Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential
Molecular dynamic simulations are performed to investigate a long-time evolution of different initial signals in nonlinear acoustic chains with realistic Exp-6 potential and with power potentials. First, in the chain with realistic potential, long-lifetime kink-shaped excitations are found in the...
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Інститут фізики конденсованих систем НАН України
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| Цитувати: | Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential / L.S. Metlov, Yu.V. Eremeichenkova // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 105-118. — Бібліогр.: 23 назв. — англ. |
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irk-123456789-1206972017-06-13T03:05:04Z Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential Metlov, L.S. Eremeichenkova, Yu.V. Molecular dynamic simulations are performed to investigate a long-time evolution of different initial signals in nonlinear acoustic chains with realistic Exp-6 potential and with power potentials. First, in the chain with realistic potential, long-lifetime kink-shaped excitations are found in the system in thermodynamic equilibrium. They give sharp peaks on high-energy tile of Gibbs distribution. Travelling along the chain, each kink collects the energy from the background atoms, and, consequently, transfers it to smallamplitude phonons. Dynamic equilibrium is observed between the processes of growth and decay of the kinks. Проведено числові експерименти за методом молекулярної динаміки з метою дослідження довгочасової еволюції різноманітних початкових сигналів в нелінійнихакустичнихланцюжкахз реалістичним потенціалом виду Ехр-6 та зі ступеневими потенціалами. Впершеулан-цюжку з реалістичним потенціалом знайдені кінкоподібні збудження, що живуть довго, у стані термодинамічної рівноваги. Вони дають гострі піки на високоенергетичному хвості розподілу Гіббса. Прямуючи вздовж ланцюжка кожний кінк збирає енергію з атомів фону, та далі віддає її малоамплітудним фононам. Спостерігається динамічна рівновага між процесами росту та загасання кінків. 2003 Article Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential / L.S. Metlov, Yu.V. Eremeichenkova // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 105-118. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 63.20.Pw, 63.20.Ry, 65.90.+i DOI:10.5488/CMP.6.1.105 http://dspace.nbuv.gov.ua/handle/123456789/120697 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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English |
| description |
Molecular dynamic simulations are performed to investigate a long-time
evolution of different initial signals in nonlinear acoustic chains with realistic
Exp-6 potential and with power potentials. First, in the chain with realistic
potential, long-lifetime kink-shaped excitations are found in the system in
thermodynamic equilibrium. They give sharp peaks on high-energy tile of
Gibbs distribution. Travelling along the chain, each kink collects the energy
from the background atoms, and, consequently, transfers it to smallamplitude
phonons. Dynamic equilibrium is observed between the processes
of growth and decay of the kinks. |
| format |
Article |
| author |
Metlov, L.S. Eremeichenkova, Yu.V. |
| spellingShingle |
Metlov, L.S. Eremeichenkova, Yu.V. Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential Condensed Matter Physics |
| author_facet |
Metlov, L.S. Eremeichenkova, Yu.V. |
| author_sort |
Metlov, L.S. |
| title |
Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential |
| title_short |
Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential |
| title_full |
Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential |
| title_fullStr |
Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential |
| title_full_unstemmed |
Spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential |
| title_sort |
spontaneously appearing discrete moving kinks in nonlinear acoustic chain with realistic potential |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2003 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120697 |
| citation_txt |
Spontaneously appearing discrete
moving kinks in nonlinear acoustic
chain with realistic potential / L.S. Metlov, Yu.V. Eremeichenkova // Condensed Matter Physics. — 2003. — Т. 6, № 1(33). — С. 105-118. — Бібліогр.: 23 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT metlovls spontaneouslyappearingdiscretemovingkinksinnonlinearacousticchainwithrealisticpotential AT eremeichenkovayuv spontaneouslyappearingdiscretemovingkinksinnonlinearacousticchainwithrealisticpotential |
| first_indexed |
2025-07-08T18:25:31Z |
| last_indexed |
2025-07-08T18:25:31Z |
| _version_ |
1837104245002731520 |
| fulltext |
Condensed Matter Physics, 2003, Vol. 6, No. 1(33), pp. 105–118
Spontaneously appearing discrete
moving kinks in nonlinear acoustic
chain with realistic potential
L.S.Metlov, Yu.V.Eremeichenkova
Donetsk Physico-Technical Institute,
National Academy of Sciences of Ukraine,
72 R.Luxemburg Str., 83114 Donetsk, Ukraine
Received September 4, 2002
Molecular dynamic simulations are performed to investigate a long-time
evolution of different initial signals in nonlinear acoustic chains with realis-
tic Exp-6 potential and with power potentials. First, in the chain with realistic
potential, long-lifetime kink-shaped excitations are found in the system in
thermodynamic equilibrium. They give sharp peaks on high-energy tile of
Gibbs distribution. Travelling along the chain, each kink collects the en-
ergy from the background atoms, and, consequently, transfers it to small-
amplitude phonons. Dynamic equilibrium is observed between the process-
es of growth and decay of the kinks.
