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The open–access journal for physics
New Journal of Physics
Hawking-like emission in kink-soliton
escape from a potential well
J A González1,2, M A García- ˜
Nustes1, A Sánchez3
and P V E McClintock2
1Centro de Física, Instituto Venezolano de Investigaciones Científicas (IVIC),
A.P. 21827, Caracas 1020-A, Venezuela
2Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
3Grupo Interdisciplinar de Sistemas Complejos (GISC), Departamento de
Matemáticas, Universidad Carlos III de Madrid, 28911 Leganés, Madrid, Spain
E-mail: jorge@ivic.ve,mogarcia@ivic.ve,p.v.e.mcclintock@lancaster.ac.uk
and anxo@math.uc3m.es
New Journal of Physics 10 (2008) 113015 (19pp)
Received 16 July 2008
Published 11 November 2008
Online at http://www.njp.org/
doi:10.1088/1367-2630/10/11/113015
Abstract. The escape of solitons over a potential barrier is analysed within the
framework of a nonlinear Klein–Gordon equation. It is shown that the creation
of a kink–antikink pair near the barrier through an internal mode instability can
be followed by escape of the kink in a process analogous to Hawking radiation.
These results have important implications in a wider context, including stochastic
resonance and ratchet systems, which are also discussed.
New Journal of Physics 10 (2008) 113015
1367-2630/08/113015+19$30.00 © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft
2
Contents
1. Introduction 2
2. Kinks and antikinks in generalized Klein–Gordon equations 3
2.1. The model equations ............................... 3
2.2. Kink dynamics in inhomogeneous media ..................... 3
2.3. Stability problem ................................. 4
2.4. Tunnelling ..................................... 6
3. Pair creation and Hawking-like emission 9
3.1. Kink–antikink pair creation ............................ 9
3.2. Hawking-like emission .............................. 11
4. Discussion 15
4.1. Inhomogeneous perturbations ........................... 15
4.2. Stochastic resonance ............................... 16
4.3. Solitonic ratchets ................................. 16
4.4. Hawking emission in Josephson junctions .................... 17
4.5. Domain wall tunnelling .............................. 18
5. Conclusion 18
Acknowledgments 18
References 19
1. Introduction
Escape from a metastable state of a dynamical system is of near universal importance. It plays
a crucial role in many classes of physical phenomena in nonequilibrium systems, including
e.g. stochastic resonance, directed diffusion in stochastic ratchets, and nucleation in phase
transitions. In the case of a classical point particle, escape over a finite barrier can occur
through the action of external perturbations, e.g. noise-assisted barrier crossing [1]–[3]. A
quantum particle can leave the well by tunnelling through the classically forbidden region,
and this is true even for macroscopic quantum objects [4]. There have been great advances
in the understanding of macroscopic quantum tunnelling, including the quantum mechanics and
tunnelling of vortices, domain walls and fluxons [5]–[10].
In cosmology and particle physics, there is an additional way for a particle to escape
from a potential well: through the so-called Hawking radiation. Classically, the gravitation is
so powerful around a black hole that nothing, not even radiation, can escape from it. Hawking
has shown, however, that quantum effects allow black holes to emit radiation. A simplified view
of this process is that vacuum fluctuations cause a particle–antiparticle pair to appear close to
the event horizon. One particle of the pair falls into the black hole, while the other escapes. To
an outside observer it appears that the black hole has emitted a particle [11,12].
In the present paper, we extend these ideas by investigating the mechanisms by which a
(classical) kink soliton can escape from a potential well created by space-dependent external
perturbations.
The paper is organized as follows: in section 2, we review well-known results about the
kink solutions in nonlinear Klein–Gordon equations, the model equations and kink dynamics
in inhomogeneous media. For the presentation of the new phenomena, it is very important
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
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to understand that inhomogeneous media can create potential wells and barriers for the kink
motion.
The main result of the paper is discussed in section 3: the Hawking-like emission of a kink
from a potential well. In section 3.1, we review the stability and instabilities of the kink modes,
as these are crucial for understanding of the main result presented in section 3.2.
In section 4, we discuss some implications of the results in a wider context and the possible
ways of observing the new phenomena in real experiments. The phenomena of stochastic
resonance and ratchet motion are well known, and are described briefly in sections 4.2 and 4.3.
However, some of the implications of our results are discussed here for the first time, e.g. a new
kind of solitonic ratchet based on the Hawking-like kink escape in section 4.3. The possibility
that the Hawking-like emission process might be observable in long Josephson junctions and
domain wall dynamics is considered in sections 4.4 and 4.5.
2. Kinks and antikinks in generalized Klein–Gordon equations
2.1. The model equations
As a model system, we take a nonlinear Klein–Gordon equation:
φtt −φx x +∂U(φ)
∂φ =F(x), (1)
where U(φ) is a potential that possesses at least two minima. The sine-Gordon and φ4equations
are particular examples of (1) [13]–[15].
The Klein–Gordon equation (1) can be derived from the Lagrangian
L=1
2φ2
t−1
2φ2
x−U(φ) +F(x)φ +k.
