# Fluid Dynamical Description of Sine-Gordon Solitons

Nematollah Riazi<EMAIL_ADDRESS>Physics Department, Shahid Beheshti
University, Evin, Tehran 19839, Iran  Marzieh Peyravi  marziyeh.peyravi@stu-
mail.um.ac.ir Department of Physics, School of Sciences, Ferdowsi University
of Mashhad, Mashhad 91775-1436, Iran

###### Abstract

We present a fluid dynamical description of a relativistic scalar field in
$1+1$ dimensions and apply the general results to the special case of Sine-
Gordon solitons. The results which include the local quantities pressure,
density and fluid velocity are compared to the standard quantities attributed
to the solitons.

Soliton, Relativistic Fluid Dynamics, Sine-Gordon System

###### pacs:

05.45.Yv, 03.75.Lm, 43.25.Rq, 47.35.-i

##  I Introduction

The energy-momentum tensor is a second rank, symmetric and conserved tensor
which is of central importance in fundamental relativistic theories like
electromagnetism and general theory of relativity (see e.g. a ; b ). For
relativistic fields, it accommodates important local quantities like density,
pressure, viscosity, fluid velocity and flux. Using the celebrated Noether’s
theorem, the energy-momentum tensor can be derived from the space and time
translation symmetries of the corresponding action or Lagrangian c ; d . In
general relativity, the energy-momentum tensor which is the source of space-
time curvature is calculated by varying the matter action ($\cal{A}$) with
respect to the metric Wein :

$T^{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{A}_{m}}{\delta
g_{\mu\nu}},$ (1)

where $T^{\mu\nu}$ is the energy-momentum tensor, $g_{\mu\nu}$111We adopt the
metric signature $(-,+,+,+)$ is the metric tensor, $\mathcal{L}_{m}$ is the
matter Lagrangian, and $g$ is the determinant of the metric tensor. In flat
space-time, the energy-momentum conservation takes the simple form RD :

$\partial_{\mu}T^{\mu\nu}=0.$ (2)

where sum over repeated indices is implied. For $\nu=0$, we can expand Eq. (2)
as RD :

$\partial_{0}T^{00}+\partial_{i}T^{i0}=0.$ (3)

which is the continuity equation $\frac{\partial\rho}{\partial
t}+\nabla\cdot\vec{j}=0$, with $\rho\equiv T^{00}$ and $j_{i}\equiv
T^{i0}=T^{0i}$. Integrating Eq. (3) over the whole Euclidean space, doing an
integration by part and assuming that $\vec{j}$ vanishes over the boundary, we
obtain RD :

$\frac{d}{dt}\int_{V}T^{00}d^{3}x+\oint\vec{j}\cdot d\vec{s}=0,$ (4)

or $\frac{dE}{dt}=0$, where $E=\int T^{00}d^{3}x$ is the total conserved
energy. It is interesting that for a single, moving soliton (kink), Eq. (3)
leads to the famous Einstein equation RD :

$E^{2}=Mc^{2}+p^{2}c^{2},$ (5)

in which $M$ and $p$ are the total rest energy (mass) and momentum of the
soliton, respectively e .

On the other hand, the relativistic energy-momentum tensor of an imperfect,
relativistic fluid is given by 6 ; RD :

$T^{\mu\nu}=\rho u^{\mu}u^{\nu}+q^{(\mu}u^{\nu)}+S^{\mu\nu},$ (6)

where $\rho$ is the co-moving energy density, $u^{\mu}$ is the fluid
4-velocity, and $q^{\mu}$ is the space-like heat vector orthogonal to
$u^{\mu}$ (i.e. $q^{\mu}u_{\mu}=0$). For sub-luminal fluid velocity
$u^{\alpha}u_{\alpha}=-1$, while luminal and super-luminal motion correspond
to $u^{\alpha}u_{\alpha}=0$ and $u^{\alpha}u_{\alpha}=+1$, respectively.
$S^{\mu\nu}$ is the stress tensor with eigenvectors defining the directions of
the principal stresses.

