Asymptotically conserved charges and 2-kink collision in quasi-integrable
potential KdV models

Harold Blas

Instituto de Física

Universidade Federal de Mato Grosso

Av. Fernando Correa, $N^{0}$ 2367

Bairro Boa Esperança, Cep 78060-900, Cuiabá - MT - Brazil

We study a particular deformation of the potential KdV model (pKdV) and
construct the quasi-conservation laws by a direct method. The charge
densities, differing from their integrable counterpart with homogeneous degree
terms, exhibit mixed scale dimension terms. The modifications of the charges
around the soliton interaction regions are examined by numerically simulating
some representative anomalies. We show numerically the elastic scattering of
two kinks for a wide range of values of the deformation parameters. It is
discussed an anomaly cancellation mechanism to define an exact conservation
law of the usual pKdV model, and a renormalization procedure is introduced for
some divergent charges by subtructing the continuous linear background
contribution. The KdV-type equations are quite ubiquitous in several areas of
non-linear science, such as the study of General Relativity in $AdS_{3}$,
Bose-Einstein condensates, superconductivity and fluid dynamics.

##  1 Introduction

Integrable models are characterized by soliton solutions and an infinite
number of conserved charges, making them mathematically elegant and physically
insightful [1, 2, 3]. However, many important physical systems with solitary
wave solutions do not fall under the integrable framework. In this context, it
has been put forward the quasi-integrability concept through the anomalous Lax
equation [4, 5] and pseudo-potential approaches [6, 7, 8], to tackle the
properties of modifications of integrable models.

The developments follow by the construction of an infinite number of
asymptotically conserved charges and the examination of the space-time
inversion symmetry property of the anomalies. These charges are asymptotically
conserved as long as the space-time integral of the corresponding anomaly
densities vanishes. The vanishing of the anomalies, in general, have been
verified numerically, except for some sub-models with particular set of
deformation parameters. Moreover, in the anomalous Lax and pseudo-potential
approaches the charge densities maintain a similar form to those of the
corresponding undeformed theories, i.e. they exhibit terms with homogeneous
scale dimensions as the integrable counterparts.

Recent research has uncovered several new towers of infinite number of
asymptotically conserved charges within deformed sine-Gordon, NLS and KdV
models [6, 9, 10, 8]. These new charges differ in form from those of the
undeformed model, as they exhibit terms with mixed scale dimensions. It is
noted that a subset of these new charges are anomalous even for the standard
integrable counterparts.

In this work we consider a deformation of the potential KdV model (pKdV) and
examine the type of charges, local and non-local, which depend explicitly on
the deformation parameters and exhibit mixed scale dimension terms. The pKdV
is one of the non-relativistic scalar field models which exhibit kink-type
topological solitons supported by non-vanishing boundary conditions (nvbc).
Regarding topological solitons it has been considered the deformations of the
relativistic SG and Bullough-Dodd models with kink solitons in [4] and [11],
respectively. So, it is interesting to consider the quasi-integrability
properties of a scalar field supporting non-relativistic topological solitons,
since such theories also appear in many areas of non-linear science, condensed
matter physics, plasma physics, Bose-Einstein condensates and, in particular,
fluid dynamics. In this context, our aim is to predict the results of solitary
wave collisions and test the quasi-integrability concept in deformed pKdV
models with nvbc. However, the nvbc inherent to the kink solitons, which may
change when the deformations are present, introduces new features when
applying the techniques developed for deformed KdV bright solitons. For
example, some quasi-conserved charges would require a renormalization
procedure by subtructing the contribution of the continuous linear background,
since the solution incorporates this vacuum solution plus the kink soliton
itself.

The space-time reflection symmetric ${\cal P}$ analytical N-kink solutions
will be obtained for the undeformed pKdV model, and the ${\cal P}$ symmetric
2-kink analytical solution for a sub-model, the so-called non-integrable
potential modified regularized long wave model (pmRLW). Using the vanishing
integrated anomaly for the 2-kink solutions we will show analytically the
quasi-integrability of the pmRLW model.

We perform the construction of the quasi-conservation laws by a direct
approach starting from the equation of motion. The numerical simulation of
two-soliton interactions in the deformed model sheds light on the dynamics of
the system. The observed elastic collisions, where solitons maintain their
shapes and velocities without significant radiation loss, hint at the
robustness and coherence of soliton behavior even in non-integrable systems.
Overall, this research contributes to our understanding of soliton dynamics in
non-integrable systems with non-relativistic kink solitons; so, highlighting
the persistence and interesting phenomena that emerge even in the absence of
strict integrability.

The paper is organized as follows. The next section presents a particular
deformation of the pKdV model. In sec. 3 some local and non-local quasi-
conservation laws are constructed through a direct approach. In sec. 4, using
the tau function method, we provide the pKdV analytical N-kink solitons, as
well as the 1-kink soliton for the deformed pKdV. A direct method allows as to
find a general 1-kink solution of the deformed pKdV. The 1-kink and 2-kink
solitons of the pmRLW theory are also uncovered. In sec. 5 it is discussed the
analytical quasi-integrability of the pmRLW theory for the 2-kink solutions.
In sec. 6 we numerically simulate the vanishing of some representative
anomalies of the deformed pKdV model, and in sec. 7 we discuss the results and
present the conclusions.

##  2 A particular deformation of the pKdV model

We will consider a deformation of the pKdV model. It involves the real scalar
field $v$ and the auxiliary field $w$ with equation of motion

$\displaystyle v_{t}+v_{x}+\frac{\alpha}{2}v_{x}^{2}+\,v_{xxx}$
$\displaystyle=$ $\displaystyle X,$ (2.1) $\displaystyle X$
$\displaystyle\equiv$
$\displaystyle-\epsilon_{2}\frac{\alpha}{4}w_{x}v_{t}+\epsilon_{1}(v_{xxt}+v_{xxx}),$
(2.2)

such that the auxiliary field satisfies

$\displaystyle v_{x}$ $\displaystyle=$ $\displaystyle w_{t}.$ (2.3)

The real numbers $\epsilon_{1}$ and $\epsilon_{2}$ are the deformation
parameters away from the usual pKdV and $\alpha$ is an arbitrary real
parameter defining the nonlinear term of the model. The model (2.1) embraces a
variety of sub-models; e.g. for $\epsilon_{1}=\epsilon_{2}=0$ one has the
integrable pKdV model, and for $\epsilon_{1}=\epsilon_{2}=1$ the non-
integrable potential modified regularized long wave model (pmRLW).

Note that defining $u\equiv\partial_{x}v$ one can write the $x-$derivative of
the eq. (2.1) as

$\displaystyle u_{t}+u_{x}+\alpha uu_{x}+\,u_{xxx}$ $\displaystyle=$
$\displaystyle\partial_{x}X,$ (2.4) $\displaystyle u$ $\displaystyle=$
$\displaystyle v_{x}=w_{t}.$

This is the so-called modified KdV model which has been studied in [12, 8] in
the context of quasi-integrability. However, the model (2.1) deserves to be
studied separately in order to examine the quasi-integrability properties and
its kink-type solitons. In fact, the undeformed pKdV model ($X=0$) exhibits a
kink-type soliton corresponding to a non-relativistic scalar model. So, it is
interesting to uncover its quasi-integrability properties, such as the quasi-
conservation laws and the kink collisions for deformed models ($X\neq 0$).

A suitable parametrization of the model (2.1) is available in order to
construct analytical or numerical soliton solutions of the model. So, let us
consider

$\displaystyle
w=-\frac{8}{\alpha}q_{x}\,\,\,\mbox{and}\,\,\,v=-\frac{8}{\alpha}q_{t}.$ (2.5)

So, substituting the expressions of $w$ and $v$ from (2.5), respectively, into
(2.1) one gets

$\displaystyle
q_{tt}+q_{xt}-4q^{2}_{xt}-2\epsilon_{2}q_{xx}q_{tt}+q_{xxxt}-\epsilon_{1}(q_{xxtt}+q_{xxxt})=0.$
(2.6)

Notice that this equation exhibits as the vacuum solution a continuous linear
background of the form

$\displaystyle q_{clb}=Kx-\Omega\,t+c_{o}.$ (2.7)

So, a general 1-soliton solution will incorporate this vacuum solution plus
the ‘kink’ soliton itself. In fact, the kink-type soliton for the field $v$
($v\sim`kink^{\prime}+\partial_{t}q_{clb}$) will interpolate two vacua of the
form (2.7).

Next, let us discuss some space-time symmetries related to soliton-type
solutions of the model. So, consider the space-time reflection around a given
fixed point $(x_{\Delta},t_{\Delta})$

$\displaystyle{\cal
P}:(\widetilde{x},\widetilde{t})\rightarrow(-\widetilde{x},-\widetilde{t});\,\,\,\,\,\,\,\,\widetilde{x}=x-x_{\Delta},\,\,\widetilde{t}=t-t_{\Delta}.$
(2.8)

In fact, the transformation ${\cal P}$ defines a shifted parity ${\cal P}_{s}$
for the spatial variable and the delayed time reversal ${\cal T}_{d}$ for the
time variable. When $x_{\Delta}=0$ ($t_{\Delta}=0$), ${\cal P}_{s}$ (${\cal
T}_{d}$) is reduced back to the pure parity ${\cal P}$ (pure time reversal
${\cal T}$).