If we define the momentum 5=∂L
∂φt, then the associated Hamiltonian will be
H=1
252+1
2φ2
x+U(φ) −F(x)φ −k,
where dk
dt=0. The energy conservation law can be written as dH
dt=0, where H=R∞
−∞ Hdx.
Equation (1) is a paradigmatic model that supports kink and antikink solutions. The
importance of kink-solitons arises in part because of their applications, not only to particle
physics, but also to diverse phenomena in condensed matter physics, e.g. domain walls in
ferromagnets and ferroelectrics, dislocations in crystals, charge density waves, fluxons in long
Josephson junctions and Josephson transmission lines [13,15].
2.2. Kink dynamics in inhomogeneous media
The effective potential V(x)for the kink is created by the external force F(x), and not by the
nonlinearity in φ[15]–[23]. However, the nonlinearity in φthat corresponds to potential U(φ)
is necessary for the existence of the kink. Suppose U(φ) possesses three extrema φ1,φ2and φ3,
where φ1and φ3are minima and φ2is a maximum. Once U(φ1)=U(φ3), the kink behaves as a
free particle governed by Newton’s first law.
The exact solution for the φ4kink, φk=tanh hx−vt−x0
2√1−v2i, is well known and shows clearly
that we cannot consider U(φ) as the potential in which the kink is moving. The differences
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
4
between the bistability in U(φ) and V(x)have been discussed in connection with soliton
stochastic resonance [20]. The fact that U(φ) should in general have at least two minima for
the existence of kinks is discussed in [21]. The case when F(x)=const 6= 0 that leads to
U(φ1)6=U(φ3)is discussed in [21,22].
It is well known [22] that when F(x)=const <0, the kink will move to the right (positive
x-direction) and when F(x)=const >0, the kink will move to the left. This behaviour depends
on the sign of the topological charge, so that for the antikink we have the opposite movements. In
general, if F(x)has a zero (say at x=x∗) and F(x)→F2>0 for x→ ∞ and F(x)→F1<0
for x→ −∞, this F(x)creates a one-well potential for the kink.
As discussed in [15]–[23] (and references therein), inhomogeneous perturbations of soliton
equations, both in the form of external forces or parametric impurities, generate effective
potentials of type V(xCM)for the motion of the kink, where xCM is the coordinate of the kink
centre of mass. There exist many methods for obtaining this effective potential.
An appropriate expression for the effective force that acts on the kink, in terms of the centre
of mass of the kink xCM, is
F(xCM)= −2
∞
Z
−∞
F(x)f0(x−xCM)dx,(2)
where f0(x)is the translational mode. The effective potential satisfies the equation
F(xCM)= −∂V(xCM)
∂xCM
.(3)
In the case of φ4-equation,
f0(x)=sech2x
2.(4)
For the antikink equation (2) should be multiplied by (−1). There are exact solutions for the
case where F(x)is not constant [22]. For instance, for
F(x)=B(4B2−1)tanh(Bx ), (5)
the exact kink solution is φk=2Btanh (Bx).
In fact, what is important about equation (5) is that there exists a zero for F(x). In this
case, for 4B2>1 the force (5) gives rise to a stable equilibrium for the kink centre of mass.
Both theoretical and numerical investigations show that perturbations of the initial position of
the kink centre of mass lead to oscillations of this variable around the point x=0.
2.3. Stability problem
There are two important techniques for investigation of kink dynamics in the presence of
perturbations. One is the collective coordinate approach [17,18] and the other is stability
analysis. Sometimes they complement each other.
The main objective of this subsection is to explain the stability conditions for the kink in
the presence of an inhomogeneous F(x)with a zero x∗such that (F(x∗)=0). In this subsection,
we will use both techniques to explain the stability conditions because both approaches can shed
light on the problem. One example of the application of the collective coordinate approach is
to use the ansatz φ=φk[x−xCM(t)]. The key idea of the technique is to derive a dynamical
equation for the collective coordinate xCM(t)that describes the dynamics of the kink centre of
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
5
mass. In [15]–[17], [23], several ways for obtaining this dynamical equation are discussed. The
effective force given by (2) is a result of such a technique.
We now present the stability problem. We consider perturbations of a kink-soliton placed
at an equilibrium position created by the force F(x). This analysis leads to the spectral
problem
ˆ
L f =0f,(6)
where φ(x,t)=φk(x)+f(x)eλt,
ˆ
L= −∂2
x+∂2U
∂φ2
φ=φk, 0 = −λ2.(7)
There are discrete eigenvalues corresponding to soliton modes (a translational mode 00, internal
modes 0i, and a continuous spectrum corresponding to phonon modes). Many interesting
phenomena can be uncovered by solving this eigenvalue problem for different forms of the
function F(x).
Let us discuss the stability conditions when F(x)possesses only one zero, say at x∗, such
that ∂F(x)
∂x|x=x∗>0. We then find that the point x=x∗is a stable equilibrium position for the
kink. Otherwise, if ∂F(x)
∂x|x=x∗<0, the equilibrium position x=x∗is unstable. An easier way to
explain this is as follows: from equation (2) we get that for intervals where F(x) > 0, the kink
will be pushed to the left, whereas for intervals where F(x) < 0, the kink will be pushed to the
right. Thus if x=x∗is a zero of F(x)and F(x) < 0 for x<x∗and F(x) > 0 for x>x∗, then
x=x∗is a stable equilibrium position.