Key words: moving kink, realistic potential, thermodynamic equilibrium,
energy concentration, Gibbs distribution
PACS: 63.20.Pw, 63.20.Ry, 65.90.+i
1. Introduction
Power expansions of interatomic potentials (taking into account anharmonisms
up to fourth order) are frequently used in order to investigate strongly nonlinear
phenomena [1–6]. However, for soliton-like excitations with large amplitudes one
cannot be sure that high-order anharmonisms may be omitted. Thus, realistic (e.g.
Lennard-Jones, Morse) potentials were used to take into account the anharmonisms
of all orders [6–8].
Several types of soliton-like excitations were found in discrete monoatomic chains
with anharmonic intersite potentials (acoustic chains).
Breathers – time-periodic space-localized one-parameter1 modes with frequency
as a parameter are revealed in Fermi Pasta Ulam (FPU) chains with 2–4 potential
1Necessary conditions for the existence of two-parameter moving breathers (frequency and ve-
locity are the parameters) and a numerical algorithm for their search are proposed in [9].
c© L.S.Metlov, Yu.V.Eremeichenkova 105
L.S.Metlov, Yu.V.Eremeichenkova
[1,10]. Other solutions – dark solitons (phase shift in stable time-periodic extended
nonlinear mode) are proved to exist in FPU chains [11,12]. In monoatomic acoustic
chains with realistic Lennard-Jones and Morse potentials, breathers are not observed
in numerical experiments, and, moreover, they are forbidden [6]. However, they arise
if time-periodic driving field is applied to the chain [7,8].
Moving kinks, i.e. time non-periodic solitons with step-shaped profiles of atom
displacements [13–17] and pulses, i.e. solitons with bell-shaped profiles [15,18,19] are
predicted in acoustic chains. These are one-parameter solitons with a velocity as a
parameter. Numerical results for pulses were obtained in [18,19] for FPU and Toda
chains. Exact non-topological moving kinks are well known in an integrable Toda
chain [13]. Existence theorems of non-topological moving kinks are also proved for
non-integrable FPU chains [14,15] and for acoustic chains with interatomic potentials
of arbitrary powers [16]. We don’t know other solutions to moving kinks in a discrete
case for acoustic chains. In continual (long wave) limit, moving kinks are observed in
the real ultrasonic experiment and are investigated numerically taking third-order
anharmonism into account (KDV equation) [17].
Thus, no soliton-like solution has been obtained in a discrete case for non-
integrable acoustic chains with realistic interatomic potentials. In this paper, spon-
taneous creation of discrete moving kinks is observed in numerical experiment for
the chain with such a potential.
The aim of this article is molecular dynamics study of a discrete acoustic chain
with realistic interatomic potential at high temperatures after thermalization. The
role of different orders of anharmonicity in power expansion of realistic potential is
investigated.