Following the quasi-integrability approach [8] let us assume that the
$v-$field solution of the deformed pKdV model evaluated on the N-soliton
solution, viz. $v_{N_{-}sol}$, under (2.8) transforms as

$\displaystyle{\cal P}(v_{N-sol})=-v_{N-sol}+const.$ (2.9)

This implies, according to (2.3) and (2.5), that

$\displaystyle{\cal P}(w_{N-sol})=-w_{N-sol}+const.;\,\,\,\,\,{\cal
P}(q_{N-sol})=q_{N-sol}+const.$ (2.10)

Therefore, one has

$\displaystyle{\cal P}(X)=X,$ (2.11)

with $X$ defined in (2.2). Moreover, below we will assume the following
boundary conditions

$\displaystyle v(x=\pm\infty)$ $\displaystyle\rightarrow$ $\displaystyle
v_{\pm};\,\,\,\,v_{x}(x=\pm\infty)\rightarrow 0.$ (2.12) $\displaystyle
w(x=\pm\infty)$ $\displaystyle\rightarrow$ $\displaystyle
w_{\pm};\,\,\,\,w_{x}(x=\pm\infty)\rightarrow 0,$ (2.13)

with $v_{\pm}$ and $w_{\pm}$ being some constant numbers.

##  3 Quasi-conservation laws: A direct approach

In order to find the quasi-conservation laws one can resort to either the
anomalous zero-curvature [12] or to the Ricatti-type pseudo-potential [8]
approaches. In these approaches the relevant quasi-conserved charges are
composed by the sum of homogeneous terms defined by a suitable scaling
symmetry which was quite useful in the past in classifying the various
conserved quantities of the undeformed systems. In fact, for the pKdV model
one can define the scaling

$\displaystyle v$ $\displaystyle\rightarrow$ $\displaystyle\lambda
v,\,\,\,\,w\rightarrow\lambda^{-1}w$ (3.1) $\displaystyle t$
$\displaystyle\rightarrow$
$\displaystyle\lambda^{-3}t,\,\,\,\,\,x\rightarrow\lambda^{-1}t.$ (3.2)

So, the deformed pKdV model (2.1) under the scaling transformation (3.1)-(3.2)
can be written as

$\displaystyle v_{t}+\frac{\alpha}{2}v_{x}^{2}+\,v_{xxx}$ $\displaystyle=$
$\displaystyle-\lambda^{-2}v_{x}-\epsilon_{2}\frac{\alpha}{4}w_{x}v_{t}+\epsilon_{1}(\lambda^{2}v_{xxt}+v_{xxx}).$
(3.3)

Since the term $v_{x}$ in (2.1) can be removed by the transformation
$x\rightarrow x-t$, one can conclude that the undeformed pKdV model
($\epsilon_{1}=\epsilon_{2}=0$) is invariant under the scaling symmetry
(3.1)-(3.2), and the relevant conserved charges will inherit this property.
Let us emphasize that in the developments of the quasi-integrability concept
applied to deformations of the integrable models such as SG, NLS, and KdV [4,
5, 6, 8], the quasi-conserved charge densities resemble to the ones of the
relevant integrable counterparts, such that their homogeneous terms do not
contain explicitly the corresponding deformation parameters. So, one can argue
that the constant asymptotic behavior of the charges is due to the quasi-
integrable deformations. Therefore, the search for quasi-conserved charge
densities which depend explicitly on the deformation parameters and terms with
mixed scale dimensions deserve to be analyzed carefully in each case. In the
case of deformations of the NLS and KdV this construction has been performed
in [9, 10] and [8], respectively.

So, here we pursue the quasi-conservation laws following a direct approach
starting from the deformed pKdV eqs. of motion (2.1)-(2.3), which allows one
to uncover the charges with mixed scaling dimension terms. So, taking the
$x-$derivative of (2.1) one can write

$\displaystyle\partial_{t}(v_{x})+\partial_{x}[v_{x}+\frac{\alpha}{2}v_{x}^{2}+v_{xxx}-X]$
$\displaystyle=$ $\displaystyle 0.$ (3.4)

This defines a first exact conservation law of the model (2.1) with conserved
charge

$\displaystyle q^{(1)}_{X}$ $\displaystyle=$ $\displaystyle\int dx\,v_{x}$
(3.5) $\displaystyle=$ $\displaystyle v_{+}-v_{-},$ (3.6)

where the b.c. (2.12) has been considered. One has that $q^{(1)}_{X}$ defines
a topological charge. Notice that this charge is conserved even for the
$X-$deformed pKdV model (2.1). By direct construction starting from the eq.
(3.4) one can write the next quasi-conservation laws

$\displaystyle\partial_{t}(v_{x}^{2})+\partial_{x}[v_{x}^{2}+\frac{2\alpha}{3}v_{x}^{3}+2v_{x}v_{xxx}-v_{xx}^{2}]$
$\displaystyle=$ $\displaystyle 2v_{x}X_{x},$ (3.7)
$\displaystyle\partial_{t}(v_{x}^{2}+\frac{1}{2}\epsilon_{1}v_{xx}^{2})+\partial_{x}[v_{x}^{2}+\frac{2\alpha}{3}v_{x}^{3}+2v_{x}v_{xxx}-(1-\epsilon_{1})v_{xx}^{2}-2v_{x}X]$
$\displaystyle=$ $\displaystyle\frac{1}{2}\alpha\epsilon_{2}v_{xx}w_{x}v_{t},$
(3.8)
$\displaystyle\partial_{t}(\frac{\alpha}{3}v_{x}^{3})+\partial_{x}[\frac{\alpha}{3}v_{x}^{3}+\frac{\alpha^{2}}{4}v_{x}^{4}]$
$\displaystyle=$ $\displaystyle-\alpha v_{x}^{2}v_{4x}+\alpha v_{x}^{2}X_{x},$
(3.9) $\displaystyle\partial_{t}(\frac{\alpha}{3}v_{x}^{3}+\epsilon_{1}\alpha
v_{x}v_{xx}^{2})+\partial_{x}[\frac{\alpha}{3}v_{x}^{3}+\frac{\alpha^{2}}{4}v_{x}^{4}]$
$\displaystyle=$ $\displaystyle(\epsilon_{1}-1)\alpha v_{x}^{2}v_{4x}+\alpha
v_{xx}\times$ (3.10)
$\displaystyle(\frac{\alpha}{2}\epsilon_{2}v_{x}w_{x}v_{t}+\epsilon_{1}v_{xt}v_{xx}),\,\,\,\,\,\,\,\,\,\,$
$\displaystyle\partial_{t}(v_{xx}^{2})+\partial_{x}[v_{xx}^{2}+\alpha
v_{xx}(v_{x}^{2})_{x}-\alpha v_{xxx}v_{x}^{2}+2v_{xx}v_{4x}-v_{xxx}^{2}]$
$\displaystyle=$ $\displaystyle-\alpha v_{x}^{2}v_{4x}+2v_{xx}X_{xx},$ (3.11)
$\displaystyle\partial_{t}(v_{xx}^{2}+\epsilon_{1}v_{xxx}^{2})+\partial_{x}[v_{xx}^{2}+\alpha
v_{xx}(v_{x}^{2})_{x}-\alpha v_{xxx}v_{x}^{2}+2v_{xx}v_{4x}-v_{xxx}^{2}-$
$\displaystyle
2v_{xx}X_{x}+2v_{xxx}X-2\epsilon_{1}v_{xxx}v_{xxt}-\epsilon_{1}v_{xxx}^{2}]$
$\displaystyle=$ $\displaystyle-\alpha
v_{x}^{2}v_{4x}+\frac{\alpha}{2}\epsilon_{2}v_{4x}w_{x}v_{t}.$ (3.12)

The r.h.s. expressions in (3.7)-(3.12) define the relevant anomalies and they
exhibit odd parities under the symmetry transformations (2.9)-(2.11). From the
above identities one can define the following quasi-conservation laws

$\displaystyle\frac{d}{dt}q^{(a)}_{j,X}\equiv\int_{-\infty}^{+\infty}dx\,\alpha^{(a)}_{j,X},\,\,\,\,\,\,a=3,5;\,\,\,\,\,\,j=1,2,...4.$
(3.13)

with

$\displaystyle q^{(3)}_{1,X}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,v_{x}^{2},$ (3.14) $\displaystyle
q^{(3)}_{2,X}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,[v_{x}^{2}+\frac{1}{2}\epsilon_{1}(v_{xx})^{2}],$
(3.15) $\displaystyle q^{(5)}_{1,X}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,\frac{\alpha}{3}v_{x}^{3},$ (3.16)
$\displaystyle q^{(5)}_{2,X}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,[\frac{\alpha}{3}v_{x}^{3}+\epsilon_{1}\alpha
v_{x}v_{xx}^{2}],$ (3.17) $\displaystyle q^{(5)}_{3,X}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,(v_{xx})^{2},$ (3.18) $\displaystyle
q^{(5)}_{4,X}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,[v_{xx}^{2}+\epsilon_{1}v_{xxx}^{2}].$
(3.19)

and

$\displaystyle\alpha^{(3)}_{1}$ $\displaystyle=$ $\displaystyle 2v_{x}X_{x},$
(3.20) $\displaystyle\alpha^{(3)}_{2}$ $\displaystyle=$
$\displaystyle\frac{\alpha}{2}\epsilon_{2}v_{xx}w_{x}v_{t},$ (3.21)
$\displaystyle\alpha^{(5)}_{1}$ $\displaystyle=$ $\displaystyle[-\alpha
v_{x}^{2}v_{4x}+\alpha v_{x}^{2}X_{x}],$ (3.22)
$\displaystyle\alpha^{(5)}_{2}$ $\displaystyle=$
$\displaystyle(\epsilon_{1}-1)\alpha v_{x}^{2}v_{4x}+\alpha
v_{xx}(\frac{\alpha}{2}\epsilon_{2}v_{x}w_{x}v_{t}+\epsilon_{1}v_{xt}v_{xx})$
(3.23) $\displaystyle\alpha^{(5)}_{3}$ $\displaystyle=$ $\displaystyle[-\alpha
v_{x}^{2}v_{4x}+2v_{xx}X_{xx}].$ (3.24) $\displaystyle\alpha^{(5)}_{4}$
$\displaystyle=$ $\displaystyle[-\alpha
v_{x}^{2}v_{4x}+\frac{\alpha}{2}\epsilon_{2}v_{4x}w_{x}v_{t}].$ (3.25)