We should add that other exact solutions for a kink moving in the presence of an external
force [22] corroborate the result that the sign of F(x)defines the direction of kink motion, and
that the points x∗where F(x)changes its sign correspond approximately to the equilibrium
positions with ∂F(x)
∂x|x=x∗>0 as the stability condition. The stability condition for the antikink
is ∂F
∂x|x=x∗<0.
As an illustration of the application of the stability analysis, consider the following
inhomogeneous force F(x)=B(4B2−1)tanh(Bx). It has only one zero x=0. The kink
solution is φk=2Btanh(Bx ).
The operator ˆ
Lin the spectral problem (6) does not explicitly depend on F(x). However,
it contains information on F(x)because it depends on φk(x); and φk(x)depends on F(x). The
calculation of the eigenvalues of ˆ
Lfor this F(x)and φk(x)leads to the following stability
condition for the translational mode
4B2>1.(8)
Note that this inequality coincides with the condition ∂F(x)
∂x|x=x∗=B2(4B2−1)
cosh2(Bx )
x=0=B2(4B2−1)> 0.
Now suppose that F(x)is qualitatively equivalent to that shown in figure 2. That is, F(x)
has a maximum at xM, a zero at x=x∗and a minimum at xm. In this case, ∂F(x)
∂x|x=x∗<0. We
have found that if |F(xM)−F(xm)|exceeds a critical value, the first internal (shape) mode
for the kink becomes unstable (01<0). The eigenfunction corresponding to this (shape) mode
is always equivalent to the function f1(x)∼sinh(B x )
cosh3(Bx ). It is the development of the instability
of this mode that leads to the break up of the kink (φk∼tanh(Bx )) and to the creation of a
kink–antikink pair.
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
6
0
0
V(x)
x
x2
≈ 0
x1
≈ −dx3
≈ G
Figure 1. Effective potential for the soliton escape problem modelled with
equation (1). As discussed in the text, V(x)is created by the force F(x).
2.4. Tunnelling
We now suppose an external force F(x)such that a point-like kink would feel an effective
potential like that shown in figure 1. When the kink-soliton is analysed as a point-like particle
(i.e. in terms of collective coordinate theory for its centre of mass [17]), the zeroes of F(x)are
equilibrium points [19,20,22], [24]–[26]. The effective force acting on the kink centre of mass
can be calculated [15,16]. When the distance between the zeroes of F(x)is comparable with the
kink’s width, however, the stability conditions for a point particle and an extended kink-soliton
clearly differ [19,20,22].
For instance, if F(x)possesses three zeroes x1,x2,x3such that ∂F
∂x|x1,3>0, ∂F
∂x|x2<0 then,
for a point-like kink, this would imply the existence of one unstable equilibrium between two
stable equilibria. However, if |x2−x1|<lkand |x3−x2|<lk, where lkis the kink’s width, the
point x2is no longer unstable for the extended kink-soliton. Thus the kink-soliton will not feel
the barrier between the two potential wells.
Gonzalez et al [19] investigated the dynamics of a kink moving in an asymmetrical
potential well with a finite barrier. For large values of barrier height and thickness, the kink
behaves as a point-like classical particle. They also revealed the existence of soliton ‘tunnelling’
through the potential barrier, a phenomenon linked to stability changes in the presence of many
zeroes of the external force [19]. When the kink width is less than those of the well and barrier,
the zero x1corresponds to a true minimum of the potential: the kink feels the barrier, and it will
not move to the right of point x=x2. When the point x=x2is not unstable, however, the kink
can move to the right, crossing the barrier even if its centre of mass is placed in the minimum
of the potential and its initial velocity is zero. The kink thus tunnels with sub-barrier kinetic
energy [19]. See also the discussion of this phenomenon in the book [16].
We now consider the double-well φ4potential U(φ) =(φ2−1)2/8 and take the force F(x)
in the form
F(x)=A(4A2−1)tanh[A(x+d)]22(x)+B(4B2−1)tanh(B x ) 21(x)
+C(4C2−1)tanh[C(x−G)]23(x), (9)
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
7
0
0
x
F(x)xMxm
Figure 2. Function F(x)(12) for 4B2<1.
where
21(x)=(tanh[D(x+(d/2))] + 1)
2,
22(x)=(1−tanh[D(x+(d/2))])
2
and
23(x)=(tanh[D(x−(G/2))] + 1)
2.
The force F(x)is chosen in the form (9) in order to give the three-zero structure discussed
above, while allowing for the possibility of exact analytic results. The effective potential for a
point-like kink is then equivalent to that shown in figure 1.
The stability problem for the kink solution in the neighbourhood of an equilibrium position
can be solved exactly. For 4A2>1, 4B2<1, 4C2>1, G>d>0, the force F(x)corresponds
to an effective potential for a point-like kink soliton that possesses a stable equilibrium around
x=x1≈ −d, an unstable equilibrium at x=x2≈0 and a stable equilibrium at x=x3≈G>0.
This point corresponds to an equilibrium of lower potential energy than that at x=x1≈
−d. Figures 3and 4show numerical simulations of a kink-soliton when d=1 and d=3,
respectively. In all simulations in this paper D=4. The initial conditions used here are
φ(x,t=0)=tanh 1
2(x+d),
φt(x,t=0)=0.