In the Eckhart formulation 7 ; 8 , the energy-momentum tensor is written in
terms of viscosity as:

$T^{\mu\nu}=\rho
u^{\mu}u^{\nu}+q^{(\mu}u^{\nu)}+\left(p-J\Theta\right)h^{\mu\nu}-2\eta\sigma^{\mu\nu}.$
(7)

where the positive quantities $J$ and $\eta$ are the bulk and shear
viscosities, respectively, $\Theta\equiv\nabla_{\mu}u^{\mu}$ is the expansion,
$h^{\mu\nu}\equiv\eta^{\mu\nu}+u^{\mu}u^{\nu}$, and $\sigma^{\mu\nu}$ is the
symmetric, traceless, shear tensor:

$\sigma^{\mu\nu}\equiv\frac{1}{2}\left(\nabla_{\alpha}u^{\mu}\eta^{\alpha\nu}+\nabla_{\alpha}u^{\nu}\eta^{\alpha\mu}\right)-\frac{1}{3}\Theta\eta^{\mu\nu}.$
(8)

For a perfect fluid in which the velocity gradients and heat flux can be
ignored, $\Theta=0$, $\sigma^{\mu\nu}=0$, $q^{\mu}=0$ and we have:

$T^{\mu\nu}=\left(\rho+p\right)u^{\mu}u^{\nu}+p\eta^{\mu\nu}.$ (9)

The canonical theory for perfect fluids, in Eulerian and Lagrangian
formulations in relation to a description of extended structures in higher
dimensions is reviewed in 11 .

On the other hand, if the system under consideration is composed of a
statistically large number of relativistic particles of the rest mass $m$ and
4-momenta $p^{\mu}$, the energy-momentum tensor can be calculated via b ; 8a :

$T^{\mu\nu}=\int\frac{f(\vec{x},\vec{p})p^{\mu}p^{\nu}d^{3}p}{p^{0}},$ (10)

in which $f$ is the number of particles per unit volume of phase space
($\vec{x},\vec{p}$), and $p^{\mu}p_{\mu}=-m^{2}$ with $m$ being the rest mass
of constituent particles ($\hbar=c=1$ is assumed).

It is well-known that solitons have a dual identity at the same time,
corresponding to particle-like and wave-like properties. By considering
solitons as particles in microscopic point of view , statistical mechanics
will be an effective and powerful method for studying their interactions.
However, their macroscopic behaviour can be represented in terms of the
hydrodynamics of soliton gas which can be interpreted as an integrable
turbulence Dh ; TB ; zak ; EL ; BD . Generalised hydrodynamics of soliton
gases has been introduced and studied for KdV solitons recently Dh ; TB ; EL ;
BD .

In this context, our main question, to be answered in this paper is the
following: what are the relationships between the field quantities like $\rho$
and $w$ (soliton velocity) and those appearing the relativistic fluid
description ($\rho$, $p$, $u^{\alpha}$), if the soliton is to be described by
a perfect fluid. Besides some expected results, there appear new interesting
relations, particularly in relation to the concept of soliton velocity, and
the possibility of super-lunimal motion.

In the next section, we apply the above-mentioned concepts and definitions to
the case of a relativistic scalar field in $1+1$ dimension, and will then
concentrate on the special case of Sine-Gordon solitons.

##  II  Scalar Fields in $1+1$ Dimensions

The action of a real scalar field in $1+1$ dimensions is given by d :

$\mathcal{A}=\int
d^{2}x\left(-\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi-V(\varphi)\right).$
(11)

where $V(\varphi)$ is the self-interaction potential. Variation of Eq. (11)
with respect to $\varphi$ leads to the equation of motion d :

$\square\varphi=\partial^{\mu}\partial_{\mu}\varphi=\frac{\partial
V}{\partial\varphi},$ (12)

while the energy-momentum tensor is given by RD :

$T^{\mu\nu}=\partial^{\mu}\varphi\partial^{\nu}\varphi+\eta^{\mu\nu}\mathcal{L},$
(13)

$\mathcal{L}$ being the Lagrangian density. We will equate this to the energy-
momentum tensor of a perfect fluid Eq. (9), as far as all the results for
$\rho$, $p$ and $u^{\alpha}$ are mutually consistent. In the case where
inconsistencies arise, we may conclude that the scalar field cannot be
described by a perfect fluid and we should resort to a more complicated fluid
model. For the time being, we keep the three possibilities
$u^{\alpha}u_{\alpha}=\varepsilon$, with $\varepsilon=-1,0,+1$ which
correspond to time-like, light-like and space-like or super-luminal motion of
the fluid, respectively.