Notice that in general the charges and anomalies defined above do not exhibit
homogeneous terms under the scaling symmetry (3.1)-(3.2). The charge
$q^{(3)}_{1}$ in (3.14) becomes an exact conserved charge for undeformed pKdV
$(X=0)$, whereas the charges $q^{(5)}_{1}$ (3.16) and $q^{(5)}_{3}$ (3.18)
define the first quasi-conserved charges even for $X=0$. Notice that
$q^{(3)}_{2}$ becomes an exact conserved charge for $\epsilon_{2}=0$.
Remarkably, any linear combination of the above charges define another quasi-
conserved charge. In particular, one can write the next linear combination

$\displaystyle q^{(5),\gamma}_{X}$ $\displaystyle\equiv$ $\displaystyle
q^{(5)}_{1,X}-\gamma\,q^{(5)}_{3,X}$ (3.26) $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,[\frac{\alpha}{3}v_{x}^{3}-\gamma\,v_{xx}^{2}],$
(3.27)

such that

$\displaystyle\frac{d}{dt}q^{(5),\gamma}_{X}=\int_{-\infty}^{+\infty}dx\,[-\alpha(1-\gamma)v_{x}^{2}v_{4x}+\alpha
v_{x}^{2}X_{x}-2\gamma v_{xx}X_{xx}],$ (3.28)

with $\gamma$ being an arbitrary real parameter. So, $q^{(5),\gamma}_{X}$
becomes a quasi-conserved charge of the deformed pKdV model. From
(3.26)-(3.28) one can see that even for undeformed pKdV ($X=0$) one has a one-
parameter family of quasi-conserved charges $q^{(5),\gamma}_{X=0}$
parametrized by $\gamma$. However, for $X=0$ and $\gamma=1$, one gets

$\displaystyle q^{(5),\gamma=1}_{pKdV}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{+\infty}dx\,[\frac{\alpha}{3}v_{x}^{3}-\,v_{xx}^{2}],$
(3.29)

which is an exact conserved charge of the pKdV model.

Let us write the quasi-conserved charge $q^{(5),\gamma}_{X=0}$ of the
undeformed pKdV model and its relevant anomaly from (3.28) as

$\displaystyle\frac{d}{dt}q^{(5),\gamma}_{X=0}\equiv-\alpha(1-\gamma)\int_{-\infty}^{+\infty}dx\,\alpha^{(5)},\,\,\,\,\,\,\,\,\,\,\,\alpha^{(5)}\equiv
v_{x}^{2}v_{4x}.$ (3.30)

Notice that for $\gamma=1$ one has an exact conservation law, whereas for
$\gamma\neq 1$ this charge behaves as an asymptotically conserved charge, i.e.

$\displaystyle
q^{(5),\gamma}_{X=0}(t=+\infty)=q^{(5),\gamma}_{X=0}(t=-\infty),$ (3.31)

provided that

$\displaystyle\int_{t=-\infty}^{t=+\infty}dt\,\int_{-\infty}^{+\infty}dx\,\alpha^{(5)}=0.$
(3.32)

Note that $X$ in the the r.h.s. of (2.1) contains a non-local term $w_{x}$,
since this can be defined from (2.3) in terms of the pKdV field $v$ as
$w_{x}=\int dt\,v_{x}$. It is interesting to search for quasi-conservation
laws with charges incorporating this type of non-local terms. So, from (2.1)
one can write

$\displaystyle\partial_{t}\Big{[}v_{x}X-v_{x}^{2}-\frac{\alpha}{3}v_{x}^{3}+v_{xx}^{2}-v_{x}v_{t}\Big{]}-\partial_{x}\Big{[}v_{x}v_{xxt}-X(v_{x}+\frac{\alpha}{2}v_{x}^{2}+v_{xxx})+$
$\displaystyle
X^{2}-v_{t}^{2}-v_{t}v_{x}-v_{t}v_{xxx}+v_{xt}v_{xx}\Big{]}=-\frac{\alpha}{2}v_{t}(v_{x}^{2})_{x}$
(3.33)

with non-local charge

$\displaystyle\hat{q}\equiv\int
dx\,[v_{x}X-v_{x}^{2}-\frac{\alpha}{3}v_{x}^{3}+v_{xx}^{2}-v_{x}v_{t}]$ (3.34)

and anomaly density

$\displaystyle\hat{\alpha}\equiv-\frac{\alpha}{2}v_{t}(v_{x}^{2})_{x}.$ (3.35)

Below we will perform numerical simulations of some of the above anomalies for
the collision of 2-kink solitons of the (deformed) pKdV model. We will
numerically verify the quasi-conservation law (3.12) and the relevant anomaly
density $\alpha^{(5)}_{4}(x,t)$ in (3.25) in the Fig. 1. Likewise, we will
examine the behavior of the anomaly (3.35) in the Fig. 2.

Next, let us consider the equivalent equation of the model (2.1) in the
$q-$field parametrization (2.6) and write the following quasi-conservation law

$\displaystyle\partial_{t}[\frac{1}{2}q_{t}^{2}-\epsilon_{2}q_{xx}q_{t}^{2}+\frac{1}{2}\epsilon_{1}q_{xt}^{2}]+\partial_{x}[\frac{1}{2}q_{t}^{2}-4q_{xt}q_{t}^{2}+q_{xxt}q_{t}-\frac{1}{2}(1-\epsilon_{1})q_{xt}^{2}-\epsilon_{1}q_{xtt}q_{t}-\epsilon_{1}q_{xxt}q_{t}]$
$\displaystyle=-2(2+\epsilon_{2})q_{xxt}q_{t}^{2}-4q_{xt}^{2}q_{t}.$ (3.36)

Then, one can define the charge and anomaly, respectively, as

$\displaystyle{\cal Q}_{1}$ $\displaystyle=$ $\displaystyle\int
dx[\frac{1}{2}q_{t}^{2}-\epsilon_{2}q_{xx}q_{t}^{2}+\frac{1}{2}\epsilon_{1}q_{xt}^{2}],$
(3.37) $\displaystyle{\cal A}_{1}$ $\displaystyle=$ $\displaystyle\int
dx[-2(2+\epsilon_{2})q_{xxt}q_{t}^{2}-4q_{xt}^{2}q_{t}].$ (3.38)

Note that the charge ${\cal Q}_{1}$ diverges when evaluated on the vacuum
solution $q_{clb}$ (2.7), due to the contribution of the term
$\frac{1}{2}q_{t}^{2}$. Therefore, the charge ${\cal Q}_{1}$ (3.37) for the
composite field $q\sim`kink^{\prime}+q_{clb}$ must be renormalized by
subtructing the contribution (which is infinite) of the continuous linear
background $q_{clb}$ in (2.7), i.e. the kink charge ${\cal Q}_{1\,kink}$ must
be renormalized. This is reminiscent of the procedure performed in the
defocusing NLS model in which the momentum associated to the dark soliton
itself has been renormalized [13].

##  4 Tau function and direct methods to construct solitons

Next we use the tau function method in order to obtain the 1-soliton and
2-soliton solutions of the model (2.1) for some particular cases of the
deformations parameters $\\{\epsilon_{1},\epsilon_{2}\\}$.