Note that, for d<2 and 1
10 <B2<1
4, the kink can tunnel past the barrier despite its initially sub-
barrier kinetic energy. However, for d>2, the kink remains trapped in the left potential well.
This behaviour can also be seen in figures 5(a) and (b) which show the different dynamical
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
8
−1
0
1
−1
0
1
−1
0
1
0
0
0 0
φ
x
0.0 2.0 4.0
6.0 8.0 10.0
12.0 14.0 16.0
Figure 3. A kink-soliton tunnelling across a potential well. The kink-soliton is
initially at d=1 from the origin. The dimensionless time is shown in each frame.
The vertical discontinuous lines correspond to the position of the minimum of
the potential (left line) and the position of the barrier (right line). The bottom
row shows the effective potential in order to make clear the phenomenon.
−1
0
1
−1
0
1
−1
0
1
0
0
0 0
0.0 2.0 4.0
6.0 8.0 10.0
12.0 14.0 16.0
φ
x
V
eff
Figure 4. A kink-soliton trapped in a potential well. Initially the kink-soliton
is at d=3 from the origin. The dimensionless time is shown in each frame.
The vertical discontinuous lines show the position of the minimum (left line) of
the potential and the barrier (right line) as in figure 3. The bottom row shows the
effective potential in order to make clear the phenomenon.
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
9
0 100 200
−5
0
5
Xcm
0 500
−5
0
5
t
(a) (b)
Figure 5. Dynamics of kink-soliton centre of mass in a potential well. (a) Kink-
soliton tunnelling across a potential barrier (d=1). (b) Trapped kink-soliton
inside the well (d=3).
regimes for the motion dynamics of the kink-soliton centre of mass. Initially, the kink-soliton
centre of mass is at the bottom of the potential well created by force F(x)given by (9) (see also
figure 1).
We stress that, in the latter situation, the kink behaves very much like a classical particle.
To see this, consider thermally activated barrier crossing for
φtt +γ φt−φx x +∂U(φ)
∂φ =F(x)+η(x,t). (10)
Here η(x,t)is a Gaussian white noise with hη(x,t)i = 0 and hη(x,t), η(x0,t0)i = 2Dδ(x−
x0)δ(t−t0). The behaviour is then very similar to that in the Kramers model [1].
Furthermore, solitonic stochastic resonance [20] can occur. Consider the equation φtt +
γ φt−φx x +∂U(φ)
∂φ =F(x)+P0cos(ωt)+η(x,t), where F(x)is defined as in (9) with A=C,
4A2>1, 1
10 <B2<1
4and G=d. This is equivalent to a symmetric bistable potential for the
kink centre of mass. In the absence of noise, the periodic force P0cos(ωt)is unable to make
the kink jump between the wells. It is therefore possible to observe a maximum in a graph of
the signal-to-noise ratio (SNR) versus the noise intensity D[20].
3. Pair creation and Hawking-like emission
3.1. Kink–antikink pair creation
As the phenomenon of kink–antikink pair creation is very important for the discussion that
follows below, we will dedicate this subsection to its explanation, using a particular example.
Let us consider the following equation:
φtt −φx x −1
2φ+1
2φ3=F(x), (11)
where
F(x)=1
2(4B2−1)sinh(Bx )/ cosh3(B x). (12)
In this case, the exact solution for a static kink placed on the equilibrium point x=0 is
φk=tanh (Bx ).
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
10
The operator ˆ
Lof the spectral problem (6) can be written explicitly in this case:
ˆ
L= −∂2
x+1−3
2 cosh2(Bx).
This is a Poschl–Teller potential. The eigenvalue problem can be solved exactly [27].
The discrete eigenvalues from the spectral problem (6) are given by the expression 0n=
−1
2+B2(3 + 23n−n2), where 3(3 + 1)=3/2B2. The first internal shape mode is unstable
when
01<0.(13)
As 01=B2(33−1)−1
2, where 3=−1+√1+6/B2
2, we obtain the condition for the instability of
the first internal shape mode:
B2<11 −√117
8.(14)
The function F(x)has a maximum at certain point x=xM, a zero at point x=x∗=0 (such
that ∂F
∂x|x=x∗<0|) and a minimum at certain point x=xm. The condition for the creation
of the instability of the first internal (shape) mode can be written in terms of the extrema
of the function: |F(xM)−F(xm)|>Fc, where Fc=3
32 [√117 −9]. Numerical calculations
show that these are good estimates for the critical value Fceven when other functions F(x)
(with a maximum, a zero and a minimum) are used. Figure 2shows the shape of the function
F(x)for 4B2<1. The point x=0 is an unstable equilibrium point for the centre of mass of the
kink. The effective potential for V(xCM)for the motion of the kink can be defined as follows:
V(xCM)= − ZxCM
0
Fef(ξ) dξ+Vc,(15)
where
Fef(ξ) = − Z∞
−∞
(4B2−1)sinh(Bx )
cosh3(Bx )sech2x−ξ
2dx,(16)
and Vcis an arbitrary constant. Figure 6shows an illustration of the effective potential V(xCM).
