{4}\right)_{t}-120w_{t}^{2}w_{tt}\left(\lambda_{5}\right)_{t}+40w_{t}w_{ttt}\left(\lambda_{4}\right)_{t}-80\lambda_{3}w_{t}^{2}\left(\lambda_{3}\right)_{t}-80w_{t}^{4}\left(\lambda_{6}\right)_{t}$
$\displaystyle+20\left(\lambda_{2}\right)_{t}\left[2w_{t}\left(\left(\lambda_{3}\right)_{t}-2\lambda_{4}w_{t}\right)+\left(\lambda_{2}\right)_{tt}+7\lambda_{3}w_{tt}\right]+20\lambda_{3}w_{t}\left(\lambda_{2}\right)_{tt}-40w_{t}^{3}\left(\lambda_{5}\right)_{tt}$
$\displaystyle+10\lambda_{2}\left[3w_{t}\left[\left(\lambda_{3}\right)_{tt}-2\lambda_{4}w_{tt}-2w_{t}\left(\left(\lambda_{4}\right)_{t}+\lambda_{5}w_{t}\right)\right]+15w_{tt}\left(\lambda_{3}\right)_{t}+\left(\lambda_{2}\right)_{ttt}+13\lambda_{3}w_{ttt}\right]$
$\displaystyle\left.+30w_{tt}\left(\lambda_{3}\right)_{ttt}+60w_{t}w_{tt}\left(\lambda_{4}\right)_{tt}+50w_{ttt}\left(\lambda_{3}\right)_{tt}+5w_{t}\left(\lambda_{3}\right)_{4t}+\left(\lambda_{2}\right)_{5t}+3\left(\lambda_{2}\right)_{t}\right\\},$
$\displaystyle w^{1}:$ $\displaystyle\,\,\lambda_{8}\,\text{arbitrary}.$

Thus, besides the fundamental singularity at $\lambda_{-1}$, we found the four
additional free parameters $\lambda_{2}$, $\lambda_{5}$, $\lambda_{6}$ and
$\lambda_{8}$. Thus, with five free parameters we match the order of the
differential equation and conclude that the system is integrable.

We compare this result of our explicit construction with the necessary
condition which arises when we assume the $\lambda_{r}$ for some as yet
unknown $r$ to be constant. We set them to some constant coefficient $\theta$.
For the unknown power $r$ the constant $\theta$ of the term $w^{r+\alpha}$
becomes undetermined when this term cancels the leading order $w^{\alpha}$.
Hence, substituting for this

$\tilde{u}(x,t)=\lambda_{0}(x,t)w(x,t)^{\alpha}+\theta w(x,t)^{r+\alpha}$ (86)

with $\lambda_{0}=-2w_{t}^{2}$ into (68), we obtain

$\frac{1}{3}\theta(r-8)(r-6)(r-5)(r-2)(r+1)w_{t}^{5}=0,$ (87)

which reproduces exactly the powers $r=2,5,6,8$found in the explicit
construction above. For the other two choices for $\lambda_{0}=0$ and
$\lambda_{0}=-6w_{t}^{2}$, we find

$\displaystyle\frac{1}{3}\theta(r-6)(r-5)(r-4)(r-3)(r-2)w_{t}^{5}$
$\displaystyle=$ $\displaystyle 0,$ (88)
$\displaystyle\frac{1}{3}\theta(r-10)(r-8)(r-6)(r+1)(r+3)w_{t}^{5}$
$\displaystyle=$ $\displaystyle 0,$ (89)

respectively. Once again we find five free parameters. We verified these
values also in an explicit construction that we do not report here.

###  3.4 xt-rotated first higher charge Hamiltonians for the KdV system, $n=4$

For the modified KdV system, the next highest charge density satisfying (19)
beyond the Hamiltonian reads

${\cal
Q}_{3}=\frac{10}{3}u^{2}u_{x}^{2}-\frac{4u^{6}}{3}-\frac{u_{xx}^{2}}{6},$ (90)

with corresponding flux

$\chi_{3}=12u^{8}+8u^{5}u_{xx}-60u^{4}u_{x}^{2}-\frac{20}{3}u^{2}u_{x}u_{xxx}+\frac{16}{3}u^{2}u_{xx}^{2}+\frac{20}{3}uu_{x}^{2}u_{xx}+\frac{1}{3}u_{4x}u_{xx}+\frac{u_{x}^{4}}{3}-\frac{u_{xxx}^{2}}{6}.$
(91)

Identifying again $u:=\varphi_{x}$, we obtain the associated Lagrangian by an
inverse Legendre transform as

${\cal
L}_{3}=\frac{1}{2}\varphi_{t}\varphi_{x}-\frac{10}{3}\varphi_{x}^{2}\varphi_{xx}^{2}+\frac{4}{3}\varphi_{x}^{6}+\frac{\varphi_{xxx}^{2}}{6}.$
(92)

Following the scheme in (18), the rotated version

${\cal
L}_{3}^{r}=\frac{1}{2}\varphi_{x}\varphi_{t}-\frac{10}{3}\varphi_{t}^{2}\varphi_{tt}^{2}+\frac{4}{3}\varphi_{t}^{6}+\frac{\varphi_{ttt}^{2}}{6},$
(93)

is used next as the starting point of the Ostrogradsky procedure. The
generalised momentum and coordinate fields are computed from (8) and (10) to

$\displaystyle\pi_{1}$ $\displaystyle=$
$\displaystyle\frac{20}{3}\left(\varphi_{t}\varphi_{tt}^{2}+\varphi_{t}^{2}\varphi_{ttt}\right)+8\varphi_{t}^{5}+\frac{\varphi_{5t}}{3}+\frac{\varphi_{x}}{2},\,\,\,\pi_{2}=-\frac{20}{3}\varphi_{t}^{2}\varphi_{tt}-\frac{\varphi_{4t}}{3},\,\,\,\pi_{3}=\frac{\varphi_{ttt}}{3},\qquad$
(94) $\displaystyle\Phi_{1}$ $\displaystyle=$
$\displaystyle\varphi,\,\,\,\Phi_{2}=\varphi_{t},\,\,\,\Phi_{3}=\varphi_{tt}.$
(95)

Once again the system is not degenerate so that the Hamiltonian density is
obtained directly from (11)

$\displaystyle{\cal H}^{\prime r}_{3}$ $\displaystyle=$
$\displaystyle\pi_{1}\Phi_{2}+\pi_{2}\Phi_{3}+\frac{3}{2}\pi_{3}^{2}+\frac{10}{3}\varphi_{t}^{2}\varphi_{tt}^{2}-\frac{1}{2}\varphi_{t}\varphi_{x}-\frac{4}{3}\varphi_{t}^{6},$
(96) $\displaystyle=$
$\displaystyle-\frac{u_{t}u_{ttt}}{3}+\frac{10}{3}u^{2}u_{t}^{2}+\frac{1}{3}uu_{4t}+\frac{20u^{3}u_{tt}}{3}+\frac{u_{tt}^{2}}{6}+\frac{20u^{6}}{3},$
(97)

with $u:=\varphi_{t}$. Hamilton’s equation of motion lead once more to many
trivially satisfied identities, that we do not report, with the equation of
motion resulting as

$\displaystyle\left(\pi_{1}\right)_{t}=-\frac{{\cal
H}^{r}_{3}}{\partial\varphi}\,\,$ $\displaystyle\Leftrightarrow$
$\displaystyle\varphi_{xt}+\frac{80}{3}\varphi_{t}\varphi_{tt}\varphi_{ttt}+40\varphi_{t}^{4}\varphi_{tt}+\frac{20}{3}\varphi_{4t}\varphi_{t}^{2}+\frac{\varphi_{6t}}{3}+\frac{20}{3}\varphi_{tt}^{3}=0,$
(98) $\displaystyle\Leftrightarrow$ $\displaystyle
u_{x}+\frac{80}{3}uu_{t}u_{tt}+40u^{4}u_{t}+\frac{20}{3}u^{2}u_{3t}+\frac{u_{5t}}{3}+\frac{20}{3}u_{t}^{3}=0,$
(99)

We easily establish that ${\cal H}^{\prime r}_{3}$ is a conserved quantity

$\displaystyle\frac{dH^{\prime r}_{3}}{dt}$ $\displaystyle=$
$\displaystyle\int\frac{d{\cal H}^{\prime r}_{3}}{dt}dx=\int
u\left[\frac{80}{3}uu_{t}u_{tt}+40u^{4}u_{t}+\frac{20}{3}u^{2}u_{3t}+\frac{u_{5t}}{3}+\frac{20}{3}u_{t}^{3}\right]dx$
(100) $\displaystyle=$ $\displaystyle-\frac{1}{2}\int(u^{2})_{x}dx=0,$ (101)

up to the validity of the equation of motion (99) and the vanishing of surface
terms.