Let us consider

$\displaystyle q_{s}(x,t)=\beta\log{\tau(x,t)},$ (4.1)

where $\tau$ defines a tau function. Next, replacing (4.1) into (2.6) one gets
the next equation

$\displaystyle\beta\tau^{3}\left[-\epsilon_{1}\tau_{xxtt}+(1-\epsilon_{1})\tau_{xxxt}+\tau_{tt}+\tau_{xt}\right]$
(4.2) $\displaystyle-$
$\displaystyle\beta\tau^{2}\left[(2\beta{\epsilon}_{2}-\epsilon_{1})\tau_{tt}\tau_{xx}+2(2\beta-\epsilon_{1})\tau_{xt}^{2}+\tau_{t}\left(-2\epsilon_{1}\tau_{xxt}+(1-\epsilon_{1})\tau_{xxx}+\tau_{x}\right)\right.$
$\displaystyle-$ $\displaystyle
2\left.\epsilon_{1}\tau_{x}\tau_{xtt}+3(1-\epsilon_{1})\tau_{xt}\tau_{xx}+3(1-\epsilon_{1})\tau_{x}\tau_{xxt}+\tau_{t}^{2}+3(1-\epsilon)\tau_{x}\tau_{xxt}\right]$
$\displaystyle+$ $\displaystyle
2\beta\tau\left[(\beta{\epsilon}_{2}-\epsilon_{1})\tau_{xx}\tau_{t}^{2}+(\beta{\epsilon}_{2}-\epsilon_{1})\tau_{tt}\tau_{x}^{2}\right.$
$\displaystyle+$
$\displaystyle\left.\tau_{x}\tau_{t}\left(4(\beta-\epsilon_{1})\tau_{xt}-3(\epsilon_{1}-1)\tau_{xx}\right)-3(\epsilon_{1}-1)\tau_{x}^{2}\tau_{xt}\right]$
$\displaystyle-$ $\displaystyle
2\tau_{t}\tau_{x}^{2}\left[(\beta({\epsilon}_{2}+2)-3\epsilon_{1})\tau_{t}-3(\epsilon_{1}-1)\tau_{x}\right]=0.$

1-soliton solution of deformed pKdV

The tau function method furnishes the first type of kink solution of (2.6) for
arbitrary values of the set $\\{\epsilon_{1},\epsilon_{2}\\}$. So,
substituting the tau function $\tau=1+e^{\Gamma_{1}}$ into (4.2), and taking
into account (4.1), one has

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{3}{(2+\epsilon_{2})\left(1+(1-\epsilon_{1})k^{2}\right)}\log{[1+e^{\Gamma_{1}}]}$
(4.3) $\displaystyle=$
$\displaystyle\frac{3}{(2+\epsilon_{2})\left(1+(1-\epsilon_{1})k^{2}\right)}\Big{\\{}\log{2}+\frac{\Gamma_{1}}{2}+\log{\cosh{(\frac{\Gamma_{1}}{2})}}\Big{\\}},$
(4.4)

with

$\displaystyle\Gamma_{1}=k_{1}x-w_{1}t+\delta;\,\,\,\,\,\,w_{1}=\frac{k_{1}+(1-\epsilon_{1})k_{1}^{3}}{1-\epsilon_{1}k_{1}^{2}}.$
(4.5)

So, the eq. (2.5) provides the first type of 1-soliton solution for the pKdV
field $v$

$\displaystyle
v_{1}=\frac{12}{\alpha}\frac{k_{1}}{(2+\epsilon_{2})\left(1-\epsilon_{1}k_{1}^{2}\right)}\mbox{Tanh}\Big{[}\frac{1}{2}\left(k_{1}x-w_{1}t+\delta\right)\Big{]}.$
(4.6)

2-soliton solution of deformed pKdV: the case $\epsilon_{1}=\epsilon_{2}=1$

The pKdV $2-$soliton solution follows similar construction as in the case of
the modified KdV for the particular case $\epsilon_{1}=\epsilon_{2}=1$. The
field $q$ takes the form [12, 14]

$\displaystyle q$ $\displaystyle=$
$\displaystyle\log{\Big{[}1+e^{\Gamma_{1}}+e^{\Gamma_{2}}+A_{12}e^{\Gamma_{1}}e^{\Gamma_{2}}\Big{]}},\,\,\,\,\Gamma_{i}=k_{i}x-w_{i}t+\delta_{i},\,\,\,w_{i}=\frac{k_{i}}{1-k_{i}^{2}},\,\,i=1,2.$
(4.7) $\displaystyle A_{12}$ $\displaystyle=$
$\displaystyle-\frac{(w_{1}-w_{2})^{2}(k_{1}-k_{2})^{2}+(w_{1}-w_{2})(k_{1}-k_{2})-(w_{1}-w_{2})^{2}}{(w_{1}+w_{2})^{2}(k_{1}+k_{2})^{2}+(w_{1}+w_{2})(k_{1}+k_{2})-(w_{1}+w_{2})^{2}}.$
(4.8)

In order to exhibit the parity symmetry we follow the approach in [8]
developed for the modified KdV. So, in order to perform the transformation
(2.8) and check the space-time parity inversion symmetry of the 2-soliton
solution we will derive a new expression for $q$ in (4.7), such that
$v_{2-sol}$ in (2.5) becomes a manifestly ${\cal P}$ invariant function. So,
let us define a new parameter $\Delta$, as $A_{12}=e^{\Delta}$, and

$\displaystyle\Gamma_{j}$ $\displaystyle=$ $\displaystyle
k_{j}\widetilde{x}-w_{j}\widetilde{t}+\eta_{0j}-\frac{\Delta}{2}\equiv\eta_{j}-\frac{\Delta}{2},\,\,\,\,\,j=1,2$
(4.9)

where

$\displaystyle\delta_{j}=-k_{j}x_{\Delta}+w_{j}t_{\Delta}+\eta_{0j}-\frac{\Delta}{2},\,\,\,j=1,2.$
(4.10)

Therefore, $q$ can be rewritten as

$\displaystyle
q=\log{\Big{[}2e^{-\Delta/4}\,e^{(\eta_{1}+\eta_{2})/2}\left(e^{\Delta/4}\cosh{(\frac{\eta_{1}+\eta_{2}}{2})}+e^{-\Delta/4}\cosh{(\frac{\eta_{1}-\eta_{2}}{2})}\right)\Big{]}}.$
(4.11)

So, using (2.5) one has

$\displaystyle v_{2-sol}$ $\displaystyle=$
$\displaystyle\frac{4}{\alpha}(w_{1}+w_{2})-\frac{8}{\alpha}\partial_{t}\log{\Big{[}e^{\Delta/4}\cosh{(\frac{\eta_{1}+\eta_{2}}{2})}+e^{-\Delta/4}\cosh{(\frac{\eta_{1}-\eta_{2}}{2})}\Big{]}}.$
(4.12)

Therefore, the 2-soliton

$\displaystyle{\bf v}_{2-sol}=v_{2-sol}\Big{|}_{\eta_{01}=\eta_{02}=0},$
(4.13)

satisfies the symmetry

$\displaystyle{\cal P}({\bf v}_{2-sol})=-{\bf v}_{2-sol}+const.$ (4.14)

Moreover, using (2.5) and following similar steps as above one can show

$\displaystyle{\cal P}({\bf w}_{2-sol})=-{\bf w}_{2-sol}+const.$ (4.15)

Due to the condition $\eta_{01}=\eta_{02}=0$, one can get the next
relationships for the coordinates $(x_{\Delta},t_{\Delta})$

$\displaystyle x_{\Delta}$ $\displaystyle\equiv$
$\displaystyle\frac{w_{2}\widetilde{\theta}_{1}-w_{1}\widetilde{\theta}_{2}}{k_{2}w_{1}-k_{1}w_{2}}$
(4.16) $\displaystyle t_{\Delta}$ $\displaystyle\equiv$
$\displaystyle\frac{k_{2}\widetilde{\theta}_{1}-k_{1}\widetilde{\theta}_{2}}{k_{2}w_{1}-k_{1}w_{2}},\,\,\,\,\,\widetilde{\theta}_{j}\equiv\frac{\Delta}{2}+\delta_{j},\,\,\,j=1,2.$
(4.17)

###  4.1 Tau function approach and ${\cal P}$ symmetric N-solitons of the
usual pKdV

In order to find the N-soliton solutions of the pKdV model satisfying the
symmetry properties (2.8) and (2.9) we closely follow the construction
performed for the KdV model in [15, 8]. So, we will construct a general
$N-$soliton solution possessing the space-time parity symmetry (2.8)-(2.9),
for any shifted point and delayed time ($x_{\Delta},\,t_{\Delta}$) in space-
time.

The usual pKdV equation of motion is defined by setting
$\epsilon_{1}=\epsilon_{2}=0$ and making the transformation $x\rightarrow x-t$
in (2.1); so, one has

$\displaystyle v_{t}+\frac{\alpha}{2}v_{x}^{2}+\,v_{xxx}=0.$ (4.18)

The Hirota’s tau function for the eq. (4.18) is introduced as

$\displaystyle v=\frac{12}{\alpha}\partial_{x}\log{\tau}.$ (4.19)

The Hirota bi-linear equation of (4.18), as well as multi-solitons, lumps, and
breather wave-solutions have recently been presented in [16]. Since the pKdV
field $v$ and the KdV field $u$ can be related by $u=\partial_{x}v$, one can
take the same tau functions for the both models in order to construct their
soliton solutions. So, let us assume the next tau function for the $N-$soliton
solution [17]

$\displaystyle\tau_{N}=\sum_{\mu}\exp\left(\sum_{j=1}^{N}\mu_{j}\Gamma_{j}+\sum_{1\leq
j<l}^{N}\mu_{j}\mu_{l}\theta_{jl}\right)$ (4.20)

where the $\mu-$summation is undertaken in all the permutations of
$\mu_{i}=0,1$, for $i=1,2,...N$, and

$\displaystyle\Gamma_{j}=k_{i}x-w_{i}t+\xi_{0j},\,\,\,\,e^{\theta_{ij}}=\left(\frac{k_{i}-k_{j}}{k_{i}+k_{j}}\right)^{2},\,\,\,\,w_{i}=k_{i}^{3}.$
(4.21)

The $\xi_{0j}^{\prime}s$ are arbitrary constants related to the space-time
translation invariance of the pKdV equation, such that each $j-$soliton
component of the $N-$soliton can be located anywhere at $\xi_{0j}$. A subset
of solutions possessing the space-time symmetry (2.8)-(2.9) is achieved by
making a particular choice of the set of parameters $\xi_{0j}$, such that the
space-time translation symmetry of the solution (4.20) is broken. So, let us
consider [8]

$\displaystyle\Gamma_{j}=k_{j}(x-x_{\Delta})-w_{j}(t-t_{\Delta})+\eta_{0j}-\frac{1}{2}\sum_{i=1}^{j-1}\theta_{ij}-\frac{1}{2}\sum_{i=j+1}^{N}\theta_{ji}\equiv\eta_{j}-\frac{1}{2}\sum_{i=1}^{j-1}\theta_{ij}-\frac{1}{2}\sum_{i=j+1}^{N}\theta_{ji}.$
(4.22)

With these redefinitions and the tau function (4.20) one can write the
$N-$soliton solution in the equivalent form

$\displaystyle
v_{N}=-\frac{6}{\alpha}(\sum_{j}k_{j})+\frac{12}{\alpha}\partial_{x}\Big{[}\log{\sum_{\nu}K_{\nu}\cosh{\left(\frac{1}{2}\sum_{j=1}^{N}\nu_{j}\eta_{j}\right)}}\Big{]},$
(4.23)

where the summation in $\nu$ involves all the permutations of $\nu_{i}=1,-1$,
$i=1,2,...N$, and $K_{\nu}=\Pi_{i>j}(k_{i}-\nu_{i}\nu_{j}k_{j})$.