Let us start with the simple case of a massless scalar field with
$V(\varphi)=0$. The scalar field then obeys a wave equation and any function
$\varphi(x\pm t)$ will be a solution ($c=1$ is assumed). Assuming the wave is
moving in the $+x$ direction, we can use the light-cone variable $\xi\equiv
x-t$. We have:

$T^{00}=(\rho+p)(u^{0})^{2}-p=\left(\frac{\partial\varphi}{\partial
t}\right)^{2}-\mathcal{L}.$ (14)

where $\mathcal{L}=\frac{1}{2}\dot{\varphi}^{2}-\frac{1}{2}\varphi^{\prime
2}$, dot and prime being derivatives with respect to $t$ and $x$,
respectively. We have:

$\displaystyle\hskip 28.45274pt\dot{\varphi}$ $\displaystyle=$
$\displaystyle\frac{\partial\varphi}{\partial t}=\frac{\partial\xi}{\partial
t}\frac{\partial\varphi}{\partial\xi}=-\varphi_{\xi},$ $\displaystyle\hskip
28.45274pt\varphi^{\prime}$ $\displaystyle=$
$\displaystyle\frac{\partial\varphi}{\partial x}=\frac{\partial\xi}{\partial
x}\frac{\partial\varphi}{\partial\xi}=\varphi_{\xi}.$ (15)

The wave equation $\Box\varphi=0$ is trivially satisfied and we have
$\mathcal{L}=0$ identically. For the energy density we obtain:

$T^{00}=\left(\frac{\partial\varphi}{\partial
t}\right)^{2}=\frac{1}{2}\left(\frac{\partial\varphi}{\partial
t}\right)^{2}+\frac{1}{2}\left(\frac{\partial\varphi}{\partial
t}\right)^{2}=\frac{1}{2}\dot{\varphi}^{2}+\frac{1}{2}\varphi^{\prime
2}=\varphi_{\xi}^{2},$ (16)

the last step being made using $\varphi^{\prime}=-\dot{\varphi}$ from Eq.
(II). Let $u^{\alpha}\equiv(\gamma u,\gamma v)$ with
$u^{\alpha}u_{\alpha}=\varepsilon=-1,0,+1$. We thus obtain:

$\displaystyle\hskip 28.45274ptT^{00}$ $\displaystyle=$
$\displaystyle\varphi_{\xi}^{2}=\left(\rho+p\right)\gamma^{2}u^{2}-p,$ (17)
$\displaystyle\hskip 28.45274ptT^{11}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial\varphi}{\partial
x}\right)^{2}=\varphi_{\xi}^{2}=\left(\rho+p\right)\gamma^{2}v^{2}+p,$ (18)
$\displaystyle\hskip 28.45274ptT^{01}$ $\displaystyle=$
$\displaystyle\partial^{0}\varphi\partial^{1}\varphi=-\frac{\partial\varphi}{\partial
t}\frac{\partial\varphi}{\partial
x}=\varphi_{\xi}^{2}=\left(\rho+p\right)\gamma^{2}uv,$ (19)

or

$T^{\mu\nu}=\left(\begin{array}[]{cc}1&1\\\ 1&1\\\
\end{array}\right)\varphi_{\xi}^{2}.$ (20)

Note that there is no spatial shear in $1+1$ dimensions and $T^{\mu}_{\mu}=0$
with $V=0$ which is also implied by Eq. (20). We have:

$T^{\mu}_{\mu}=\left(\rho+p\right)u^{\mu}u_{\mu}+2p=\left(\rho+p\right)\varepsilon+2p=\begin{cases}2p&\textrm{($u^{\mu}u_{\mu}=0$)}\\\
\rho+3p&\textrm{($u^{\mu}u_{\mu}=+1$)}\\\
-\rho+p&\textrm{($u^{\mu}u_{\mu}=-1$)},\\\ \end{cases}$ (21)

while Eq. (13) leads to:

$T^{\mu}_{\mu}=\partial^{\mu}\varphi\partial_{\mu}\varphi+2\left(-\frac{1}{2}\partial^{\mu}\varphi\partial_{\mu}\varphi\right)=0,$
(22)

We therefore obtain:

$p=-\frac{\epsilon}{\epsilon+2}\rho.$ (23)

Equations (17)-(19) constitute three equation for four unknowns $\rho$, $p$,
$u$, $v$ once $\varphi(x-t)$ is known as a uniformly moving solution of the
wave equation. The remaining equation comes from one of the conditions
$\varepsilon\equiv u^{\mu}u_{\mu}=\gamma^{2}(-u^{2}+v^{2})=0,\pm 1$.

For the general case of a scalar field with potential ($V(\varphi)\neq 0$), we
rewrite our fundamental equations in the following form:

$\displaystyle\hskip 28.45274pt\pi\gamma^{2}u^{2}-p$ $\displaystyle=$
$\displaystyle K+V,$ (24) $\displaystyle\hskip 28.45274pt\pi\gamma^{2}v^{2}+p$
$\displaystyle=$ $\displaystyle K-V,$ (25) $\displaystyle\hskip
28.45274pt\pi\gamma^{2}uv$ $\displaystyle=$ $\displaystyle-\psi,$ (26)
$\displaystyle\hskip 28.45274ptu^{\mu}u_{\mu}$ $\displaystyle=$
$\displaystyle\gamma^{2}(v^{2}-u^{2})=\varepsilon,$ (27)

where $K\equiv\frac{1}{2}\varphi^{\prime 2}+\frac{1}{2}\dot{\varphi}^{2}$,
$\pi\equiv p+\rho$ and $\varepsilon=0,\pm 1$. We also have $\mathcal{L}=-S-V$,
where $S\equiv\frac{1}{2}\varphi^{\prime 2}-\frac{1}{2}\dot{\varphi}^{2}$,
$\psi\equiv\dot{\varphi}\varphi^{\prime}$. From Eqs. (24), (25) and (27) we
obtain:

$\pi\left(2\gamma^{2}u^{2}+\varepsilon\right)=2K,$ (28)

and since $\pi^{2}\gamma^{4}u^{2}v^{2}=\psi^{2}$, we obtain

$\frac{\psi^{2}}{u^{2}\left(u^{2}+\varepsilon/\gamma^{2}\right)}\left(2u^{2}+\varepsilon/\gamma^{2}\right)^{2}=4K^{2},$
(29)

Eq. (24) minus Eq. (25) gives:

$-\pi\varepsilon-2p=2V,$ (30)

solving Eq. (29) for $u^{2}$ gives:

$\gamma^{2}u^{2}=-\frac{\epsilon}{2}\pm\frac{1}{\sqrt{1-\psi^{2}/K^{2}}}.$
(31)

Once $\gamma^{2}u^{2}(x,t)$ is obtained using Eq. (31) in terms of the
solution $\varphi(x,t)$, $\gamma^{2}\pi=\frac{2K}{2u^{2}+\varepsilon}$ or
$\pi=-\frac{\psi}{u\sqrt{u^{2}+\varepsilon}}$ is immediately calculated, from
which $p$ is obtained using $p=K-V-\pi\left(u^{2}+\varepsilon\right)$, $\rho$
from $\rho=\pi-p$ and $v^{2}=u^{2}-\varepsilon$. For the null case, equations
simplify considerably since $\varepsilon=0$ and $u^{2}=v^{2}$.