####  3.4.1 Integrability from Painlevé test

Next we carry out a Painlevé test for the PDE (99). The leading order
contributions from each of the terms are in this case $u_{x}\sim
w^{\alpha-1}$, $uu_{t}u_{tt}\sim w^{2\alpha-1}$, $u^{4}u_{t}\sim
w^{5\alpha-1}$, $u^{2}u_{ttt}\sim w^{\alpha-3}$, $u_{5t}\sim w^{\alpha-5}$,
$u_{t}^{3}\sim w^{3\alpha-3}$. Matching the powers of the third and fifth or
sixth term, $5\alpha-1=\alpha-5$, we find that $\alpha=-1$. Using this value
in the expansion (83) we read off the coefficients in the powers of $w$ and
set them to zero

$w^{-6}:\quad-20\lambda_{0}w_{t}\left(\lambda_{0}^{2}+2w_{t}^{2}\right)\left(2\lambda_{0}^{2}+w_{t}^{2}\right)=0.$
(102)

For the solution $\lambda_{0}=iw_{t}/\sqrt{2}$ to (102) we find at the next
orders

$\displaystyle w^{-5}:$
$\displaystyle\,\,\frac{20}{3}w_{t}^{4}\left(4\lambda_{1}w_{t}+i\sqrt{2}w_{\text{tt}}\right)=0\quad\Rightarrow\quad\lambda_{1}=-\frac{iw_{\text{tt}}}{2\sqrt{2}w_{t}},$
(103) $\displaystyle w^{-4}:$
$\displaystyle\frac{20iw_{t}^{2}}{3}\left[4i\left(4w_{tt}+w_{t}\partial_{t}\right)\left(\lambda_{1}+\frac{iw_{tt}}{2\sqrt{2}w_{t}}\right)+6\sqrt{2}w_{t}^{2}\left(\lambda_{1}^{2}+\frac{w_{tt}^{2}}{8w_{t}^{2}}\right)\right]\Rightarrow\lambda_{2}\,\text{arbitrary},$
$\displaystyle w^{-3}:$
$\displaystyle\,\,\Rightarrow\lambda_{3}\,\text{arbitrary},$ $\displaystyle
w^{-2}:$
$\displaystyle\,\,\lambda_{4}=\frac{1}{320w_{t}^{7}}\left\\{80w_{t}^{3}\left[-2w_{t}\left(w_{\text{tt}}\left(\lambda_{2}\right)_{t}+2\lambda_{2}w_{ttt}\right)-2w_{t}^{2}\left(\left(\lambda_{2}\right)_{tt}+2\lambda_{3}w_{tt}\right)\right.\right.,$
$\displaystyle\left.\left.\quad-4w_{t}^{3}\left(\lambda_{3}\right)_{t}+5\lambda_{2}w_{tt}^{2}\right]\right\\}-i\sqrt{2}\left\\{4w_{t}\left[w_{t}\left(10w_{4t}w_{tt}+200\lambda_{2}^{2}w_{t}^{4}-2w_{t}\left(w_{5t}+3w_{x}\right)\right.\right.\right.$
$\displaystyle\quad\left.\left.\left.+5w_{ttt}^{2}\right)-25w_{\text{tt}}^{2}w_{ttt}\right]+45w_{\text{tt}}^{4}\right\\},$
$\displaystyle w^{-1}:$ $\displaystyle\,\,\lambda_{5}\,\text{arbitrary},$
$\displaystyle w^{0}:$ $\displaystyle\,\,\lambda_{6}\,\text{arbitrary}.$

This in this case we found the four free parameters $\lambda_{2}$,
$\lambda_{3}$, $\lambda_{5}$, and $\lambda_{6}$, so that together with the
fundamental singularity at $\lambda_{-1}$, we match the order of the
differential equation and deduce that the system is integrable. Once again we
can compare with the necessary condition. The Ansatz (86) with solution
$\lambda_{0}=\pm iw_{t}/\sqrt{2}$ to the first constraint yields

$\frac{1}{3}\theta(r-6)(r-5)(r-3)(r-2)(r+1)w_{t}^{5}=0,$ (104)

which identifies the same powers $r=2,3,5,6$ that we found in our explicit
construction. For the other two solutions $\lambda_{0}=0$ and $\lambda_{0}=\pm
i\sqrt{2}w_{t}$, we find

$\displaystyle\frac{1}{3}\theta(r-5)(r-4)(r-3)(r-2)(r-1)w_{t}^{5}$
$\displaystyle=$ $\displaystyle 0,$ (105)
$\displaystyle\frac{1}{3}\theta(r-8)(r-6)(r-5)(r+1)(r+3)w_{t}^{5}$
$\displaystyle=$ $\displaystyle 0,$ (106)

respectively. In all cases we find five free parameters that may be verified
by an explicit construction.

##  4 Exact benign and malevolent solutions and their classical energies

Next we construct solutions for the modified rotated KdV equations (41) and
those resulting from interpreting the higher charges of the rotated system as
Hamiltonians (68) and (99). With regard to the question of whether the
characteristic features of HTDT show up at the classical level, we are
especially interested in periodic benign solutions in time with sufficiently
many free parameters to satisfy the initial conditions of the rotated Cauchy
problem. For some examples we will also constrain the real $x$-axis and thus
include boundaries, so that the Cauchy problem is extended to an initial-
boundary value problem. Assuming travelling wave solutions these equations may
be integrated out and with appropriate assumptions periodic solutions can be
obtained. Here we use instead, except for the linear $(n=2)$-case, the
extended Jacobian elliptic function expansion method [43] that yields exact
periodic solutions in a straightforward manner. Important for our purposes
here is that this approach also exhibits the different types of dispersion
relations.

###  4.1 Exact solutions for the rotated $n=2$-mKdV equations of motion

In constructing solutions to the equations of motion (41) we start with the
$(n=2)$-case, which is straightforward to solve as it is linear and can be
solved explicitly by elementary methods, but is still instructive in
exhibiting some key characteristics. Factorising $u(x,t)=f(x)g(t)$, equation
(41) is converted into the two equations

$\frac{f_{x}}{f}=-\lambda,\qquad\frac{2g_{t}}{g}+\frac{g_{ttt}}{g}=-\lambda,$
(107)

with $\lambda\in\mathbb{C}$ being a constant. Solving both equations, the
general solution then simply becomes

$u_{2}(x,t)=\left(c_{1}e^{-m_{1}t}+c_{2}e^{-m_{2}t}+c_{3}e^{-m_{3}t}\right)c_{4}e^{-\lambda
x},$ (108)

with $m_{i}$ denoting the three roots of $t^{3}+2t+\lambda=0$ and $c_{i}$ the
required integration constants. A similar solution is obtained when assuming a
travelling wave solution of the form $u(x,t)=u(\zeta)$ where $\zeta:=x-ct$. We
can then integrate out the equation directly and subsequently once more when
multiplied by $u_{\zeta}$ obtaining

$\frac{du}{d\zeta}=\sqrt{\frac{2}{c^{3}}\left[\left(\frac{1}{2}-c\right)u^{2}-c_{1}u-c_{2}\right]},$
(109)

with integration constants $c_{1},c_{2}$. Equation (109) is easily integrated
out producing a similar solution as reported in (108).