Then, from (4.23) let us define the $N-$soliton solution with space-time
translation symmetry broken as

$\displaystyle{\bf v}_{N}=v_{N}|_{\eta_{0j}=0}.$ (4.24)

It is a matter of direct verification that this solution ${\bf v}_{N}$ will
exhibit the symmetry (2.8)-(2.9), i.e.

$\displaystyle{\cal P}({\bf v}_{N})=-{\bf v}_{N}+const.$ (4.25)

Next, we present the above constructions for the cases $N=1,2,3$, and describe
the main properties of the corresponding solitons.

Case $N=1$. One has $\tau_{1}=1+e^{\Gamma_{1}}$, which can be written as

$\displaystyle\tau_{1}=2e^{-\frac{\eta_{1}}{2}}\Big{[}\cosh{\frac{\eta_{1}}{2}}\Big{]},\,\,\,\,\eta_{1}=k_{1}(x-x_{\Delta})-w_{1}(t-t_{\Delta})+\eta_{01}.$
(4.26)

Next, one has

$\displaystyle\log{(\tau_{1})}=\log{2}-\frac{\eta_{1}}{2}+\log{\Big{[}\cosh{\frac{\eta_{1}}{2}}\Big{]}}.$
(4.27)

Therefore, inserting (4.27) into (4.19) one can write

$\displaystyle{\bf
v}_{1}=-\frac{6k_{1}}{\alpha}+\frac{12}{\alpha}\partial_{x}\log{\Big{[}\cosh{\frac{\eta_{1}}{2}}\Big{]}}|_{\eta_{01}=0}.$
(4.28)

This expression satisfies the symmetry transformation ${\cal P}({\bf
v}_{1})=-{\bf v}_{1}+const.$ This soliton can be written as

$\displaystyle{\bf
v}_{1}=-\frac{6k_{1}}{\alpha}\,\Big{\\{}1-\mbox{Tanh}\Big{[}\frac{k_{1}(x-x_{\Delta})-w_{1}(t-t_{\Delta})}{2}\Big{]}\Big{\\}}.$
(4.29)

This is the topological 1-kink solution of the pKdV model (4.18).

Case $N=2$. The tau function becomes

$\displaystyle\tau_{2}=1+e^{\Gamma_{1}}+e^{\Gamma_{2}}+e^{\Gamma_{1}+\Gamma_{2}+\theta_{12}},$
(4.30)

which, following the above construction, can be rewritten as

$\displaystyle\tau_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{k_{1}-k_{2}}e^{(\eta_{1}+\eta_{2})/2}\Big{[}(k_{1}-k_{2})\cosh{\left(\frac{\eta_{1}+\eta_{2}}{2}\right)}+(k_{1}+k_{2})\cosh{\left(\frac{\eta_{1}-\eta_{2}}{2}\right)}\Big{]}.$
(4.31) $\displaystyle\eta_{i}$ $\displaystyle=$ $\displaystyle
k_{i}(x-x_{\Delta})-w_{i}(t-t_{\Delta})+\eta_{0i},\,\,\,\,i=1,2.$ (4.32)

Then, inserting (4.27) into (4.19) and taking into account (4.23) and (4.24)
it is straightforward to construct a ${\cal P}$ invariant 2-soliton as

$\displaystyle{\bf
v}_{2}=-\frac{6}{\alpha}(k_{1}+k_{2})+\frac{12}{\alpha}\partial_{x}\log{\Big{[}(k_{1}-k_{2})\cosh{\left(\frac{\eta_{1}+\eta_{2}}{2}\right)}+(k_{1}+k_{2})\cosh{\left(\frac{\eta_{1}-\eta_{2}}{2}\right)}\Big{]}}\Big{|}_{\eta_{01}=\eta_{02}=0}.$
(4.33)

Thus, this pKdV 2-soliton solution transforms under the parity transformation
as ${\cal P}({\bf v}_{2})=-{\bf v}_{2}+const.$

Case $N=3$. It can be constructed following similarly steps. So, the tau
function $\tau_{3}$ becomes

$\displaystyle\tau_{3}=1+e^{\Gamma_{1}}+e^{\Gamma_{2}}+e^{\Gamma_{3}}+e^{\Gamma_{1}+\Gamma_{2}+\theta_{12}}+e^{\Gamma_{1}+\Gamma_{3}+\theta_{13}}+e^{\Gamma_{2}+\Gamma_{3}+\theta_{23}}+e^{\Gamma_{1}+\Gamma_{2}+\Gamma_{3}+\theta_{12}+\theta_{13}+\theta_{23}},$
(4.34)

which can be rewritten as

$\displaystyle\tau_{3}$ $\displaystyle=$
$\displaystyle\frac{2e^{(\eta_{1}+\eta_{2}+\eta_{3})/2}}{(k_{1}-k_{2})(k_{1}-k_{3})(k_{2}-k_{3})}\Big{[}C(x,t)\Big{]}$
(4.35) $\displaystyle C(x,t)$ $\displaystyle\equiv$
$\displaystyle(k_{1}-k_{2})(k_{1}-k_{3})(k_{2}-k_{3})\cosh{[(\eta_{1}+\eta_{2}+\eta_{3})/2]}+$
$\displaystyle(k_{1}+k_{2})(k_{1}+k_{3})(k_{2}-k_{3})\cosh{[(-\eta_{1}+\eta_{2}+\eta_{3})/2]}+$
$\displaystyle(k_{1}+k_{2})(k_{1}-k_{3})(k_{2}+k_{3})\cosh{[(\eta_{1}-\eta_{2}+\eta_{3})/2]}+$
$\displaystyle(k_{1}-k_{2})(k_{1}+k_{3})(k_{2}+k_{3})\cosh{[(\eta_{1}+\eta_{2}-\eta_{3})/2]},$
$\displaystyle\eta_{i}$ $\displaystyle=$ $\displaystyle
k_{i}(x-x_{\Delta})-w_{i}(t-t_{\Delta})+\eta_{0i},\,\,\,\,i=1,2,3.$

Similarly, it is straightforward to construct a ${\cal P}$ invariant 3-soliton
as

$\displaystyle{\bf
v}_{3}=-\frac{6}{\alpha}(k_{1}+k_{2}+k_{3})+\frac{12}{\alpha}\partial_{x}\log{C(x,t)}\Big{|}_{\eta_{01}=\eta_{02}=\eta_{03}=0}.$
(4.36)

Clearly, this pKdV 3-soliton solution transforms as ${\cal P}({\bf
v}_{3})=-{\bf v}_{3}+const.$

So, the ${\cal P}$ invariant N-soliton solutions would allow us to show that
the $(x,t)-$integrated anomalies of the integrable pKdV model vanish. In
particular, for the quasi-conserved charges $q^{(5)}_{1}$ (3.16) and
$q^{(5)}_{3}$ (3.18) defined for $X=0$, one can show the vanishing of the
$(x,t)-$integrated anomaly $-\alpha v_{x}^{2}v_{4x}$ (3.22) (or (3.24)) using
the parity ${\cal P}$ argument.

###  4.2 A direct method and a second type of 1-kink solutions

Next we present a direct method in order to find a second type of 1-soliton
solution for any set $\\{\epsilon_{1},\epsilon_{2}\\}$, which generalizes the
one obtained above through the tau function method. In fact, a direct method
provides a general 1-soliton solution of (2.6) by assuming the form

$\displaystyle q_{I}$ $\displaystyle=$ $\displaystyle
q_{0}\Big{\\{}\log{\cosh{[\frac{\zeta}{2a}]}}+b\,\zeta+c\Big{\\}},\,\,\,\,\,\zeta=kx-
w_{1}t+\delta.$ (4.37)

Note that we have incorporated the term $q_{0}(b\,\zeta+c)$ corresponding to
the continuous linear background (2.7). A direct substitution of $q_{I}$ into
(2.6) provides the relationships

$\displaystyle
w_{1}=\frac{a^{2}k+(1-\epsilon_{1})k^{3}}{a^{2}-\epsilon_{1}k^{2}};\,\,\,\,\,q_{0}=\frac{3a^{2}}{(a^{2}+(1-\epsilon_{1})k^{2})(2+\epsilon_{2})},$
(4.38)

such that $a,\,b$ and $c$ are arbitrary real parameters. So, through (2.5) one
has the second type of 1-soliton solution for $v$

$\displaystyle
v_{I}=-q_{o}bw_{1}+\frac{12a}{\alpha}\frac{k}{(a^{2}-\epsilon_{1}k^{2})(2+\epsilon_{2})}\mbox{Tanh}\Big{[}\frac{1}{2a}(kx-
w_{1}t+\delta)\Big{]}.$ (4.39)

This is a new general form of a 1-kink soliton plus a constant background
which can not be found by the tau function method. Clearly, the two types of
solutions $v_{1}$ in (4.6) and the kink sector of $v_{I}$ in (4.39) become the
same for $a^{2}=1$ and for arbitrary values of the set
$\\{\epsilon_{1},\epsilon_{2}\\}$. However, the solution $v_{I}$ describes a
family of kinks parametrized by an additional free parameter $a$, which can be
related to the width of the kink $\sim\frac{2a}{k}$. As the initial condition
of our numerical simulations for 2-kink collision below we will consider the
superposition of two solitons of the general type (4.39) located some distance
apart.