##  III Hydrodynamics of Sine-Gordon Solitons

We are now in a position to do the same calculations for the interesting case
of the well-known Sine-Gordon solitons. The Sine-Gordon system is the
prototype of a relativistic integrable system. The self-interaction potential
of this system in dimensionless units is given by c ; e ; 10 ; man ; Raj ;
Vach ; Ryder ; horita :

$V(\varphi)=1-\cos(\varphi),$ (32)

leading to the field equation:

$\Box\varphi=\sin(\varphi).$ (33)

Single soliton solutions of this equation which are called kinks (and anti-
kinks) are given by:

$\varphi(x,t)=\pm
4\arctan\Big{(}\exp\left(\gamma_{w}\left(x-wt\right)\right)\Big{)},$ (34)

where $w$ is the soliton velocity and
$\gamma_{w}\equiv\left(1-w^{2}\right)^{-1/2}$. Multi-soliton solutions of the
SG equation can be derived using various methods like inverse scattering 10
or Bäcklund transformations c . Besides the single soliton (kink) solution Eq.
(34), we concentrate only on the kink-antikink and breather solutions c ; e ;
kink-antikink 9 :

$\varphi(x,t)=4\arctan\left[\frac{\sinh\left(\gamma_{w}wt\right)}{w\cosh\left(\gamma_{w}x\right)}\right],$
(35)

and breather 9 :

$\varphi(x,t)=4\arctan\left[\frac{\frac{\sin(wt)}{\sqrt{1+w^{2}}}}{\frac{w\cosh(x)}{\sqrt{1+w^{2}}}}\right].$
(36)

As can be seen from the figures, the first solution (Fig. 1), describes an
anti-kink moving from left and a kink moving from right toward each other,
interacting around $x\sim 0$ and $t\sim 0$, and then moving apart with the
same velocities. The breather solution (Fig. 2) describes a bound state of
kink-antikink, swinging around their “center of mass” indefinitely.

###  III.1 Uniformly Moving Kink

For the uniformly moving kink solution Eq. (34), we obtain:

$\frac{K}{\psi}=-\frac{1+w^{2}}{2w},$ (37)

$\gamma^{2}u^{2}=1/(1-w^{2})=\gamma_{w}^{2}$ for the + solution and
$\epsilon=-1$. We then have $\gamma^{2}v^{2}=w^{2}\gamma_{w}^{2}$. It is seen
that we are led to the expected result $u=1$ and $v=w$, only if we adopt
$\epsilon=-1$ (time-like solution), and select the + solution. In what
follows, we keep using these choices for two-soliton solution (kink-antikink
and kink-kink).

###  III.2 Two-soliton Solutions

The exact solution Eq. (35) describes the scattering of a kink with an anti-
kink, with opposite topological charges222 Topological current is defined by
$J^{\mu}=\frac{1}{2\pi}\partial^{\mu}\varphi$, from which
$Q=\int_{-\infty}^{+\infty}J^{0}dx=\frac{1}{2\pi}\left[\phi(+\infty)-\phi(-\infty)\right].$
and interaction region around $t\approx 0$ and $x\approx 0$ (see Fig. 1).

Figure 1: The kink-antikink solution of the SG system. The two oppositely
charged solitons interact non-linearly around $t\approx 0$ and $x\approx 0$,
and then leave the interaction region without any change in shape.

Figure 2: The breather solution of the SG system during $-40\leq t\leq 40$ and
in the interval $-10<x<10$.

Both of the two-soliton solutions Eqs. (35) and (36) are in the following
form:

$\varphi(x,t)=4\arctan\left(\frac{f(t)}{g(x)}\right).$ (38)

which implies:

$\displaystyle\hskip 28.45274pt\dot{\varphi}$ $\displaystyle=$
$\displaystyle\frac{4g\dot{f}}{f^{2}+g^{2}},$ (39) $\displaystyle\hskip
28.45274pt\varphi^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{4fg^{\prime}}{f^{2}+g^{2}},$ (40) $\displaystyle\hskip
28.45274ptK$ $\displaystyle=$
$\displaystyle\frac{8\left(g^{2}\dot{f}^{2}+f^{2}g^{\prime
2}\right)}{\left(f^{2}+g^{2}\right)^{2}},$ (41) $\displaystyle\hskip
28.45274pt\psi$ $\displaystyle=$
$\displaystyle-\frac{16fg\dot{f}g^{\prime}}{(f^{2}+g^{2})^{2}},$ (42)
$\displaystyle\hskip 28.45274ptS$ $\displaystyle=$
$\displaystyle\frac{8\left(f^{2}g^{\prime
2}-g^{2}\dot{f}^{2}\right)}{\left(f^{2}+g^{2}\right)^{2}},$ (43)
$\displaystyle\hskip 28.45274pt\pi$ $\displaystyle=$ $\displaystyle
2S=\frac{16\left(f^{2}g^{\prime
2}-g^{2}\dot{f}^{2}\right)}{\left(f^{2}+g^{2}\right)^{2}},$ (44)
$\displaystyle\hskip 28.45274ptp$ $\displaystyle=$ $\displaystyle S-V,$ (45)
$\displaystyle\hskip 28.45274pt\rho$ $\displaystyle=$ $\displaystyle S+V,$
(46) $\displaystyle\hskip 28.45274pt\gamma^{2}u^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}+\frac{g^{2}\dot{f}^{2}+f^{2}(g^{\prime})^{2}}{|g^{2}\dot{f}^{2}-f^{2}(g^{\prime})^{2}|},$
(47) $\displaystyle\hskip 28.45274pt\gamma^{2}v^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}\dot{f}^{2}+f^{2}(g^{\prime})^{2}}{|g^{2}\dot{f}^{2}-f^{2}(g^{\prime})^{2}|}.$
(48)

As a check for the correctness of these relations, we apply them to the case
of moving kink with $\varphi(x,t)=4\arctan\left(\exp(\gamma(x-wt))\right)$,
for which $f(t)=\exp(-\gamma wt)$, $g(x)=\exp(-\gamma x)$,
$\gamma_{w}=\left(1-w^{2}\right)^{-1/2}$, $w$ being the kink velocity
($|w|<1$). Straightforward calculation yields $\dot{f}=-\gamma wf$,
$g^{\prime}=-\gamma g$

$\displaystyle\hskip 28.45274ptS$ $\displaystyle=$
$\displaystyle\frac{8f^{2}g^{2}}{\left(f^{2}+g^{2}\right)^{2}}=\frac{\pi}{2},$
(49) $\displaystyle\hskip 28.45274ptp$ $\displaystyle=$
$\displaystyle\frac{8f^{2}g^{2}}{\left(f^{2}+g^{2}\right)^{2}}-V=S-V=\left(\frac{1-w^{2}}{1+w^{2}}\right)K-V,$
(50) $\displaystyle\hskip 28.45274pt\rho$ $\displaystyle=$
$\displaystyle\frac{8f^{2}g^{2}}{\left(f^{2}+g^{2}\right)^{2}}+V=S+V=\left(\frac{1-w^{2}}{1+w^{2}}\right)K+V,$
(51) $\displaystyle\hskip 28.45274pt\gamma^{2}u^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{1-w^{2}}=\gamma_{w}^{2},$ (52) $\displaystyle\hskip
28.45274pt\gamma^{2}v^{2}$ $\displaystyle=$
$\displaystyle\gamma_{w}^{2}-\varepsilon=\gamma_{w}^{2}w^{2},\qquad(\varepsilon=-1)$
(53)

Note that $u^{\alpha}=\left(\gamma,\gamma w\right)$ as expected, only if
$\varepsilon=-1$. For the static kink ($w=0$, $\gamma=1$), we recover the
expected results $\rho=K+V$ and $p=K-V$.

We can now get back to the general results Eqs. (43)-(48), set
$\varepsilon=-1$, and insert appropriate function for $f(t)$ and $g(x)$ for
kink-antikink and breather solutions.
Kink-antikink scattering:

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\sinh\left(\gamma_{w}wt\right),$ $\displaystyle g(x)$
$\displaystyle=$ $\displaystyle w\cosh\left(\gamma_{w}x\right).$ (55)

Breather:

$\displaystyle f(t)$ $\displaystyle=$
$\displaystyle\frac{\sin(wt)}{\sqrt{1+w^{2}}},$ $\displaystyle g(x)$
$\displaystyle=$ $\displaystyle\frac{w\cosh(x)}{\sqrt{1+w^{2}}}.$ (56)