Depending on the values of $\lambda$ we encounter malevolent exponentially and
oscillatory divergent solutions as well as benign complex solutions that stay
finite when time evolves, see figure 1. Dissipative solutions of similar type
were previously found for extended versions of the Pais-Uhlenbeck oscillators
on the classical and quantum level [48, 49, 50]. In section 5 we will comment
on the field theoretical version.

Figure 1: Classical solutions for the $n=2$ rotated modified KdV equation at
different locations $x$ as functions of time $t$ with $c_{1}=3$,
$c_{2}=c_{3}=2$, $c_{4}=1$, $m_{1}\in\mathbb{R}$,
$m_{2}=m^{*}_{3}\in\mathbb{C}$ for $\lambda=1$, $\lambda=-1$, $\lambda=i$ in
panels (a), (b) and (c), respectively. In panel (c) real parts are depicted as
solid lines and imaginary parts as dotted lines.

Notice that even having a benign solution and despite having as many constants
in the solution (108) as the order of the differential equation in $t$, we can
not construct from them an exact solution to the rotated Cauchy problem, as
the initial profile functions $u_{2}(x,0)$, $(u_{2})_{t}(x,0)$ and
$(u_{2})_{tt}(x,0)$ can not be chosen arbitrarily, but are linearly dependent
of the form $\sim e^{-\lambda x}$.

The energies of the solutions are computed from the densities in (52). For
real values of $\lambda$ the energies become infinite when integrating over
the entire real axis and when $\lambda\in i\mathbb{R}$ we obtain vanishing
energies on finite intervals $E=\int_{-\pi/2}^{\pi/2}{\cal H}^{\prime
r}(u_{2})dx=0$.

###  4.2 Exact solutions for the rotated $n=3$-mKdV equations of motion

Next we consider the standard KdV equation (23) with $n=3$ in order to exhibit
the key differences between the original and the rotated version. Making an
Ansatz for the solution of the original version in (23) in terms of the Jacobi
elliptic function $\text{sn}(z|m)$ with parameter $m\in[0,1]$, and at this
point unknown constants $\alpha$, $k$, $\omega$,

$u_{3}(x,t)=\alpha\,\text{sn}^{2}(kx+t\omega|m),$ (110)

we obtain, upon substitution into original KdV equation (23), the constraining
equation

$\displaystyle
2\alpha\text{cn}(kx+t\omega|m)\text{dn}(kx+t\omega|m)\text{sn}(kx+t\omega|m)$
(111)
$\displaystyle\qquad\qquad\times\left[\omega-4k^{3}(1+m)+6k\left(\alpha+2k^{2}m\right)\text{sn}^{2}(kx+t\omega|m)\right]=0.$

The last factor vanishes when we set $\alpha=-2mk^{2}$ and the dispersion
relation to $\omega(k)=4k^{3}(1+m)$. We will make use of the real and complex
periodicities

$u_{3}\left(x,t\right)=u_{3}\left(x+\frac{2}{k}K,t\right)=u_{3}\left(x+\frac{2}{k}iK^{\prime},t\right)=u_{3}\left(x,t+\frac{2}{\omega}K\right)=u_{3}\left(x,t+\frac{2}{\omega}iK^{\prime}\right),$
(112)

with $K:=K(m)$, $K^{\prime}:=K(1-m)$ denoting the complete elliptic integrals
of the first kind. In the limit $m\rightarrow 1$ this solution reduces to the
well-known solution
$u_{3}(x,t)=-2k^{2}\text{tanh}^{2}\left(8k^{3}t+kx\right)$. Similar solutions
may be obtained from different starting points akin to (110) involving other
Jacobi elliptic functions.

For the rotated KdV equation the Ansatz (110), when substituted into (41),
yields the constraint

$\displaystyle
2\alpha\text{cn}(kx+t\omega|m)\text{dn}(kx+t\omega|m)\text{sn}(kx+t\omega|m)$
(113)
$\displaystyle\qquad\qquad\times\left[k-4(1+m)\omega^{3}+6\omega\left(\alpha+2m\omega^{2}\right)\text{sn}^{2}(kx+t\omega|m)\right],$

so that when solving the dispersion relation for $\omega$, we obtain three
distinct solutions

$u_{3r}^{(n)}(x,t)=\alpha^{(n)}\,\text{sn}^{2}\left[kx+t\omega^{(n)}|m\right],\quad\alpha^{(n)}:=-2m\left[\omega^{(n)}\right]^{2},\quad\omega^{(n)}:=\left[\frac{k}{4+4m}\right]^{1/3}\tau^{n}\
$ (114)

for $n=1,2,3$ with $\tau:=e^{2\pi i/3}$ denoting the third primitive root of
unity.

The complex solutions
$u_{3r}^{(1)}(k>0)=u_{3r}^{(2)}(k<0)=[u_{3r}^{(2)}(k>0)]^{*}=[u_{3r}^{(3)}(k<0)]^{*}$
are malevolent as they have singularities at $(kx+t\omega^{(1,2)})=\ell
iK^{\prime}$ for $\ell\in\mathbb{Z}$, i.e. at the points

$\displaystyle\hat{x}_{\ell,\mu}^{(n)}$ $\displaystyle=$
$\displaystyle-\frac{\ell}{k}\cot\left(\frac{2\pi
n}{3}\right)K^{\prime}+\frac{2\mu
K}{k},\qquad\qquad\qquad\qquad\qquad\,\,n=1,2\,\,\mu,\nu\in\mathbb{Z},$ (115)
$\displaystyle\hat{t}_{\ell,\nu}^{(n)}$ $\displaystyle=$
$\displaystyle\frac{2^{2/3}}{k^{1/3}}(m+1)^{1/3}K\csc\left(\frac{2\pi
n}{3}\right)\left[\ell K^{\prime}+2\nu\tan\left(\frac{2\pi
n}{3}\right)K\right],$ (116)

that are reached in finite time $t$. Figure 2 shows how a typical malevolent
solution runs into a singularity as time evolves.

Figure 2: Classical malevolent solutions $u_{3r}^{(1)}(x,t)$ at different
times for the $n=3$ rotated modified KdV equation with $k=1.2$, $m=0.25$ and
singularities at the points
$(\hat{x}_{-1,0}^{(1)},t_{-1,0}^{(1)})=(1.0376,4.0070)$,
$(\hat{x}_{-1,-1}^{(1)},t_{-1,0}^{(1)})=(-1.7720,4.0070)$.

In contrast, the real solutions $u_{3r}^{(3)}(k>0)=u_{3r}^{(1)}(k<0)$ are
oscillatory and of a benign nature. However, this does of course not yet mean
that we can solve the rotated Cauchy problem as for this we need accommodate
three arbitary independent function $u$, $u_{t}$ and $u_{tt}$ as initial
profiles at $t=0$. In order to obtain the right amount of free functions we
have to combine three of the solutions. As our equation is nonlinear we can of
course not simply add them, but need to do this by means of a generalised
superposition as obtained from a Bäcklund transformation. In appendix A we
explain how to achieve that.

Figure 3: Classical superimposed solutions (158) for the $n=3$ rotated
modified KdV equation with $\omega=\omega^{(3)}$, $\alpha_{1}=1/2$,
$\alpha_{2}=3/2$, $m=0.3$, $\lambda=0$ for $k=1/2$ and $k=3/2$ in panels (a)
and (b), respectively.

As seen in figure 3 when taking $\omega=\omega^{(3)}$ the solutions are
benign. Using the three parameters $\lambda$, $\alpha_{1}$, $\alpha_{2}$, we
can adjust these solutions to generate many initial value profiles for the
rotated Cauchy problem thus creating subset in the functional parameter space
with exact solutions of benign nature. In [10] it was argued that for the KdV-
system some of the exact solutions would blow up as they evolve in time and
doubt was raised about the existence of benign solutions. Here we have
explicitly shown that benign solutions can easily be constructed and combined
in such a way that the Cauchy problem is solved for some specific initial
profile functions. It remains unclear whether the Cauchy problem can be solved
in complete generality.