The methods used are described in section 2. Numerical results and discussions
for thermalized gas of spontaneously appearing moving kinks are given in section 3.
2. Methods and approximations
Equations of motion
mr̈i =
1
2
∂/∂ri
∑
ij
V (|ri − rj|) (1)
are integrated numerically with periodic boundary conditions for an acoustic chain
containing 100 atoms. The nearest neighbor approximation is used. Similar results
are obtained when six neighbors and 1500 atoms are taken into account.
We use the system of units in which energy is measured in [K], mass in atomic
mass units [a.m.u.], and distance is measured in the units of equilibrium interatomic
distance [d0]. The unit of time is equal to the period of harmonic zone boundary
phonon mode (T0 = 8.3716 · 10−13 sec). It corresponds to the case of neon with
atom mass m = 20.18 a.m.u. Velocities are measured in the units of sound velocity
vs = 12.72 [(K/a.m.u.)1/2].
106
Kinks in nonlinear acoustic chain
Sixth-order symplectic Yoshida algorithm [20] is applied in numerical simula-
tions. Time step is equal to 0.008 T0 which provides energy conservation with the
accuracy 4E/E = 10−6 during the whole simulation time (t = 106 T0).
Realistic interatomic Exp-6 potential
V exp−6(|ri − rj|) = A0 exp(−α(xij − 1)) −
αA0
6
x−6
ij ; xij = |ri − rj|/d0 (2)
is used with the parameters A0 = 35.9335 K, α =13.6519 obtained ab−initio for the
dimer of neon [21]. Single empirical parameter d0 = 3.091 Å is fitted to equilibrium
interatomic distance in neon dimer [22].
Four types of initial conditions are used.
• Zone boundary mode with wave vector k = π (π-mode). Equal opposite initial
velocities |V0| = 0.24 vs were assigned to neighboring atoms.
• White noise. Random initial velocities were fixed.
• Shock waves. Initial velocities |V0| = 0.96 vs directed inside the chain were
given to three atoms at each end of the chain.
• Exact breathers. Initial displacements of atoms were fixed according to the
breather exact form A(. . . , 0,−1/2, 1, 1/2, 0, . . .) for odd-parity breather and
A(. . . , 0,−1, 1, 0, . . .) for even-parity breather [3]. The amplitude is A=0.279
d0.
In all the cases the same energy E = 90 K/atom is fed into the system by the
initial conditions. This energy is two times larger than cohesive energy of neon dimer.
Numerical experiments with initial energies of 300 K/atom and 1000 K/atom are
also fulfilled to study the temperature dependence of a number of kinks.
To investigate the role of different order anharmonisms, the simulations have been
performed on acoustic chains with power interatomic potentials. These potentials
are defined the ones that expand the Exp-6 potential
V exp−6(|ri − rj|) = V0 +
K2
2
(xij − 1)2 +
K3
3
(xij − 1)3 +
K4
4
(xij − 1)4 + . . . (3)
and take higher powers into account, consequently.
3. Results and discussions
Excluding the cases specially mentioned, an acoustic chain with Exp-6 potential
is considered with different initial conditions in all the subsections. Chains with
power and Toda potentials are also reported.
107
L.S.Metlov, Yu.V.Eremeichenkova
Figure 1. Energy distribution of atoms E(ε) = εn(ε). Solid curve corresponds to
Gibb’s form of the n(ε), the circles denote the distribution E(ε) obtained in the
numerical experiment. To get “experimental” distribution, the energies of atoms
are averaged for the time 11 T0. The distribution corresponds to one of the most
intensive kinks (t = 188947 T0).
3.1. π -mode
Up to the time ≈ 50 T0, initial π-mode perturbed by numerical noise is destroyed
by period-doubling instability (see also [6]). In the following, the motion of atoms
becomes more and more chaotic. Simultaneously, the process of space concentration
of energy takes place. Well-localized solitons with the width compatible to a lattice
constant are spontaneously created since ≈ 600 T0. Thus, the solitons in the chain
with realistic potential appear after thermalization. On the contrary, breathers in
FPU chain arise directly as a result of π-mode destruction, and the system is ther-
malized after the decay of breathers.