The effective potential V(xCM )can be seen as a potential barrier for the motion of the kink.
The effective force is exponentially small everywhere except for a localized zone that contains
an unstable equilibrium position that is a maximum of the effective potential. Figure 7shows
the evolution of the instability of the shape mode and the creation of a kink–antikink pair (in
addition to the already existing kink).
Thus, if the kink is very close to a potential barrier with these properties, then kink–antikink
pairs can be created. This scenario is also valid for any other F(x)with these properties, i.e. it is
not restricted to the explicit function used in this simulation. Moreover, the same phenomenon
takes place also for the sine-Gordon equation
φtt −φx x + sin(φ) =F(x), (17)
where F(x)=2(B2−1)sinh(Bx )/cosh2(B x). The first internal (shape) mode is unstable
for B2<2/3∗(3∗+ 1), where 3∗=(5 + √17/2). The critical value for |F(xM)−F(xm)|is
different for the sine-Gordon, but the shape of F(x)is similar to that in equation (12).
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
11
0
0
xCM
V(xCM)
Figure 6. Effective potential barrier V(xCM)created by the force (12).
−1
0
1
−1
0
1
−1
0
1
−40 0
−1
0
1
−40 0 −40 0 40
0.0 2.0 4.0
6.0 8.0 10.0
12.0 14.0 16.0
18.0 20.0 22.0
φ
x
Figure 7. Kink–antikink pair creation via the internal mode instability. The
dimensionless time is shown in each frame.
3.2. Hawking-like emission
We now show that a kink can be emitted from the potential well via an internal mode instability,
even when the height and thickness of the barrier are very large. A kink–antikink pair can be
created near the potential barrier due to the internal mode instability of the original kink. The
antikink will be attracted by the potential barrier (a stable equilibrium for the antikink), and the
kink then escapes in a process resembling Hawking emission.
Let us consider the situation where G>d>2, such that kink tunnelling is impossible,
but where the absolute value of the derivative of F(x)at the point x=0 is such that new
instabilities can develop. Thus the equilibrium points are sufficiently separated but
∂F(x)
∂x
x=0
can take relatively large values. A stability analysis [22] considering perturbations around a
kink-soliton placed at the equilibrium position x=0[φ (x,t)=φk(x)+f(x)eλt] leads to the
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
12
eigenvalue problem
ˆ
L f =0f,(18)
where ˆ
L= −∂x2+6B2−1
2−6B2
cosh2(Bx ),0= −λ2. There are both discrete and continuous
spectra. The discrete eigenvalues correspond to soliton modes: a translational mode 00and an
internal mode 01. The continuous spectrum (0k,k>1) corresponds to phonon modes [22]. In
the present case,
00=2B2−1
2and 01=5B2−1
2.(19)
The stability condition for the equilibrium point x=0 of the kink is defined by the eigenvalue
of the translational mode [ f0(x)=(1/cosh2(Bx ))], i.e. 4 B2>1. When the kink equilibrium
position x=0 is unstable (i.e. 4B2<1), but 10B2>1, the kink can move away from the
point x=0 without large deformations. The reason is that the internal (shape) mode f1(x)=
(sinh(Bx ))/(cosh2(B x)) is still stable.
For 10B2<1, however, the internal mode becomes unstable, leading to decay of the kink
into an antikink and two kinks (see figure 9). For a kink equilibrated on a single barrier like
F(x)=B(4B2−1)tanh(Bx )such that the internal mode is unstable, the initial stage of kink
deformation is described by the solution
φ(x,t)=tanh x
2−f10
sinh(Bx )
cosh2(Bx )eλt.
Note that, even if the initial perturbation of the kink shape is very small ( f101), the slope of
φ(x)at its centre of mass will eventually be negative, so that the kink will break up, producing
an additional kink–antikink pair. Note also that the amplitude of the deformation cannot increase
indefinitely because of energy conservation. In fact, the final state is an antikink at point x=0,
and two kinks moving in opposite directions (where φ(x,t)is finite everywhere).
The whole dynamics of the kink–antikink pair creation and the kink escape in the two-well
potential produced by F(x)defined as in (9) can be described using the dynamics of the kink
and antikink configurations, the translational mode and the internal (shape) mode [22,23]. It can
be proved that there exists a stable equilibrium stationary solution (see figure 8) that corresponds
approximately to the final state that is represented in figure 9.
We have constructed a solution that describes this stable stationary state:
φf(x)=2Atanh[A(x+d)]22(x)+ 2Ctanh[C(x−G)]23(x)−f1tanh[Qx ]21(x)24(x), (20)
where 24(x)=1−tanh[D(x−(G/2))]
2,Q=Band f1=2C.
The energy balance equation is satisfied approximately H(initial conditions) =H(final
state). We should recall that the perturbed Klein–Gordon equation (1) is not completely
integrable and part of the energy is radiated in the form of ‘phonon’ waves. That is, the whole
energy His conserved, but not all the energy is solitonic for t→ ∞.
Due to the properties of functions 2i(x), which for D1 behave as Heaviside functions,
the stability problem of solution (20) can be reduced to three simpler spectral problems similar
to that formulated in equation (18). The solution is stable for 4A2>1,4B2<1,4C2>1.