Next we compute the energies of these solutions in one period
$x\in[a-K/k,a+K/k]$ for arbitrary $a$ associated to the Hamiltonians $H_{0}$
and $H^{\prime r}$. Setting $a=0$, we find real energies for the solution
$u_{3}$ of the original KdV equation

$\displaystyle E_{3}(k,m)$ $\displaystyle=$
$\displaystyle\int_{-K/k}^{K/k}{\cal H}_{0}(u_{3})dx,$ $\displaystyle=$
$\displaystyle\frac{16}{5}k^{5}\left[\left(m^{2}+2m+2\right)K-\left(2m^{2}+3m+2\right)K^{\prime\prime}\right],$

with $K^{\prime\prime}=E(m)$ denoting the complete elliptic integrals of the
second kind. As expected, since $H_{0}$ is conserved in time the energies for
particular solutions are indeed time-independent.

For the solutions $u_{3r}^{(n)}$ of the rotated KdV solution we obtain

$E_{3r}^{(n)}(k,m)=\int_{-K/k}^{K/k}{\cal H}^{\prime
r}\left[u_{3r}^{(n)}\right]dx=\frac{1}{3}k\left[2K^{\prime\prime}-\frac{2+m}{1+m}K\right].$
(118)

Notice that the energies are identical, real for all values of $n$ and finite,
except in the limit $m\rightarrow 1$, for which we have

$\lim_{m\rightarrow
1}E_{3}(k>0,m),-E_{3}(k<0,m),-E_{3r}^{(n)}(k>0,m),E_{3r}^{(n)}(k<0,m)\rightarrow\infty.$
(119)

The reality of the energy is easily understood by noting that ${\cal
H}^{\prime r}\left[u_{3r}^{(n)}(x,t=0)\right]\in\mathbb{R}$ and the fact that
the energy is a conserved quantity in time. Alternatively, by noting that for
two different solutions $u_{1}$ and $u_{2}$ the modified ${\cal
CPT}$-symmetry: $u_{1}(x,t)\rightarrow u_{2}^{*}(-x,-t)$, $H^{\prime
r}(u_{1})=[H^{\prime r}(u_{2})]^{*}$ is realised pairwise for $u_{3r}^{(n)}$
for the same energy solutions, we can utilise the arguments from [44] to
ensure the reality of the energy.

###  4.3 Exact solutions for the rotated $n=4$-mKdV equations of motion

In order to solve the rotated modified KdV equation (41) with $n=4$ we have to
slightly alter the Ansatz (110) and reduce the order of the Jacobi elliptic
function. Proceeding in the same manner, substituting

$u_{4r}(x,t)=\alpha\,\text{cn}(kx+t\omega|m),$ (120)

into (41) yields the constraint

$\alpha\text{dn}(kx+t\omega|m)\text{sn}(kx+t\omega|m)\left[6\omega\left(m\omega^{2}-2\alpha^{2}\right)\text{cn}(kx+t\omega|m)^{2}-k+(1-2m)\omega^{3}\right],$
(121)

leading to the solutions

$u_{4r}^{(n)}(x,t)=\alpha^{(n)}\,\text{cn}\left(kx+t\omega^{(n)}|m\right),\quad\alpha^{(n)}:=\frac{\sqrt{m}\omega^{(n)}}{\sqrt{2}},\quad\omega^{(n)}:=\left(\frac{k}{1-2m}\right)^{1/3}\tau^{n}.$
(122)

The complex solutions are once again of malevolent nature, whereas the real
solutions $u_{4r}^{(1)}$ for $m>1/2$ and $u_{4r}^{(3)}$ for $m<1/2$ are
benign. Combining the solutions $u_{4r}$ according to the Miura transformation
(58) produces indeed a new solution to the rotated $n=3$-mKdV equation.

For the standard solution $u_{4}$ of the original modified KdV equation we
compute

$E_{4}(k,m)=\int_{-K/k}^{K/k}{\cal
H}_{0}(u_{4})dx=-\frac{1}{6}k^{3}\left\\{[m(3m-4)+1]K+(2m-1)K^{\prime\prime}\right\\},$
(123)

which for the $\operatorname{sech}$-solution becomes $\lim_{m\rightarrow
1}E_{4}(k,m)=-1/6k^{3}$. In turn for the solutions $u_{4r}^{(n)}$ of the
rotated mKdV equation we obtain

$E_{4r}^{(n)}(k,m)=\int_{-K/k}^{K/k}{\cal H}^{\prime
r}\left[u_{4r}^{(n)}\right]dx=\frac{\omega^{(n)}}{4m-2}\left\\{\left[(4-3m)m-1\right]K+(1-2m)K^{\prime\prime}\right\\},$
(124)

which are never real for all $n$. We have either $E_{4r}^{(3)}\in\mathbb{R}$
and $E_{4r}^{(1)}=[E_{4r}^{(2)}]^{*}\notin\mathbb{R}$ when $k>0,m<1/2$;
$k<0,m>1/2$ or $E_{4r}^{(1)}\in\mathbb{R}$ and
$E_{4r}^{(2)}=[E_{4r}^{(3)}]^{*}\notin\mathbb{R}$ when $k>0,m>1/2$;
$k<0,m<1/2$. In the trigonometric limit we obtain $\lim_{m\rightarrow
1}E_{4r}^{(n)}(k,m)=-\tau^{n}(-k)^{1/3}$. In the case the ${\cal
CPT}$-symmetry is broken for the complex solutions $u_{4r}^{(n)}$ and leads to
complex conjugate energy solutions.

To address the rotated Cauchy problem and find more possibilities for initial
profile functions we could in principle carry out a similar analysis as in the
previous subsection by constructing exact nonlinearly superpositioned
solutions involving several parameters from a rotated mKdV Bäcklund
transformation following [45]. However, in that case the analogue equations to
(153) and (154) can not be solved analytically when the seed solution is taken
to be a Jacobi elliptic function. A similar problem occurs when trying to
build such a solution form an alternative version of the mKdV Bäcklund
transformation when following [46]. Alternatively, one could combine several
solutions using Darboux–Crum transformations [47], but instead we will
consider here a numerical study of the corresponding initial-boundary value
problem. Thus, we consider (41) on the domain $t\in[0,\infty)$, $x\in[-X,X]$
with

$u(x,0)=f_{1}(x),\,\,\,u_{t}(x,0)=f_{2}(x),\,\,\,u_{tt}(x,0)=f_{3}(x),\,\,\,u(\pm
X,t)=a_{\pm}(t),$ (125)

where the functions $f_{i}(x)$ are arbitrary. Consistency of the constraints
requires that $a_{\pm}(0)=f_{1}(\pm X)$, $(a_{\pm})_{t}(0)=f_{2}(\pm X)$ and
$(a_{\pm})_{tt}(0)=f_{3}(\pm X)$. As a compatible set of solutions for the
initial-boundary value functions we take here for instance

$f_{1}(x)=\sin(x),\,\,\,f_{2}(x)=\cos(2\pi
x/X),\,\,\,f_{3}(x)=0,\,\,\,a_{\pm}(t)=\sin(t)\pm\sin(X).$ (126)

In figure 4 we depict the numerical solutions for $u(x,t)$ for a particular
interval together with some typical timelines.

Figure 4: Classical solutions to the initial-boundary value problem for the
$n=4$ rotated modified KdV equation with boundary value functions (126) in
panels (a) and selected timelines in panels (b) and (c).

We observe that solutions appear to remain benign and do not develop any
singularities as time evolves. This confirms the findings in [10] for a
different set of boundary value functions.