It is now easy to insert the expressions for $K$ and $\psi$ for each case and
calculate the corresponding fluid velocity. For the scattering of a kink and
an antikink, two solitons approach each other during $t\in(-\infty,0)$,
interact around $t\sim 0$, and scatter off each other with almost constant
velocity during $t\in(0,+\infty)$. For the asymptotic situation
$t,x\rightarrow\infty$, we can use the approximation $\sinh\gamma x\simeq 1$
and $\sinh(\gamma wt)\sim 1$, which neatly leads to $v\simeq w$. This shows
that the fluid velocity coincides with the collective soliton velocity as long
as the solitons are far from the interaction region. The interesting question
is: what is the fluid velocity in the interaction region? In order to answer
this question, we plot the velocity field of the fluid $v$ as a function of
$x$ for two values of $t$ in Figs. 4 and 5 for kink-antikink scattering, and
Figs. 6 and 7 for the breather solution. Straightforward calculation leads to:

$\gamma^{2}v^{2}=-\frac{1}{2}+\frac{1}{2}\frac{w^{2}+\tanh^{2}(\gamma_{w}x)\tanh^{2}(\gamma_{w}wt)}{\left|w^{2}-\tanh^{2}(\gamma_{w}x)\tanh^{2}\gamma_{w}(wt)\right|},$
(57)

for the kink-antikink scattering, and

$\gamma^{2}v^{2}=-\frac{1}{2}+\frac{1}{2}\frac{w^{2}+\tan^{2}(wt)\tanh^{2}(x)}{\left|w^{2}-\tan^{2}(wt)\tanh^{2}(x)\right|}$
(58)

for the breather solution. It is also interesting to find the asymptotic value
of the soliton velocity $v$, far from the interaction region. Putting
$\tanh(\gamma x)$ and $\tanh(\gamma wt)$ in Eq. (57) equal to one, we obtain
$v=\pm w$, which correspond to two solitons approaching (or receding) each
other from infinity, with velocities $\pm w$. Of course, the velocities do not
remain constant as the solitons approach each other.

If we define the soliton velocity as the velocity of the energy density peak,
we clearly see from Fig. 3 that it is nearly constant at large distances,
approaches the velocity of light and eventually exceeds it near the center of
mass! There is nothing to worry about, as this velocity is not necessarily
equal to the group velocity which should not be greater than $c=1$. Also drawn
on the same figure, is the fluid velocity contours, which display the same
behaviour. The fluid velocity defined via Eq. (58), is constant at large
distances and diverges in the vicinity of the interaction point. This
observation, leads to the conclusion that the kink and antikink become space-
like (cross the light cone) very near to the interaction region.

Figure 3:  The fluid velocity ($v$) and the energy density ($\rho$) peak for
kink-antikink scattering versus the light cone in $t-x$ plane.

Instead of writing down the explicit forms of $\rho$ and $p$ as functions of
$t$ and $x$ which are too lengthy, we prefer to plot the results for the
typical value $w=1/2$ and $\gamma=2/\sqrt{3}$. Plots of $p$, $\rho$, as
functions of $x$ are shown in Figs. 8 and 9. $t\sim 0$ and $x\sim 0$ are the
collision time and location, respectively. The existence of regions of high
negative pressure can be attributed to the attracting force between solitons
of opposite topological charges. Peaks in the density $\rho$ indicate the
instantaneous position of the soliton. The behavior of kink-antikink
scattering in the Sine-Gordon system is totally different from that of non-
integrable models like the $\phi^{4}$, $\phi^{6}$ and $\phi^{8}$ systems hos ;
ba ; mo1 ; mo2 ; mo3 ; mo4 ; mo5 .

Figure 4:  The fluid velocity $v$, as a function of $x$ for $t=-10$, before
collision. The kink is moving with the velocity $w=0.5$, while the anti-kink
is moving with velocity $w=-0.5$, toward each other. The interaction region is
$x\sim 0$, around which the fluid velocity becomes largely different from the
soliton velocity, and eventually becomes super-luminal.