A more physically motivated explanation of the process may also be useful. The initial
condition is a kink placed close to the point x=0. Due to the conditions discussed above, the
internal mode of this kink is unstable, leading to the creation of a kink–antikink–kink structure.
After that, the first kink will be placed on the left side of the point x=0 (the barrier). Here,
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
13
−d 0 G
−1.5
−1.0
−0.5
0
0.5
1.0
1.5
x
φf(x)
Figure 8. When the internal mode of the kink near the barrier is unstable, the
stable stationary field configuration is a kink (inside the left well), an antikink
(near the barrier), and a kink that has escaped to the right well.
the function F(x)as defined by equation (9) is positive. So this kink will effectively be pushed
to the left and it will be finally captured by the zero of function F(x)near x= −d. This is a
stable equilibrium for the kink. The antikink will be close to the point x=0. As ∂F
∂x|x=0<0,
this is a stable equilibrium for the antikink. So the antikink remains close to this point. The
‘third particle’ of the trio (the other kink) will be placed on the right side of point x=0, where
function F(x)as defined by equation (9) is negative. Thus, this kink will be pushed to the right
until it is finally captured by the equilibrium situated near the point x=G.
This scenario leads unambiguously to the Hawking emission of a kink via kink–antikink
pair creation. A numerical simulation of this process is shown in figure 9.
The initial conditions in these simulations are φ(x,t=0)=tanh 1
2(x+ 1)
,φt(x,t=0)=0.
Effectively, during this process, a kink–antikink pair is created due to the instability of the
original kink internal mode. Note that topological charge is conserved. The point x=0 (where
|∂F(x)
∂x|x=0<0 ) is an attractive equilibrium position for the antikink. Thus, the potential barrier
will capture the antikink, whereas the newly created kink escapes to the right potential well.
The original kink remains inside the left potential well. Figure 9shows this phenomenon with
an initial condition consisting of a kink inside the potential well (i.e. the kink centre of mass is
at a point x0<0), but very close to the barrier that is the source of instability.
The emission of a kink-soliton from the potential well by the decay of a kink into an
antikink and two kinks can also occur under less strict conditions. Suppose e.g. that the
conditions for the internal mode instability are satisfied near the barrier, but that the kink centre
of mass is at the bottom of the potential well, i.e. x0= −d. The kink is an extended object. So
even when its centre of mass is at the potential minimum, the kink’s ‘wings’ feel the action of
the potential wells. If the values of [F(x)] for x<−dare sufficiently large, the kink is pushed
to the right, closer to the barrier, and an instability will develop leading to kink emission as
described above. This phenomenon can also be observed in the perturbed sine-Gordon equation,
where U(φ) =1−cos φ. In fact, we can construct a function F(x)that corresponds to a bistable
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
14
−1
0
1
−1
0
1
−1
0
1
−d0 G
−1
0
1
−d0 G−d0 G
φ
x
0.0 2.0 4.0
8.0 10.0
12.0 14.0 16.0
18.0 20.0 22.0
6.0
Figure 9. Kink emission via the internal mode instability. The dimensionless
time is shown in each frame.
potential for the sine-Gordon kink:
F(x)=2(A2−1)(sinh[A(x+d)])
cosh2[A(x+d)]22(x)+ 2(B2−1)(sinh[Bx ]/cosh2[B x])21(x)
+2(C2−1)(sinh[C(x−G)])
cosh2[C(x−G)]23(x), (21)
where the 2i(x)functions are defined as in equation (9).
For B2<2/[3∗(3∗+ 1)], with 3∗=(5 + √17)/2 the shape mode becomes unstable and
the kink–antikink pair is created. We have checked this phenomenon numerically. In general,
the relevant features needed for it to occur are the following: U(φ) should have at least two
minima φ1and φ3such that for U(φ1)=U(φ3)the kink can exist (a monotonic function such
that φ→φ1, as x→ −∞,φ→φ3as x→ ∞); and F(x)should generate a barrier such that
the first internal shape mode is unstable. The shapes of the kink and the internal mode are very
similar in all equations of type (1). In the case of sine-Gordon, the exact kink solution of the
unperturbed equation is φk(x,t)=4 arctan{exp[(x−vt−x0)/√1−v2]}.
Now, returning to the φ4-equation, if F(x)is such that |F(xM)−F(xm)|>Fc, the barrier
can create a kink–antikink pair even if the original kink is very far inside the potential well
(see figure 10). Here, the initial conditions are φ(x,t=0)=tanh 1
2(x+ 10),φt(x,t=0)=
0. Figures 5,9and 10 are the results of numerical simulations of the partial differential
equation (1). We have used a standard implicit finite difference method with open boundary
conditions φx(−l,t)=φx(l,t)=0 and a system length 2l=2000. However, the phenomena
described do not depend on the numerical method. In the simulations shown in figures 9and 10,
the initial conditions used were the exact stationary kink solutions of the unperturbed φ4-
equation: φk=tanh x−x0
2, where x0defines the initial position of the kink centre of mass.