###  4.4 Exact solutions for the rotated higher charge $n=3,4$-mKdV systems

Periodic solutions to the $n=3$ equation of motion (68) obtained from the
higher charge Hamiltonian (66) can be obtained in the same way resulting to

$\hat{u}_{3r}^{(n)}(x,t)=\alpha^{(n)}\,\text{cn}^{2}\left(kx+t\omega^{(n)}|m\right),\,\,\alpha^{(n)}:=2m\omega^{(n)},\,\,\omega^{(n)}:=\left(\frac{3k}{8(7m-7m^{2}-2)}\right)^{1/5}\hat{\tau}^{n},$
(127)

for $n=1,\ldots,5$ with $\hat{\tau}:=e^{2\pi i/5}$ denoting the fifth
primitive root of unity. As above, the complex solutions diverge at finite
time, whereas the real solutions
$\hat{u}_{3r}^{(5)}(x,t),k<0=\hat{u}_{3r}^{(2)}(x,t),k>0$ are benign.

The energies resulting from the higher charge Hamiltonian (66) for these
solutions are computed to

$\displaystyle\hat{E}_{3r}(k,m)$ $\displaystyle=$
$\displaystyle\int_{-K/k}^{K/k}{\cal
H}_{3}^{r}\left[\hat{u}_{3r}^{(n)}\right]dx$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!=\frac{16\left[\omega^{(n)}\right]^{8}}{9k}\left[(m-1)\left(45m^{3}-73m^{2}+40m-8\right)K+4(2m-1)\left(7m^{2}-7m+2\right)K^{\prime\prime}\right],$

which is real only for the real solutions.

Solutions to (99) are obtained in the same way resulting to

$\hat{u}_{4r}^{(n)}(x,t)=\alpha^{(n)}\,\text{cn}\left(kx+t\omega^{(n)}|m\right),\,\,\alpha^{(n)}:=\sqrt{\frac{m}{2}}\omega^{(n)},\,\,\omega^{(n)}:=\left(\frac{3k}{6m-6m^{2}-1}\right)^{1/5}\hat{\tau}^{n},$
(129)

for $n=1,\ldots,5$, with
$\hat{u}_{4r}^{(5)}(x,t),k>0=-\hat{u}_{4r}^{(2)}(x,t),k<0$ being real benign
solutions and the remaining ones complex divergent in time.

The energies resulting from the higher charge Hamiltonian (97) for these
solutions are

$\displaystyle\hat{E}_{3r}(k,m)$ $\displaystyle=$
$\displaystyle\int_{-K/k}^{K/k}{\cal
H}_{3}^{r}\left[\hat{u}_{4r}^{(n)}\right]dx$ $\displaystyle=$
$\displaystyle\frac{\left[\omega^{(n)}\right]^{6}}{126k}\left\\{\left[7\left(14m^{2}-14m-1\right)-4(2m-1)\left(6m^{2}-6m-19\right)\left[\omega^{(n)}\right]^{2}\right]K^{\prime\prime}\right.$
$\displaystyle+\left.(m-1)\left[7\left(30m^{2}-22m-1\right)+2\left(12m^{2}+45m-38\right)\left[\omega^{(n)}\right]^{2}\right]K\right\\},$

which is also real only for the real solutions.

So, in the $n=3$-theory the energies, resulting from the Hamiltonians are
always real, even for the complex solutions, but for the $n=3$ higher charges
and all charges in the $n=4$-theory, only the real solutions posses real
energies.

##  5 Quantization

We follow here the approach elaborated on in [33] for higher time-derivative
generalizations of the Klein-Gordon theory. The initial step of the scheme
consists of Fourier transforming the scalar fields

$\varphi(x,t)=\frac{1}{2\pi}\int dk\phi(t,k)e^{ikx},$ (131)

and subsequently quantizing the theory involving the fields $\phi(t,k)$ for
fixed wave numbers $k$. For the scheme to be applicable the resulting
$k$-dependent Lagrangian is required to be of the general form

$L_{k}=\frac{1}{2}\sum_{n=0}^{N}C_{n}(k)\left(\frac{d^{n}\phi(t,k)}{dt^{n}}\right)^{2}.$
(132)

In general, the higher time-derivative Lagrangians considered above contain
products of three or more of the fields and their derivatives, such that they
are not of the form (132). However, the Fourier transformed Lagrangian of the
rotated version (26) for $n=2$ reads

$L^{r}_{k}=\frac{1}{2}\left(ik\phi_{t}\phi+2\phi_{t}^{2}-\phi_{tt}^{2}\right)=\frac{1}{2}\sum_{i,j=0}^{2}\tilde{C}_{ij}\frac{d^{i}\phi}{dt^{i}}\frac{d^{j}\phi}{dt^{j}}=\frac{1}{2}\Phi^{\intercal}\tilde{C}\Phi,$
(133)

with

$\tilde{C}:=\begin{pmatrix}0&\frac{ik}{2}&0\\\ \frac{ik}{2}&2&0\\\
0&0&-1\end{pmatrix},\qquad\Phi:=\begin{pmatrix}\phi\\\ \phi_{t}\\\
\phi_{tt}\end{pmatrix},$ (134)

which can be transformed to the variant (132) by means of an orthogonal
transformation $S$. The transformed Lagrangian in the fields $\psi$ is derived
as

$L^{r}_{k}=\frac{1}{2}\left\\{-\psi^{2}+\left[1-\sqrt{1-\frac{k^{2}}{4}}\right]\psi_{t}^{2}+\left[1+\sqrt{1-\frac{k^{2}}{4}}\right]\psi_{tt}^{2}\right\\}=\sum_{n=0}^{2}\frac{C_{n}}{2}\left(\frac{d^{n}\psi}{dt^{n}}\right)^{2}=\frac{1}{2}\Psi^{\intercal}C\Psi,$
(135)

with

$C:=S\tilde{C}S^{\intercal},\qquad S=\left(\begin{array}[]{ccc}0&0&1\\\
\frac{2iC_{2}}{k\sqrt{2-\frac{8C_{2}}{k^{2}}}}&\frac{1}{\sqrt{2-\frac{8C_{2}}{k^{2}}}}&0\\\
\frac{2iC_{1}}{k\sqrt{2-\frac{8C_{1}}{k^{2}}}}&\frac{1}{\sqrt{2-\frac{8C_{1}}{k^{2}}}}&0\end{array}\right),\qquad\Psi:=\begin{pmatrix}\psi\\\
\psi_{t}\\\ \psi_{tt}\end{pmatrix}=S\Phi,$ (136)

and $C_{n}:=C_{nn}$, $SS^{\intercal}=1$. The Euler-Lagrange equation resulting
from (135) is then

$\sum_{n=0}^{2}C_{n}(-1)^{n}\frac{d^{2n}\psi}{dt^{2n}}=0.$ (137)

Using the mode expansion

$\psi=\sum_{j=1}^{2}a_{j}e^{-i\omega_{j}t}+b_{j}e^{i\omega_{j}t},$ (138)

the equation of motion acquires the form

$\sum_{n=0}^{2}C_{n}(\omega_{i})^{2n}=0.$ (139)

The two canonical fields $\psi_{n}$ and their corresponding canonical momenta
$\pi_{n}$ computed from (10) and (9), respectively, satisfying the canonical
Poisson bracket relation (10) are then elevated to obey the quantum
commutation relations

$\left[\psi_{n},\pi_{m}\right]=i\delta_{nm}.$ (140)

As verified in detail by Weldon [33], these relations hold provided the Fock
space generators $a_{i},b_{i}$ commute to the residues $R_{i}$

$\left[a_{i},b_{j}\right]=\delta_{ij}R_{i},\qquad\text{with}\quad
R_{i}:=\lim_{z\rightarrow\omega_{i}}\frac{z-\omega_{i}}{P(z)}=\frac{1}{P^{\prime}(\omega_{i})},\quad
P(z):=\sum_{n=0}^{2}C_{n}(z)^{2n}.$ (141)

The Hamiltonian $H_{k}^{r}$ for the fields $\psi$ following from
Ostrogradsky’s scheme for scalar field theories as laid out in section 2.2
becomes