##  5 Analytical quasi-integrability of the pmRLW theory
($\epsilon_{1}=\epsilon_{2}=1$)

Let us consider a sub-model of (2.1) defined by the special case
$\epsilon_{1}=\epsilon_{2}=1$. So, one has the sub-model

$\displaystyle
v_{t}+v_{x}+\frac{\alpha}{2}v_{x}^{2}+\frac{\alpha}{4}w_{x}v_{t}-v_{xxt}=0.$
(5.1)

This model can be considered as the potential modified regularized long wave
model (pmRLW). The non-integrable mRLW model has been discussed before, see
[8] and references therein. The analytical form of a 2-soliton solution of
(5.1) and its ${\cal P}$ invariant representation was obtained in
(4.13)-(4.14).

So, taking into account that the 2-soliton (4.13) satisfies the parity
symmetry (4.14), and the symmetry of the auxiliary field $w$ in (4.15), one
can show analytically the vanishing of the anomalies belonging to the various
quasi-conservation laws. Thus, it is an analytical proof of the quasi-
integrability of the pmRLW theory (5.1), i.e. the deformed pKdV for
$\epsilon_{1}=\epsilon_{2}=1$. Notice that similar argument has been used in
order to present this proof for the KdV-like quasi-conservation laws in [8].
Here, we are generalizing this proof for the new quasi-conservation laws
presented above. Then, it is worth to mention that this adds a new strong
result on the analytical proof, not only numerical, of the quasi-integrability
of a (non-integrable) theory, discussed in [12, 8].

##  6 Numerical treatment of the anomalies

Here we numerically simulate the vanishing of some representative anomalies of
the deformed pKdV model, which have been discussed above through the symmetry
arguments to advance the, so far, only plausible explanation for the anomalous
charges in the soliton collision of the deformations of the integrable models,
such as the SG [6], NLS [9, 10] and KdV [8] models, respectively. The above
properties will be qualitatively reproduced in our numerical simulations of
the 2-kink interaction of the deformed pKdV model, for a variety of soliton
configurations and a wide range of values of the set of deformation parameters
$\\{\epsilon_{1},\epsilon_{2}\\}$.

We have presented in (4.39) a general one kink-type solution of the model for
any set of the deformation parameters $\\{\epsilon_{1},\epsilon_{2}\\}$, and
it exhibits a general dispersion relation (4.38). We will use two of this
exact one soliton solution located far apart as the initial condition in order
to simulate two-soliton collision for the deformed model. Our initial
condition must be a field configuration which reduces the emission of
radiation in order to accurately simulate the various quantities, such as the
charges and anomalies.

Since an analytical solution for 2-soliton for any values of $\epsilon_{1}$
and $\epsilon_{2}$ of (2.1) is not known; as mentioned above, we will take as
an initial condition the superposition of two solitons of the general type
(4.39) located some distance apart. So, let us consider a linear superposition
of two expressions of type (4.37)

$\displaystyle q_{2s}(x,t)$ $\displaystyle=$ $\displaystyle
q_{1}\Big{\\{}\log{\cosh{[\frac{\zeta_{1}}{2a_{1}}]}}+b_{1}\zeta_{1}+c_{1}\Big{\\}}+q_{2}\Big{\\{}\log{\cosh{[\frac{\zeta_{2}}{2a_{2}}]}}+b_{2}\zeta_{2}+c_{2}\Big{\\}},$
(6.1) $\displaystyle w_{j}$ $\displaystyle=$
$\displaystyle\frac{a_{j}^{2}k_{j}+(1-\epsilon_{1})k_{j}^{3}}{a_{j}^{2}-\epsilon_{1}k_{j}^{2}};\,\,\,\,\,q_{j}=\frac{3a_{j}^{2}}{[a_{j}^{2}+(1-\epsilon_{1})k_{j}^{2}](2+\epsilon_{2})};\,\,\,\zeta_{j}=k_{j}x-w_{j}t+\delta_{j},\,\,\,\,j=1,2;$
(6.2)

such that $a_{j},\,b_{j}$ and $c_{j},\,j=1,2,$ are arbitrary real parameters.
The expression of each component of the field $q_{2s}(x,t)$ is presented in
(4.37). We closely follow the numerical techniques presented in the Appendix
of [8].

We plot the function $q_{2s}(x,t_{i})$ (upper green line in top left panel of
Fig.1), and the kink $v_{2s}(x,t_{i})$ (bottom green line in the top left
panel of Fig.1) for initial time $t_{i}=0$. The field $v_{2s}(x,t_{i})$
(green) represents the initial configuration of our numerical solution of the
model (2.1) for two-soliton collision. Notice that the auxiliary field
$q_{2s}(x,t_{i})$ (green) undergoes significant changes only around the kink
regions from an approximately linear behavior in regions far away from the
kinks; so, realizing the continuous linear background solution in (2.7) as the
vacuum of the theory. Whereas, the field $v_{2s}(x,t_{i})$ behaves as a two-
kink-like function and approaches constant values asymptotically for
$x\rightarrow\pm\infty$. These patterns and properties will be useful when
imposing the relevant initial and boundary conditions of our numerical
simulations. So, in the bottom lines of the top left panel of Fig. 1 we plot
the field configurations of the 2-soliton of the pKdV model. One has the kink
$v_{2sol}$ (bottom lines) and the field $q_{2sol}$ (upper lines) for three
successive times, i.e. the initial (green), collision (orange) and final
(magenta) times, respectively, for $\epsilon_{1}=1.2,\epsilon_{2}=0.9$..

We examine numerically the quasi-conservation law (3.12) and the relevant
anomaly density $\alpha^{(5)}_{4}(x,t)$ in (3.25). The Fig.1 (top right) shows
the anomaly density $\alpha^{(5)}_{4}(x,t)$ for three successive times,
initial (green), collision (orange) and final (magenta), respectively. The
bottom figures show the plots $(\int_{-L}^{+L}dx\,\alpha^{(5)}_{4}(x,t))$ and
$(\int_{t=t_{i}}^{t}dt^{\prime}\,\int_{-L}^{+L}dx^{\prime}\,\alpha^{(5)}_{4}(x^{\prime},t^{\prime})$.
Notice the vanishing of the $(x,t)-$integrated anomaly at the order of
$\approx 10^{-3}$. So, one can regard $q^{(5)}_{4,X}$ in (3.19) as an
asymptotically conserved charge within numerical accuracy, for the 2-kink
collision.

Figure 1: (color online) Numerically simulated two-kink collision, $v$ field
of (2.1) (bottom of top left figures), and the field $q$ of (2.6) (upper of
top left figures). The top Figs. show three successive times, initial (green),
collision (orange) and final (magenta), respectively, for
$\epsilon_{2}=0.9,\epsilon_{1}=1.2$. The top right Fig. shows the anomaly
density $\alpha^{(5)}_{4}(x,t)$ in (3.25) for three successive times, initial,
collision and final, respectively. The bottom left Fig. shows the
$x-$integrated anomaly and the bottom right shows the $(x,t)-$integrated
anomaly. Notice that the right bottom figure vanishes within numerically
accuracy.

Next, we examine numerically the non-local quasi-conservation law (3.33) and
the relevant anomaly density $\hat{\alpha}$ in (3.35). The plot in Fig.2 (top
Fig.) shows the anomaly density $\hat{\alpha}$ for three successive times,
initial (green), collision (orange) and final (magenta), respectively. The
middle and bottom figures show the plots
$(\int_{-L}^{+L}dx\,\hat{\alpha}(x,t))$ and
$(\int_{t=t_{i}}^{t}dt^{\prime}\,\int_{-L}^{+L}dx^{\prime}\,\hat{\alpha}(x^{\prime},t^{\prime})$.
Notice the vanishing of the $(x,t)-$integrated anomaly at the order of
$\approx 10^{-9}$. So, one can regard $\hat{q}$ in (3.34) as a non-local
charge which is asymptotically conserved within numerical accuracy, for the
2-kink collision.

Figure 2: (color online) Numerically simulated two-kink collision for
$\epsilon_{2}=0.9,\epsilon_{1}=1.2$. The top Fig. shows the anomaly density
$\hat{\alpha}$ in (3.35) for three successive times, initial (green),
collision (orange) and final (magenta), respectively. The middle Fig. shows
the $x-$integrated anomaly and the bottom shows the $(x,t)-$integrated
anomaly. Notice that this figure vanishes within numerically accuracy $\approx
10^{-9}$.