Figure 5:  The fluid velocity $v$, as a function of $x$ for $t=10$, after
collision. The kink is moving with the velocity $w=0.5$, while the anti-kink
is moving with velocity $w=-0.5$, receding each other. The interaction region
is $x\sim 0$, around which the fluid velocity becomes largely different from
the free soliton velocity.

Figure 6:  The fluid velocity $v$ of breather, as a function of $x$ for
$t=0.1$ and $w=0.5$.

Figure 7:  The fluid velocity $v$ of breather, as a function of $x$ for $t=1$
and $w=0.5$.

Figure 8: Plots depict the pressure as a function of $x$ during time from
$t=-10$ to $t=10$. (a) For the kink-antikink solution and (b) For the breather
solution.

Figure 9: Plots depict the energy density as a function of $x$ during the time
from $t=-10$ to $t=10$; (a) For the kink-antikink solution and (b) For the
breather solution.

##  IV Continuity Equation

One of the basic equations of hydrodynamics is the continuity equation, which
in its classical (non-relativistic) form is associated with the conservation
of mass. In its relativistic form, it takes the covariant form
$\partial_{\mu}j^{\mu}=0$, governing the current $4$-vector $j^{\mu}$. To this
end, we start from the energy-momentum conversation
$\partial_{\mu}T^{\mu\nu}=0$ with $\nu=0$. One is tempted to define
$j^{\mu}\equiv T^{\mu 0}$ as the relativistic current $4$-vector as mentioned
in the introduction. However, one should note that $T^{\mu 0}$ is not
necessarily a $4$-vector, but part of a tensor. The equation
$\partial_{\mu}T^{\mu 0}=0$ is usually integrated over space, defining total
energy as $E=\int T^{00}d^{N}x$, assuming vanishing flux density over the
boundary, and reaching the conservation of total energy $\frac{dE}{dt}=0$. To
achieve a consistent relation, one defines:

$\displaystyle j^{\mu}$ $\displaystyle\equiv$ $\displaystyle
T^{\mu\nu}u_{\nu}=\left(\rho+p\right)u^{\mu}u^{\nu}u_{\nu}+u_{\nu}\eta^{\mu\nu}p,$
(59) $\displaystyle=$
$\displaystyle\left[\rho+\left(1+\varepsilon\right)p\right]u^{\mu}.$

This relation reduces to $j^{\mu}=\rho u^{\mu}$ for the time-like case
$\varepsilon=-1$. However, $j^{\mu}$ is not conserved locally since:

$\partial_{\mu}j^{\mu}=\left(\partial_{\mu}T^{\mu\nu}\right)u_{\nu}+T^{\mu\nu}\partial_{\mu}u_{\nu}=T^{\mu\nu}\partial_{\mu}u_{\nu}.$
(60)

The $4$-current $j^{\mu}$ is conserved, provided that $\partial_{\mu}u_{\nu}$
is purely anti-symmetric or $\partial_{[\mu}u_{\nu]}=0$. For the bi-
dimensional case $u^{\mu}=(u,v)$, this condition translates in to
$\partial_{0}u=0$, $\partial_{1}v=0$ and $\partial_{1}u=-\partial_{0}v$.

##  V concluding remarks

We attempted a fluid dynamical description of a relativistic scalar field and
in particular the Sine-Gordon solitons. The main question we were to answer
was the following: Is it possible to assign -in a consistent way-
hydrodynamical, perfect fluid quantities pressure $p$, density $\rho$ and
fluid velocity $u^{\alpha}$ to a single or double soliton solution? Is the
fluid velocity $v$ necessary equal to the soliton velocity $w$? Is $\rho$
equal to $T^{00}$? We ended with the conclusion that the answers are not
positive in a general sense. However, interesting relationships were found
which may extend our knowledge about relativistic solitons. The procedure can
straightforwardly be applied to non-perfect fluid models and solitons in
dimensions higher than $1+1$, which is left for a future communication.

###### Acknowledgements.

N. R acknowledges the support of Shahid Beheshti University Research Council.
M. P acknowledges the support of Ferdowsi University of Mashhad.

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