$H_{k}^{r}=-\frac{1}{2}\left(C_{0}\psi^{2}+C_{1}\psi_{t}^{2}+2C_{2}\psi_{t}\psi_{ttt}-C_{2}\psi_{tt}^{2}\right),$
(142)

with $\pi_{1}=C_{1}\psi_{t}-C_{2}\psi_{ttt}$, $\pi_{2}=C_{1}\psi_{tt}$,
$\Phi_{1}=\psi$, $\Phi_{2}=\psi_{t}$. Using the mode expansion (138) the
Hamiltonian can be cast into the form

$H_{k}^{r}=\frac{1}{2}\sum_{i=1}^{2}\frac{\omega_{i}}{R_{i}}\left(a_{i}b_{i}+b_{i}a_{i}\right).$
(143)

To illustrate the dilemma of HTDT let us first discuss the real scenario for
which we assume that $b_{i}=a_{i}^{\dagger}$. Then one may define the vacuum
state with energy $E_{0}=(\omega_{1}+\omega_{2})/2$ by the relation
$a_{i}\left|vac\right\rangle=0$, following from
$H\left|vac\right\rangle=E_{0}\left|vac\right\rangle$. The norm of the one-
particle state $a_{i}^{\dagger}\left|vac\right\rangle$

$\left\langle vac\right|a_{i}a_{i}^{\dagger}\left|vac\right\rangle=R_{i},$
(144)

with energy $\omega_{i}+E_{0}$ can therefore only be positive when the residue
$R_{i}$ is positive. The energy of an n-particle state constructed from Fock
states is $n\omega_{i}+E_{0}$. Thus for a theory with positive norm states and
a spectrum that is bounded from below we require $R_{i}>0$ and $\omega_{i}>0$.
However, as argued in [33] one may only satisfy here one or the other
inequality.

Before discussing $L^{r}_{k}$, we illustrate this feature with a simpler toy
model. Considering for instance a Lagrangian of the form (132)

$L_{t}=\frac{1}{2}\left[-\psi^{2}+\left(1+k^{2}\right)\psi_{t}^{2}-\frac{1}{4}\left(1+k^{2}\right)\psi_{tt}^{2}\right],$
(145)

the corresponding equations of motion (139) has the solutions

$\omega_{1}^{\pm}=\pm\sqrt{2+\frac{2|k|}{\sqrt{1+k^{2}}}},\qquad\omega_{2}^{\pm}=\pm\sqrt{2-\frac{2|k|}{\sqrt{1+k^{2}}}},$
(146)

where $\omega_{i}^{+}=-\omega_{i}^{-}>0$. Either two of these solutions may
play the role of the $\omega_{j}$ in (138). For the related residues we find

$R_{1}^{\pm}=-\frac{1}{2\omega_{1}^{\pm}\sqrt{k^{4}+k^{2}}},\qquad
R_{2}^{\pm}=\frac{1}{2\omega_{2}^{\pm}\sqrt{k^{4}+k^{2}}},$ (147)

where $R_{1}^{+}=-R_{1}^{-}<0$ and $R_{2}^{+}=-R_{2}^{-}>0$. Thus we may
either pick two states with positive definite norm states and positive
residues $R_{1}^{-}$, $R_{2}^{+}$ at the cost of having an unbounded spectrum
from below since $\omega_{1}^{-}<0$ or we can require boundedness from below
with energy states build from $\omega_{1}^{+}$, $\omega_{2}^{+}$ at the prize
of having one negative norm state in the theory.

Returning to the discussion of the spectrum of $H_{k}^{r}$, we find that the
corresponding equations of motion (139) has the solutions

$\displaystyle\omega_{1}^{\pm}$ $\displaystyle=$ $\displaystyle\pm
2\sqrt{\frac{1}{2-\sqrt{4-k^{2}}-\sqrt{24-k^{2}+4\sqrt{4-k^{2}}}}},\qquad$
(148) $\displaystyle\omega_{2}^{\pm}$ $\displaystyle=$ $\displaystyle\pm
2\sqrt{\frac{1}{2-\sqrt{4-k^{2}}+\sqrt{24-k^{2}+4\sqrt{4-k^{2}}}}}.$ (149)

All of these energies are complex with non-vanishing complex part throughout
the entire range of $k\in\mathbb{R}$. These energies correspond to resonances
with energies $E=E_{r}-i\Gamma/2$, where $E_{r}\in\mathbb{R}$ corresponds to
the measurable real part of the energy and $\Gamma\in\mathbb{R}^{+}$ to the
decay width. Thus we need to identify the $\omega_{j}$ in (138) in such a way
that the imaginary parts of the energies are negative for the entire spectrum.
The only possible choice that achieves this is to define
$E_{i}^{n}:=E_{0}-n\omega_{i}$ with $E_{0}:=-\omega_{1}-\omega_{2}$. None of
these spectra is bounded from below, as is also illustrated in figure 5. Note
that any other choice of the $\omega_{i}$ would lead to spectra that do not
have a positive decay width at least in part of their spectra.

Figure 5: Quantum spectrum of the $n=2$ rotated modified KdV equation.

##  6 Conclusions

The Lorentz invariant formulation of Ostrogradsky’s method generalised to
scalar field theories produced a consistent set of Hamilton’s equations when
applied to space-time rotated mKdV systems, but it failed to yield the correct
time evolution or a conserved Hamiltonian. Instead, employing the
nonrelativistic scheme achieved the latter outcomes, which is expected
considering the lack of Lorentz invariance in the mKdV system. We outlined an
alternative general scheme that circumvents Ostrogradsky’s method by
concealing time-derivatives beyond the first order in newly defined scalar
fields. This approach enables the use of conventional methods to derive
canonical variables and associated Hamiltonians. We have demonstrated that
mKdV systems can be formulated equivalently in this alternative way. However,
it remains an open question whether this equivalence can always be
established. The conversion relies on the computation of the first two
integrals in (17), and in particular when the $t$-integration can not be
carried out trivially, the procedure becomes technically very difficult and
may even become impossible to be carried out. In our examples this was the
case for the next highest charge not reported here.

We derived the time-evolution and have constructed consistent sets of equal-
time commutation relations for the canonical coordinate and momentum fields of
the mKdV systems for generic values of $n$, which should aid future
quantisation procedures of the theories. For the special cases of $n=3$ and
$n=4$ we also derived the alternative multi-field version of the theories and
the consistent time-evolution for the canonical fields of the first higher
charge Hamiltonian. These type of Hamiltonians were constructed from inverse
Legendre transformed higher charges, which were subsequently space-time
rotated before converted into Hamiltonians by means of the Ostrogradsky
method. By carrying out a Painlevé test we established that the systems
obtained in this way are integrable.

For the $n=2,3,4$-theories we derived several exact real as well as complex
benign and malevolent solutions. The classical energies of these systems were
shown to be real when the solutions respect a generalised ${\cal
CPT}$-symmetry and occur in complex conjugate pairs when that symmetry is
broken. By including multi-soliton solutions we showed that one may generate a
large set of initial value functions for the Cauchy problem and in the case of
$n=4$ we also extended our investigations to the initial-boundary value
problem, revealing benign solutions. It remains an open issue whether these
type of boundary value problems can be solved in complete generality for the
systems considered. Nonetheless, we established that in their functional
parameter space of initial boundary value functions they certainly possess
islands of benign nature.

We have quantised the $n=2$-theory and for a special toy model were led to the
usual dilemma in HTDT that we have to make a choice between two physically
undesirable features without being able to avoid both at the same time. We
have either a theory with proper normalisable states of positive norm, at the
cost of having a spectrum that is not bounded from below, or the reverse,
allowing for negative norm states in the system would lead to a spectrum that
is bounded from below. In our $n=2$-mKdV theory the choice was dictated by
selecting the correct sign in the decay width, leading to a theory that is
unbounded from below. We leave the quantisation of the $n\neq 2$-theories for
future work.

Acknowledgments: BT is supported by a City, University of London Research
Fellowship.