##  7 Discussions and some conclusions

We have examined the quasi-integrability properties of a particular
deformation of the potential KdV model (pKdV) (2.1) using a direct approach in
order to construct the quasi-conservation laws, and through a numerical
simulation of 2-kink collisions. For the undeformed pKdV
($\epsilon_{1}=\epsilon_{2}=0$) we have constructed the parity ${\cal P}$
symmetric N-soliton solutions, and shown that the $(x,t)-$integrated anomalies
of the usual pKdV model vanish. As representative quasi-conservation laws of
the pKdV model we have examined the quasi-conserved charges $q^{(5)}_{1}$
(3.16) and $q^{(5)}_{3}$ (3.18) for $X=0$, and shown the vanishing of the
common anomaly $-\alpha v_{x}^{2}v_{4x}$ using its odd parity property under
${\cal P}$ symmetry. In addition, we have discussed an anomaly cancellation
mechanism in order to get the exact conserved charge
$q^{(5),\gamma}_{X=0}=q^{(5)}_{1}-\gamma\,q^{(5)}_{3}$ in (3.30) for the
parameter value $\gamma=1$.

Some charges, such as ${\cal Q}_{1}$ in (3.37), when evaluated for the
composite field $q\sim`kink^{\prime}+q_{clb}$ must be renormalized by
subtructing the infinite contribution of the continuous linear background
$q_{clb}$ in (2.7), i.e. the kink charge ${\cal Q}_{1\,kink}$ must be
renormalized. This is analog to the renormalization procedure in the
defocusing NLS model in which the normalization and momentum charges
associated to the dark soliton itself have been renormalized by subtructing a
continuous wave background [13].

Taking into account that the 2-soliton (4.13) satisfies the parity symmetry
(4.14) it has been shown the analytical vanishing of the anomalies. Therefore,
it provides an analytical proof of the quasi-integrability of the pmRLW theory
(5.1), i.e. a sub-model of deformed pKdV in the particular case
$\epsilon_{1}=\epsilon_{2}=1$. So, this adds a new strong result on the
analytical proof, not only numerical, of the quasi-integrability of a (non-
integrable) theory, discussed in [12, 8] for deformed KdV models.

We have considered numerically the behavior of the quasi-conservation laws
(3.12) and (3.33) with the relevant anomaly densities $\alpha^{(5)}_{4}(x,t)$
in (3.25) and $\hat{\alpha}(x,t)$ in (3.35), respectively. The Fig.1 (top
right) and Fig. 2 (top) show the anomaly densities for three successive times,
initial (green), collision (orange) and final (magenta), respectively. The
figures show the $x-$integrated and the $(x,t)-$integrated anomalies. The
$(x,t)-$integrated anomalies in the both cases vanish within numerical
accuracy. So, one can regard the charges $q^{(5)}_{4,X}$ in (3.19) and
$\hat{q}$ in (3.34) as asymptotically conserved charges within numerical
accuracy, for the 2-kink collision.

Some qualitative aspects of the Liouville’s theorem remain true for continuous
integrable systems admitting a Lax pair representation [1]. So, one must have
an infinite number of conservation laws whose conserved charges are in
involution [1, 2]. In this context, the presence of the novel towers of
asymptotically conserved charges as above, even in the standard pKdV model,
are restricted to special field configurations satisfying the symmetry (2.8)
and (2.9). Therefore, one can not use these charges, even though they are
infinitely many, in order to match to the number of degrees of freedom of the
pKdV model. Of course, the true conserved charges hold for general field
configurations, i.e. being solitonic or not. We believe, that an anomaly
cancellation mechanism is required in order to get the set of true conserved
charges of the standard pKdV model. For example, in order to get an exact
conservation law in (3.28) for $X=0$ one must have $\gamma=1$, which is the
condition for the anomaly cancellation. In fact, subtructing the quasi-
conservation laws (3.9) and (3.11) (for $X=0$) one can get (3.28) to be an
exact conservation law provided that $X=0$ and $\gamma=1$, since their
anomalies $-\alpha v_{x}^{2}v_{4x}$ cancel to each other for $\gamma=1$. Of
course, the charge $q^{(5)\gamma=1}_{X=0}$ in (3.28) will be conserved for
general solutions of the pKdV.

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# Individual identity in songbirds: signal representations and metric learning
for locating the information in complex corvid calls

Dan Stowell1, Veronica Morfi1, Lisa F. Gill2

1Machine Listening Lab, Centre for Digital Music,
Queen Mary University of London, UK
2Max Planck Institute for Ornithology, Seewiesen, Germany
{dan.stowell<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Bird calls range from simple tones to rich dynamic multi-harmonic structures.
The more complex calls are very poorly understood at present, such as those of
the scientifically important corvid family (jackdaws, crows, ravens, etc.).
Individual birds can recognise familiar individuals from calls, but where in
the signal is this identity encoded? We studied the question by applying a
combination of feature representations to a dataset of jackdaw calls,
including linear predictive coding (LPC) and adaptive discrete Fourier
transform (aDFT). We demonstrate through a classification paradigm that we can
strongly outperform a standard spectrogram representation for identifying
individuals, and we apply metric learning to determine which time-frequency
regions contribute most strongly to robust individual identification.
Computational methods can help to direct our search for understanding of these
complex biological signals.

Index Terms: bird, LPC, aDFT, metric learning, corvid, animal communication

##  1 Introduction

Bird vocalisations are highly complex. They are often analysed as sinusoidal
or harmonic sounds, or as spectrotemporal “patches” [1], but in general this
can obscure their rich structure: unlike humans, songbirds have two sets of
vocal folds, which they can use simultaneously; and they also have muscles
specialised for rapid pitch modulation [2]. Songbird species make use of these
abilities in different ways, not all of which are fully understood. These
complexities pose problems for signal processing paradigms borrowed from the
study of speech or music. Yet bird vocalisations are an important area of
scientific study: from behavioural studies we know that they can contain
information about species, about individual identity, and more, and so we seek
signal processing methods that can represent that information in a form
suitable for analysis. Corvids, a family of songbirds that includes ravens,
crows, jays and other species, have been the focus of much research, due to
their social complexity and remarkable cognitive skills. But unlike in other
songbirds, their vocalisations have not been as extensively studied, maybe
because they mainly produce short, non-tonal vocalisations (calls) that are
often structurally complex and may involve the two-voice phenomenon. However,
their vocal complexity, in combination with high levels of sociality and
cognition has made corvids a suitable target for studying vocal recognition
[3]. Jackdaws (C. monedula) are highly vocal, group-living corvids that breed
in colonies and form strong, lifelong pair bonds. They are highly vocal and
use a variety of vocalisations to maintain contact and communicate with their
conspecifics [4]. Recently, it has been suggested that pair members are able
to recognise each other’s contact calls [5]. But are we able to discriminate
individuals by analysing these challenging vocal signals? And which parts of
the signals carry the individual information? To explore these questions, we
analysed a dataset of calls coming from 20 individually recorded jackdaws,
using alternative signal representations together with automatic
classification and metric learning.

Figure 1: Spectrogram of a single jackdaw call (left) and of its LPC residual
(right).

##  2 Method

###  2.1 Data collection

The dataset came from 20 adult, hand-raised, group-housed jackdaws (in
accordance with the 2010/63/EU European directives for the protection of
animals used for scientific purposes). Individual vocalisations were recorded
using a microphone (TC20, Earthworks, USA) and solid-state recorder (PMD661,
Marantz, Japan, at a sampling rate of 48000 Hz) while the birds were
temporarily held in social isolation (for max. 32 minutes) in a sound chamber
(2.0 x 2.0 x 1.3 m, fitted with white acoustic foam), as part of a different
study. Contact calls were classified and clean recordings selected by visual
and acoustic assessment. Here, we analysed a total of 1156 calls (3 to 93 per
individual, median = 60), with a median duration of 0.272 seconds (standard
deviation: 0.047, min: 0.160, max: 0.510 seconds). An example call is shown in
Figure 1 (left).

###  2.2 Signal processing

Our aim was to produce spectrogram representations that allowed individual
identity information to surface most readily. For standard spectrograms we
used a frame size of 1024 with Hann windowing and 75% overlap.

As a potential improvement on this baseline we evaluated linear prediction
(LPC) analysis, widely used in speech processing as a step which aims to
separate the effects of the glottal source (for songbirds, the syringeal
sources) from the vocal tract filter [6]. To each call we applied LPC analysis
of order 10 to the whole audio clip. Songbirds might encode individuality in
fixed/gestural aspects of the syrinx oscillation and/or the vocal tract
setting. We therefore used the LPC residual (which is an approximation to the
syringeal source signal) in the same way as we would use the original raw
audio, for spectrogram processing (see Figure 1 right), and we also used the
LPC spectrum (an approximation to the vocal tract filter) separately as an
alternative spectral representation of each call.

Separately we explored the use of the adaptive Discrete Fourier Transform
(aDFT) proposed in [7]. This was originally introduced as an alternative,
adaptive method to compute a spectrogram and the sinusoidal parameters used by
the “adaptive harmonic model” (aHM) in human speech analysis and synthesis.
The aDFT is similar to the Discrete Fourier Transform (DFT) but uses a
frequency basis that can completely follow the variations of the fundamental
frequency $F_{0}$ throughout a recording, in contrast to the constant
frequency basis of DFT. This time-varying $F_{0}$ is estimated as part of the
algorithm, which thus offers the potential of a more accurate time-frequency
representation. The harmonics can be easily traced and are typically more
prominent even in mid/high frequencies, compared to those in the DFT
spectrogram.