If we use different functions as initial conditions, but with a kink-like shape and the same
topological charge, the initial deformations in the time evolution of the field φ(k,t)will of
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
15
−1
0
1
−1
0
1
−1
0
1
−d0 G
−1
0
1
−d0 G −d0 G
φ
x
0.0 2.0 4.0
8.0 10.0
14.0 16.0
18.0 20.0 22.0
6.0
12.0
Figure 10. A kink–antikink pair is created outside the potential well barrier even
when the initial kink is very deep inside the well. The antikink is attracted to the
barrier and the kink escapes to the right potential well. The dimensionless time
is shown in each frame.
course differ. However, the main qualitative features of the described phenomena will be exactly
the same.
All of these phenomena can be perceived as non-local effects attributable to the spatially
extended character of the kink. When the kink feels the instability of the barrier, it suffers
deformations producing fluctuations in the field φ(x,t)leading, ultimately, to development of
the kink–antikink pair. We remark that these phenomena are robust. They can also occur for
other forces F(x)that create three equilibria for the kink, provided that the slope of F(x)at
the unstable equilibrium is sufficiently large. Furthermore, F(x)can also be such that there are
other equilibria separated from this three-equilibria basic structure.
4. Discussion
We now discuss some properties of the inhomogeneous perturbations used in the present paper.
Then we consider applications to stochastic resonance and ratchets, and discuss the possibility
that the Hawking-like emission process might be observable in long Josephson junctions.
4.1. Inhomogeneous perturbations
When the centre of mass of a kink is close to the equilibrium position created by
F(x)∼tanh (Bx ), the dynamics of the kink translational mode is equivalent to a harmonic
oscillator [20]. Our construction of F(x)in the paper is equivalent to a Kramers model, where
a bistable potential with two minima and a maximum is constructed using three parabolas. As
already pointed out above, our model is analytically solvable: we can solve the stability problem
exactly for both the translational mode and all the internal modes.
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
16
Using the geometrical theory of dynamical systems we can extend the results to all models
that are topologically equivalent to the present one [20]. This means all the models where
(locally) F(x)possesses three zeroes that generate a bistable effective potential for the motion
of kink centre of mass can produce this kind of dynamics. Note that the force F(x)does not
have to take large values in order to observe these phenomena. For instance, the F(x)given
above for the sine-Gordon equation has the property F(x)→0 as x→ ±∞, and it is small
everywhere. This F(x)can also be used for the φ4equation. The important features are that
F(x)should have three zeroes and that, near the central zero, the first internal mode should be
unstable.
The case F(x)=const is very familiar to experimentalists [15]. This is important because
the phenomena presented in this paper can be observed using a F(x)constructed with piece-
wise-constant forces as follows:
F(x)=F1<0,for x<x1<0,
F(x)=F2>0,for x1<x<x2=0,
F(x)=F3<0,for x2<x<x3,
F(x)=F4>0,for x>x3,
where −F1=F4and x3>−x1. We have a complete theory and simulations. There are some
necessary conditions for 1=F2−|F3|, and for the distance between the equilibrium points
(i.e. where F(x)changes sign). Under some conditions, the kink remains trapped inside the left
well; under other conditions it can tunnel through the barrier to the right well, or it can escape
via a Hawking-like emission.
The piece-wise-constant F(x)could readily be constructed in experiments with domain
walls (using e.g. constant external fields) and in long Josephson junctions ([15,16] and refer-
ences therein). We comment, firstly, that the F(x)given in these papers is very much like the
F(x)described above in the preceding paragraphs. It is almost constant everywhere. However,
it has the advantage that it is smooth even though it changes sign three times. Secondly,
Hawking-like emission can occur even when the absolute values of F(x)are small everywhere.
4.2. Stochastic resonance
As mentioned above, a kink can [20] be involved in a motion equivalent to the original stochastic
resonance paradigm of a particle in a bistable potential [28]. But this is only possible when
the equilibrium positions of the potential are sufficiently separated such that the kink feels the
bistability and the kink internal modes are stable in the neighbourhood of all equilibrium points.
Otherwise, the picture shown in the final frames of figures 9and 10 applies, i.e. an antikink will
be stabilized in the central equilibrium positions and each of the wells will be occupied by a
kink. We have checked numerically that the time series of the centre of mass of this structure
does not then have a maximum in the SNR versus Dplot, i.e. stochastic resonance does not
occur.
4.3. Solitonic ratchets
These phenomena can also give rise to ratchet behaviour. For the equation φtt +γ φt−
φx x +∂U(φ )
∂φ =F(x)+η(x,t), ratchet motion of a single kink is possible provided certain
conditions are satisfied by the force F(x)and the noise η(x,t). An example is the force
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
17
F(x)= −Asin(B x )+ 2 µAsin(2Bx +θ ), where µ=1
4,θ=(π/2). In this case, η(x,t)should
be coloured noise [29,30]. Ratchet motion will be affected in cases where the kink does not
feel all irregularities of the effective potential, and also by phenomena related to the complex
dynamics and stability of kink internal modes. Furthermore, in some cases the kink can be
trapped inside the potential well created by F(x)in such a way that the kink translational mode
does not exist. In such cases, kink motion is obviously very difficult [19,23,24].