##  Appendix A Superposition principle for the solutions of the rotated KdV

In this appendix we derive a generalised superposition principle for the
solutions of the rotated KdV equation based on Bäcklund transformations and
apply it to our solutions (114). Our starting point consists of two different
solutions to the rotated time-independent Schrödinger equation with different
energies $E_{1}$, $E_{2}$ for the same potential $V_{0}$

$-\left(\psi_{1}\right)_{tt}+V_{0}\psi_{1}=E_{1}\psi_{1},\qquad-\left(\psi_{2}\right)_{tt}+V_{0}\psi_{2}=E_{2}\psi_{2}.$
(150)

It is then easy to see that the two functions $\hat{\psi}_{1}=1/\psi_{1}$,
$\hat{\psi}_{2}=1/\psi_{2}$ satisfy the space-independent Schrödinger
equations

$-\left(\hat{\psi}_{1}\right)_{tt}+\hat{V}_{1}\hat{\psi}_{1}=E_{1}\hat{\psi}_{1},\qquad-\left(\hat{\psi}_{2}\right)_{tt}+\hat{V}_{2}\hat{\psi}_{2}=E_{2}\hat{\psi}_{2},$
(151)

with potentials

$\hat{V}_{i}=V_{0}-2\left(\ln\psi_{i}\right)_{tt},\qquad i=1,2.$ (152)

So far the above are simply the initial steps of a rotated version of a
Darboux transformation, see e.g. [51]. The connection to the solutions of the
KdV equation is made by recalling that the standard Lax operator for the KdV
system $L=-\partial_{x}^{2}-u$ can be interpreted as a Hamilton operator with
potential $V=-u$ and $u$ being a solution of the KdV equation. This is
trivially adapted to the current situation with $x\leftrightarrow t$.

Introducing next the quantities $u_{0}=-V_{0}$, $u_{1}=-\hat{V}_{1}$,
$u_{2}=-\hat{V}_{2}$ and $u_{i}=(w_{i})_{t}$ the equations above yield

$\displaystyle u_{0}+u_{1}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(w_{0}-w_{1}\right)^{2}-2E_{1},$ (153)
$\displaystyle u_{0}+u_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(w_{0}-w_{2}\right)^{2}-2E_{2}.$ (154)

In the spirit of Bianchi’s theorem of permutability [52], we can set up a
Lamb-diagram [53] by assuming further that there exists a function $u_{12}$
with $u_{12}=(w_{12})_{t}$ satisfying

$\displaystyle u_{1}+u_{12}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(w_{1}-w_{12}\right)^{2}-2E_{2},$ (155)
$\displaystyle u_{2}+u_{12}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left(w_{2}-w_{12}\right)^{2}-2E_{1}.$ (156)

We can then combine these four equations to the nonlinear version of the
superposition principle

$w_{12}=w_{0}-4\frac{E_{1}-E_{2}}{w_{1}-w_{2}},$ (157)

so that

$u_{12}=u_{0}-4\frac{(E_{1}-E_{2})(u_{1}-u_{2})}{(w_{1}-w_{2})^{2}}.$ (158)

Next we specialise (157) to our concrete case. We notice that our solutions
(114) can be interpreted as rotated version of a shifted Lamé potential.
Adapting the solutions for such potentials reported in [54, 55], it is
convenient to define the functions $H$, $\Theta$, $Y$ and $Z$

$H\left(z\right):=\vartheta_{1}\left(z\kappa,q\right),\quad\Theta\left(z\right):=\vartheta_{4}\left(z\kappa,q\right),\quad
Y(z):=\frac{H_{z}\left(z\right)}{H\left(z\right)},\quad
Z(z):=\frac{\Theta_{z}\left(z\right)}{\Theta\left(z\right)},$ (159)

defined in terms of Jacobi’s theta functions $\vartheta_{i}(z,q)$ with
$i=1,2,3,4$, $\kappa:=\pi/(2K)$ and nome $q:=\exp(-\pi K^{\prime}/K)$. Then
the combination of the functions

$\psi_{1}^{\alpha}(t)=\frac{H(\omega t+p+\alpha)}{\Theta(\omega
t+p)}e^{-(\omega
t+p)Z(\alpha)},\qquad\text{and}\qquad\psi_{2}^{\alpha}(t)=\frac{\Theta(\omega
t+p+\alpha)}{\Theta(\omega t+p)}e^{-(\omega t+p)Y(\alpha)},$ (160)

satisfy the space-independent Schrödinger equations (150) for the same
potential

$V_{0}=2m\omega^{2}\text{sn}^{2}(\omega t+p|m),$ (161)

with different energies

$E_{1}=\omega^{2}\left[1+m-m\text{sn}^{2}(\alpha|m)\right],\qquad
E_{2}=\omega^{2}\left[1+m-\text{sn}^{-2}(\alpha|m)\right],3$ (162)

respectively, for $\omega=\omega^{(n)}$ as specified in (129) and free
parameters $\alpha,p$. We have now various options to derive two potentials
from the functions in (160), either from $\psi_{1}^{\alpha}(t)$ and
$\psi_{2}^{\alpha}(t)$, but as the energies depend on $\alpha$ we may also use
$\psi_{1}^{\alpha_{1}}(t)$, $\psi_{1}^{\alpha_{2}}(t)$ or
$\psi_{2}^{\alpha_{1}}(t)$, $\psi_{2}^{\alpha_{2}}(t)$ with
$\alpha_{1}\neq\alpha_{2}$. Here we present the second option. Then specifying
$p=kx+\lambda$ we obtain from (152) the solutions

$u_{i}(x,t):=\left.-2m\omega^{2}\text{sn}^{2}(z|m)+\left[\frac{H^{2}_{z_{i}}(z_{i})}{H^{2}(z_{i})}-\frac{H_{2z_{i}}(z_{i})}{H(z_{i})}-\frac{T^{2}_{z}(z)}{T^{2}(z)}+\frac{T_{2z}(z)}{T(z)}\right]\right|_{{\footnotesize\begin{matrix}z=kx+\omega
t+\lambda,\\\ z_{i}=z+\alpha_{i},\qquad\end{matrix}}}$ (163)

for $\omega=\omega^{(n)}$. Integrating this over $t$ we obtain

$w_{i}(z)=\left.-2\omega\left[z-\mathcal{E}(z|m)-\frac{H_{z_{i}}(z_{i})}{H(z_{i})}+\frac{T_{z}(z)}{T(z)}+Z(\alpha_{i})\right]\right|_{{\footnotesize\begin{matrix}z=kx+\omega
t+\lambda,\\\ z_{i}=z+\alpha_{i},\qquad\end{matrix}}}$ (164)

where $\mathcal{E}(z|m)=\int_{0}^{z}\text{dn}^{2}(w|m)dw$ denotes Jacobi’s
epsilon function. Substituting these expressions into (158) leads to a
superposition with the three free parameters $\lambda$, $\alpha_{1}$ and
$\alpha_{2}$.