In order to apply the aDFT to our dataset only one modification of the
original algorithm was needed. The $F_{0}$ variations for human speech are
considered to be between 40 and 700 Hz, but songbirds have an even higher and
wider range. Hence, an $F_{0}$ range between 80 and 2000 Hz was used for the
estimation of the $F_{0}$ variations throughout each recording. The values of
the $F_{0}$ curve that was used as the frequency basis for aDFT were then
divided by 2, because of the potential importance of sub- and inter-harmonics.
Furthermore, we tested this aDFT with and without the adaptive iterative
refinement algorithm, also proposed in [7], which iteratively refines the
$F_{0}$ estimate used to produce the aDFT spectrogram. We therefore tested
‘unrefined’ and ‘refined’ versions.

The aDFT is designed to improve the characterisation of frequency-modulated
harmonic sounds, and so is a candidate for analysis of songbird vocalisations
(although it does not directly model two-voice phenomena). The aDFT produces
spectrograms with a varying rather than regular temporal frame rate. In order
to perform pairwise comparisons between spectrograms, and to maximise
comparability against the standard spectrogram representation, we resampled
the aDFT spectrograms onto a regular grid (by nearest-neighbour interpolation)
at the same frequency and time resolution as the standard spectrogram (Figure
2).

Figure 2: aDFT spectrogram of a single call: using unrefined (left) and
refined (right) $F_{0}$ estimate.

###  2.3 Classification tests

We used a classification paradigm to evaluate the various signal
representations. Our aim was not to achieve the best classification possible,
but to probe which representations gave the clearest indications of individual
identity. For this reason, we did not use an arbitrarily powerful classifier,
but the well-known and simple $k$-nearest-neighbours (kNN) classifier (with
$k=3$). Data that work well with a kNN classifier will have their classes in
compact, well-separated clusters. However, the kNN classifier makes few
assumptions about class distributions beyond that: it is tolerant of
multimodal classes, and of classes having different variances, as long as they
remain compact and well-separated. kNN also naturally encompasses multi-class
data and not just binary-labelled data. Since we were working with multi-class
data in a high-dimensional space (spectrogram pixels) about which relatively
little is known (e.g. class multimodality), kNN was preferred over other
simple discriminators such as linear discriminant analysis.

The classifier was applied to pairwise distances between spectrograms,
evaluated using four different distance metrics: Euclidean and Manhattan, each
applied to magnitudes and to log-magnitudes. The sound files were all aligned
by their onset. In order to reduce any problems due to relative misalignment
when measuring pairwise distances, we allowed a relative movement of $\pm 20$
ms and used the alignment that gave the minimum distance.

We therefore obtained classification results using data preprocessed in
various ways: with/without LPC analysis, with a standard spectrogram or aDFT
(and aDFT could be with/without refinement of its $F_{0}$ estimate),
with/without log-transformed magnitudes, with Euclidean or Manhattan distance.
We evaluated the contributions of these factors using a mixed-effects
statistical model (GLMM, with call recording ID as the random factor), to
determine whether each made a significant difference to performance, and to
check for interactions between the factors. In order to help understand the
effects of the various signal-processing choices, we also visualised the
pairwise distances using t-SNE [8].

The above investigations help to ensure that we choose signal processing
methods that elucidate individual-identifying information if it is present,
and through LPC help us to consider one facet of the question of ‘where’ the
information lies. It is natural to ask to what extent information lies in the
onset or the decay of the call, in the fundamental or the harmonics—the latter
question being particularly pertinent when a call can simultaneously contain
energy from the two syringeal sources. Note that the kNN classifier depends
fundamentally on the distance metric used, and in particular that the relative
weighting of features can make a dramatic difference to results. In the first
test we have described, we varied the metric but did not vary the feature
weightings, in effect telling the classifier to pay equal attention to every
pixel. We therefore complemented this approach by using the large margin
nearest neighbours (LMNN) framework (a type of metric learning) to derive an
analysis of the relative importance of each spectrogram pixel. The LMNN
process learns a linear transformation of the data space in such a way as to
maximise the separation of classes, specifically to optimise kNN performance
[9]. It can therefore improve on the basic kNN classification results; however
our focus was not on the improved classification score but on the linear
projection that was learnt. After LMNN was trained (using the Python
metric_learn package) we mapped the projection matrix back onto the
spectrogram pixels, giving an overall importance weight for each pixel. (Note
that for implementation reasons we could not include the $\pm 20$ ms variable-
alignment in the LMNN test.) To visualise these results, we applied the
ranking transformation to the importance weights (to normalise their dynamic
range) and then plotted them in the same format as spectrograms.

##  3 Results

\-

Figure 3: t-SNE distance plots for jackdaw calls under two different signal
representations, in this case using Manhattan distance. Left: raw audio;
right: LPC residual. Each marker type/colour represents a different
individual. The axes have no direct interpretation. Figure 4: kNN
classification results. Chance level is 8.0%. For each signal representation,
results are shown for each of the four different distance measures.

Classification results show clearly that a dramatic improvement over a
standard spectrogram representation is possible (Figure 4). Our GLMM analysis
found a significant effect of all our signal-processing interventions, but
also significant interactions between all of them (as can be seen in the
rather bimodal results of Figure 4), so here we will not focus on the GLMM
results in detail. In general the Manhattan metric gave best results. When
using Manhattan distance, we found that the LPC residual and/or aDFT led to a
strong improvement from around 74% to 90% in individual identification; yet
the joint application of LPC and aDFT did not strongly improve results beyond
that. The aDFT spectrograms showed strong performance with either Manhattan or
Euclidean distance. The LPC filter estimate, however, did not lead to strong
classification, performing noticeably worse than the standard spectrogram.

The t-SNE plots give visual indications of the characteristics of the high-
dimensional spaces produced by the variant analyses. Figure 3 shows the
strongest-performing metric (Manhattan distance) for the raw audio and the LPC
residual; the LPC residual can be seen to enhance the per-individual
clustering structure. (Plots for aDFT were qualitatively similar to that for
the LPC residual.) Some of the classes appear multimodal, which may for
example reflect an individual’s use of differing call types.

Figure 5: Feature importance maps derived from LMNN, for raw audio (left) or
LPC residual (right). Darker pixels are higher-ranked in LMNN weighting.

Feature importance weightings derived from LMNN help to understand the
relative importance of specific time-frequency regions, and they also offer
clues as to what sort of benefit the LPC preprocessing gives (Figure 5). For
both raw audio and LPC residual, the strongest-weighted pixels are
concentrated soon after call onset, and place heavy emphasis on regions
containing the fundamental (which in many cases includes a downward sweep from
approx. 1 kHz) and its first one or two harmonics – more broadly speaking, the
region 1–3 kHz. Features which attain more importance after LPC transformation
are the higher frequencies near the onset. LPC analysis inherently produces a
residual with a whiter spectrum, and so for these sounds amplifies the upper
harmonics (Figure 1). However this does not inherently lead the LMNN analysis
to apply more weight to them; if the higher harmonics did not contain useful
information we would expect LMNN to suppress them.

##  4 Discussion

The results here do not directly tell us about the production or perception
processes involved in jackdaw contact calls, but they do give us information
which helps to guide investigations into those processes, and also which helps
us to design systems to extract information automatically from such
vocalisations. We have demonstrated that signal processing interventions can
dramatically improve the automatic identification of individuals with a low-
complexity classification algorithm, which tells us that they can transform
the signal into a format in which individuals’ calls have more stable and
repeatable character.

Our feature importance analysis suggests that the identifying information is
concentrated soon after onset, and spreads across the fundamental(s) and
harmonics. However, we note that for this particular analysis we could not
vary the relative alignment of the signals, meaning that discriminative
information in the tail-end of calls may not be apparent since calls are
variable in length.

We found that both LPC and aDFT lead to representations that facilitate
classification. These are two rather different interventions, yet the
improvements are not strongly additive: results when using both are only a
little stronger than when using either. LPC is a well-known technique, and
relatively efficient to apply, whereas aDFT and related representations are
not very well studied, and take much more computation than a standard
spectrogram. The cost-benefit ratio therefore speaks in favour of the LPC
residual for the moment, though we have shown that advanced adaptive
spectrograms are worth exploring for their surfacing of information present in
the signal.

The LPC residual, rather than the LPC filter, showed the strongest connection
with individual identity in our tests. Linear prediction is often used for
source-filter analysis, with the residual interpreted as the glottal or
syringeal source signal. However, the LPC residual is also likely to normalise
away any differences in the channel between bird and microphone, and so one
should be cautious before interpreting the LPC filter component as purely
representing “the vocal tract”. It also means that the improved recognition
using the LPC residual could be interpreted as a normalisation rather than a
decomposition. However—contrary to that line of thought—in the present dataset
recordings are controlled and the channel effects are stable, although factors
such as the orientation of a bird’s head with respect to the microphone can
still affect the received signal. Nevertheless, our results suggest that the
signal component from the syringeal source contains sufficient information for
recognising individuals in this species, and that this information is not just
in the fundamental frequencies but also involves the overall harmonic
structure of the source signal—a suggestion which should be compared against
physiological and perceptual evidence in future.

##  5 Acknowledgements

This work was supported by EPSRC Early Career research fellowship
EP/L020505/1. LFG was funded by the Max Planck Society. We also thank Auguste
von Bayern for discussions and support during the experiments.

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