The effect discussed in this paper could lead to a new kind of solitonic ratchet in systems
where the kink cannot propagate: the kink is trapped in a potential well but, when it is driven by
the external perturbation, it approaches the barrier of the potential. A kink–antikink pair is then
created, and a kink is emitted. This kink can be trapped in the contiguous well and the process
can start again. As a result of this process, kink propagation in one direction is possible even
though the original kink has not abandoned its potential well.
4.4. Hawking emission in Josephson junctions
Before considering the possibility of Hawking-like emission in Josephson junctions, we remark
that noise-assisted Hawking kink emission is also possible. Consider the case where the
properties of F(x)are such that the kink can sit stably at the minimum of the left potential
well without being pushed to the barrier by the potential wall, and the values of F(x)near
the barrier are such that the kink internal model should be unstable. The presence of noise can
then make the kink approach the barrier and break up, producing the same phenomenon as that
depicted for the noise-free case in figure 9.
Long Josephson junctions are very good physical objects for the observation of the soliton
dynamics. Akoh et al [31] constructed a device in which details of the dynamics of individual
fluxons could be observed. They created a local inhomogeneity and they were able to capture
a fluxon using this inhomogeneity as in a potential well. Then they applied a constant bias
current to the system. The effect of all these actions is equivalent to having a kink-soliton inside
a potential well with a finite barrier on the (say) right side. They observed the escape of the
fluxon from the potential well. The paper in question is not specific about the concepts of soliton
tunnelling and soliton Hawking emission, probably because one needs a theoretical framework
in order to introduce such concepts. However this experiment could now be repeated using a
similar setup, and the Hawking-like emission could be specifically sought.
In order to be definite, we will provide some details about a possible experiment. We
consider a device in the form of an NbN–oxide–NbN junction with the dimensions of 2.5×
150 µm2. A propagating fluxon can be captured at any desired position in the effective potential
well created by placing a resistor on the surface of a portion of the long Josephson junction.
The surface resistor has a length larger than the Josephson penetration depth λJ. The NbN
film used in the experiment has a London penetration depth λLof about 290 nm. Meanwhile
λJis estimated to be 4 µm. The dimensions of the surface resistor used to capture the fluxon
are 2.5×20 µm2. So the length of the resistor should be 20 µm. The current density J0of
this Josephson junction is 2 kA cm−2. It is well known that the fluxon dynamics is very well
described by the behaviour of kink-solitons [15]. A Josephson pulse generator at the input side
of the junction generates a fluxon. A latching current Irsupplied from a SQUID connected to
the resistor can re-drive the captured fluxon. Once a fluxon has been captured in the potential
well, a fluxon should be observable on the output side after supplying a current of Ir=0.11 mA.
This would indicate that the fluxon had escaped from the potential well.
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
18
The length of the resistor defines the width of the potential well. We estimate that for
resistor lengths of the order of 6 µm and a supply current Irof the order of 0.1 mA, fluxon
tunnelling should be observable. On the other hand, for resistor lengths of the order of 30 µm,
tunnelling would be impossible. In this latter case, for supply currents of the order of 1 mA it
should be possible to observe an escaping fluxon due to a fluxon–antifluxon pair creation.
The supply current here is proportional to the absolute value of the slope of the potential
V(x)in figure 1for x<−dor, in other terms, to the absolute value of F(x)given by
equation (9) for x<−d.
4.5. Domain wall tunnelling
Recently, there has been much interest in the macroscopic tunnelling of domain walls
[9,10,32], e.g. those trapped by crystal defects in a magnet. Usually tunnelling phenomena
are studied as purely quantum effects that should occur at very low temperatures. In general,
calculations show the probability to be very small [9,10,32]. Our results show however that,
under the right conditions, escape will occur with certainty.
One experimental manifestation of the macroscopic tunnelling of domain walls is that the
rate of the magnetization relaxation processes does not decrease to zero when the temperature
is reduced, but maintains a finite value, independent of temperature [3,7].
The present results suggest that very large domain walls should show not only the
macroscopic tunnelling but also the Hawking emission. Other systems with topologically
coherent structures, e.g. vortices and spiral waves [33], may be expected to behave similarly.
Such phenomena should in principle be observable. In particular, it is in principle possible
to build a long Josephson junction, with an inhomogeneous perturbation like the one studied
here, where the kink escape could be observed [34]. The perturbation F(x)can be readily
implemented through an applied magnetic field to push the domain wall off the defect [10].
This experimental setup has already been used to study ratchets with a moving kink in a medium
with potential wells and barriers. Thus, the implementation of the appropriate barrier will not
be a problem. The escape of the kink will be reflected in the electrical performance of the
junction.
5. Conclusion
We have shown that the Hawking radiation phenomenon is not restricted to the vicinity of black
holes, as had been assumed. Rather, it is a more widespread phenomenon that can occur even in
condensed matter systems. Our theory of the process is dynamical, and we can follow in detail
what happens in simulations. We point to an experiment with a long Josephson junction where
we believe that this phenomenon can be observed and where we suspect that it may already have
been observed [31], albeit without full appreciation of what was occurring.
Acknowledgments
We acknowledge support of the Engineering and Physical Sciences Research Council (UK).
One of us (JAG) is grateful to the Royal Society of London for funding his visit to Lancaster.
New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
19
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New Journal of Physics 10 (2008) 113015 (http://www.njp.org/)