## References

  * [1] M. Ostrogradsky,  Mémoire sur les équations différentielles relatives an probléme des isopérimétres, volume VI 4,  1850\.
  * [2] R. P. Woodard,  Ostrogradsky’s theorem on Hamiltonian instability,  Scholarpedia 10(8), 32243 (2015).
  * [3] K. S. Stelle,  Renormalization of higher-derivative quantum gravity,  Phys. Rev. D 16(4), 953 (1977).
  * [4] L. Modesto and I. L. Shapiro,  Superrenormalizable quantum gravity with complex ghosts,  Phys. Lett. B 755, 279–284 (2016).
  * [5] A. Salvio and A. Strumia,  Quantum mechanics of 4-derivative theories,  The EPJ C 76, 1–15 (2016).
  * [6] C. M. Bender and P. D. Mannheim,  No-ghost theorem for the fourth-order derivative Pais-Uhlenbeck oscillator model,  Phys. Rev. Lett. 100(11), 110402 (2008).
  * [7] M. Raidal and H. Veermäe,  On the quantisation of complex higher derivative theories and avoiding the Ostrogradsky ghost,  Nucl. Phys. B 916, 607–626 (2017).
  * [8] A. Smilga,  Benign vs. malicious ghosts in higher-derivative theories,  Nucl. Phys. B 706(3), 598–614 (2005).
  * [9] A. Smilga,  On exactly solvable ghost-ridden systems,  Phys. Lett. A 389, 127104 (2021).
  * [10] T. Damour and A. Smilga,  Dynamical systems with benign ghosts,  Phys. Rev. D 105, 045018 (2022).
  * [11] A. Smilga,  Benign ghosts in higher-derivative systems,  in J. of Phys.: Conf. Series, 2028 012023 (2021).
  * [12] S. W. Hawking and T. Hertog,  Living with ghosts,  Phys. Rev. D 65(10), 103515 (2002).
  * [13] T. Biswas, T. Koivisto, and A. Mazumdar,  Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity,  JCAP 2010, 008 (2010).
  * [14] H. A. Weldon,  Finite-temperature retarded and advanced fields,  Nucl. Phys. B 534(1-2), 467–490 (1998).
  * [15] S. Mignemi and D. L. Wiltshire,  Black holes in higher-derivative gravity theories,  Phys. Rev. D 46(4), 1475 (1992).
  * [16] V. O. Rivelles,  Triviality of higher derivative theories,  Phys. Lett. B 577(3-4), 137–142 (2003).
  * [17] D. S. Kaparulin, S. L. Lyakhovich, and A. A. Sharapov,  BRST analysis of general mechanical systems,  J. of Geo. and Phys. 74, 164–184 (2013).
  * [18] M. S. Plyushchay,  Massless point particle with rigidity,  Mod. Phys. Lett. A 4(09), 837–847 (1989).
  * [19] M. S. Plyushchay,  Massless particle with rigidity as a model for the description of bosons and fermions,  Phys. Lett. B 243(4), 383–388 (1990).
  * [20] M. Dine and N. Seiberg,  Comments on higher derivative operators in some SUSY field theories,  Phys. Lett. B 409(1-4), 239–244 (1997).
  * [21] A. Smilga,  Ultraviolet divergences in non-renormalizable supersymmetric theories,  Phys. of Part. and Nucl. Lett. 14, 245–260 (2017).
  * [22] M. Pavšič,  Stable self-interacting Pais–Uhlenbeck oscillator,  Mod. Phys. Lett. A 28(36), 1350165 (2013).
  * [23] D. S. Kaparulin, S. L. Lyakhovich, and A. A. Sharapov,  Classical and quantum stability of higher-derivative dynamics,  EPJ C 74, 1–19 (2014).
  * [24] M. Avendano-Camacho, J. A. Vallejo, and Y. Vorobiev,  A perturbation theory approach to the stability of the Pais-Uhlenbeck oscillator,  J. Math. Phys. 58(9) (2017).
  * [25] N. Boulanger, F. Buisseret, F. Dierick, and O. White,  Higher-derivative harmonic oscillators: stability of classical dynamics and adiabatic invariants,  EPJ C 79, 1–8 (2019).
  * [26] C. Deffayet, S. Mukohyama, and A. Vikman,  Ghosts without runaway instabilities,  Phys. Rev. Lett. 128(4), 041301 (2022).
  * [27] C. Deffayet, A. Held, S. Mukohyama, and A. Vikman,  Global and local stability for ghosts coupled to positive energy degrees of freedom,  JCAP 2023(11), 031 (2023).
  * [28] A. Fring and B. Turner,  Higher derivative Hamiltonians with benign ghosts from affine Toda lattices,  J. Phys. A: Math. Theor. 56, 295203 (2023).
  * [29] A. Fring and B. Turner,  Integrable scattering theory with higher derivative Hamiltonians,  Eur. Phys. J. Plus 138(12), 1136 (2023).
  * [30] H. Motohashi and T. Suyama,  Quantum Ostrogradsky theorem,  JHEP 2020(9), 1–10 (2020).
  * [31] F. J. De Urries and J. Julve,  Ostrogradski formalism for higher-derivative scalar field theories,  J. Phys. A: Math. and Gen. 31(33), 6949 (1998).
  * [32] R. Thibes,  Natural higher-derivatives generalization for the Klein–Gordon equation,  Mod. Phys. Lett. A 36(28), 2150205 (2021).
  * [33] H. A. Weldon,  Quantization of higher-derivative field theories,  Ann. Phys. 305(2), 137–150 (2003).
  * [34] Y. Nutku,  Hamiltonian formulation of the KdV equation,  J. Math. Phys. 25, 2007–2008 (1984).
  * [35] G. Bowtell and A. E. G. Stuart,  A particle representation for the Korteweg-de Vries solitons,  J. Math. Phys. 24, 969–981 (1983).
  * [36] R. M. Miura,  The Korteweg-de Vries Equation: A Survey of Results,  SIAM Review 18, 412–459 (1976).
  * [37] P. Painlevé,  Mémoire sur les équations différentielles dont l’intégrale générale est uniforme,  Bull. Soc. Math. France 28, 201–261 (1900).
  * [38] M. Ablowitz, A. Ramani, and H. Segur,  A connection between nonlinear evolution equations and ordinary differential equations of P-type. II,  J. Math. Phys. 21, 1006–1015 (1980).
  * [39] J. Weiss, M. Tabor, and G. Carnevale,  The Painlevé property for partial differential equations,  J. Math. Phys. 24, 522–526 (1983).
  * [40] J. Weiss,  The Painlevé property for partial differential equations. II: Bäcklund transformation, Lax pairs, and the Schwarzian derivative,  J. Math. Phys. 24, 1405–1413 (1983).
  * [41] M. Kruskal, N. Joshi, and R. Halburd,  Analytic and Asymptotic Methods for Nonlinear Singularity Analysis,  Lect. Notes Phys. 638, 175–208 (2004).
  * [42] B. Grammaticos and A. Ramani,  Integrability- and How to detect it,  Lect. Notes Phys. 638, 31–94 (2004).
  * [43] C. Liang, X. Yong, L. Zhongfei, and H. Jiahua,  The extended Jacobian Elliptic function expansion method and its application to nonlinear wave equations,  Fizika A 12(4), 161–170 (2003).
  * [44] A. Fring and T. Taira,  Complex BPS solitons with real energies from duality,  J. of Phys. A: Mathe. and Theor. 53(45), 455701 (2020).
  * [45] M. Wadati,  Bäcklund transformation for solutions of the modified Korteweg-de Vries equation,  JPSJ 36(5), 1498–1498 (1974).
  * [46] J. Schiff,  Bäcklund transformations of MKdV and Painleve equations,  Nonlinearity 7(1), 305 (1994).
  * [47] A. Arancibia, F. Correa, V. Jakubskỳ, J. M. Guilarte, and M. S. Plyushchay,  Soliton defects in one-gap periodic system and exotic supersymmetry,  Phys. Rev. D 90(12), 125041 (2014).
  * [48] V. V. Nesterenko,  Instability of classical dynamics in theories with higher derivatives,  Phys. Rev. D 75(8), 087703 (2007).
  * [49] A. Szegleti and F. Márkus,  Dissipation in Lagrangian formalism,  Entropy 22(9), 930 (2020).
  * [50] J. Sanders,  A dual-oscillator approach to complex-stiffness damping based on fourth-order dynamics,  Nonlin. Dan. 109, 285–301 (2022).
  * [51] V. B. Matveev and M. A. Salle,  Darboux transformation and solitons,  (Springer, Berlin) (1991).
  * [52] L. Bianchi,  Vorlesungen über Differentialgeometrie,  (Teubner, Leipzig) (1927).
  * [53] G. L. Lamb Jr,  Analytical descriptions of ultrashort optical pulse propagation in a resonant medium,  Rev. Mod. Phys. 43, 99–124 (1971).
  * [54] G. Dunne and J. Feinberg,  Self-isospectral periodic potentials and supersymmetric quantum mechanics,  Phys. Rev. D 57(2), 1271 (1998).
  * [55] J. Cen, F. Correa, and A. Fring,  Degenerate multi-solitons in the sine-Gordon equation,  J. Phys. A: Math. Theor. 50, 435201 (2017).