The thermalization is defined here as a state when the energy distribution of the
atoms obtained in numerical simulation is close to Gibb’s distribution. The system
persists in such a state for a long time. The deviations from Gibbs distribution
caused by solitons are possible at high-energy tile only.
In the figure 1 energy distribution of atoms E(ε) = εn(ε) is given. The E(ε)
is defined so that dE(ε) = E(ε)dε is total energy of atoms which fall into energy
interval (ε, ε+ dε). The n(ε)dε is the number of atoms with the energies residing in
the same interval. Solid curve E(ε) corresponds to Gibbs distribution n(ε) obtained
in the harmonic approximation. The circles denote the distribution E(ε) observed
in the numerical experiment, when one of the most intensive solitons appears. This
soliton gives a local peak on the tile of experimental E(ε). The position of the peak
108
Kinks in nonlinear acoustic chain
on the energy scale manifests that the energy concentrated in the soliton is ≈ 15
times as large as the average energy of an atom in the system. At all the rest energies,
experimental distribution agrees with the Gibb’s one. It indicates that the majority
of the atoms move chaotically.
Figure 2. Fractional deviation of “ex-
perimental” distribution n(ε) from the
Gibbs one, nexper(ε)/nGib(ε), at t =
2316248 T0 (1 - black circles connected
by the line; right scale). The digit 2 de-
notes “experimental” (grey circles) and
Gibbs (solid line) distributions E(ε) at
the same t.
Fractional deviation of experimental
distribution n(ε) from Gibb’s one is re-
ported in figure 2 (numbered 1). Exper-
imental and Gibb’s distributions E(ε)
are also given here (numbered 2). The
value of experimental n(ε) is ≈ 36 times
as much as Gibb’s one near the peak at
the tile, while nexper/nGib ∼ 1 at all the
rest energies.
A typical pattern of map of tracks
of solitons is plotted in figure 3 where
a single powerful soliton is seen clearly.
Time periodicity in the motion of atoms
is not observed. Velocity of the soliton
(the slope of the curve of energy maxi-
mum position via the time) is equal to
v = 2.6 vs. Numerical experiment shows
that the velocities and the amplitudes
of the observed solitons are connected
uniquely. Therefore, these solitons are
one-parameter ones. Besides, they each
pass through the others without loss of
their individuality like the solitons in in-
tegrable systems (figure 4).
The form of the obtained solitons
can be determined from an enlarged
fragment of the map of tracks plotted
in figure 6. The velocities of atoms are given here. The atom in the soliton moves
in the same direction as the soliton does and transfers its momentum to the next
neighbor. It can be said with good accuracy that this soliton is localized on two
sites only. Space dependence of the atom’s velocity has a bell-shaped form at each
moment of time. Therefore, the displacements of atoms have a step-shaped form,
and the solitons can be identified as non-topological moving kinks.
Let us use the parameter
C0 = N
N∑
i=1
E2
i /(
N∑
i=1
Ei)
2 (4)
as quantitative indicator of energy concentration [4]. The C0 = 1 if all the atoms
have the same energies, C0 = 1.75 for equilibrium Gibb’s distribution in harmonic
109
L.S.Metlov, Yu.V.Eremeichenkova
Figure 3. Map of tracks of the moving kink from the numerical experiment of
figure 1. Horizontal axis indicates the position along the chain, vertical axis corre-
sponds to the time (time is going downward). The energies and velocities of atoms
are denoted as short vertical lines with the length proportional to the magnitude
(positive direction is downward).
110
Kinks in nonlinear acoustic chain
Figure 4. Interaction of the moving kinks.
111
L.S.Metlov, Yu.V.Eremeichenkova
Figure 5. Parameter of energy concentration via the time (black line). White line
is average value 〈C0〉.
Figure 6. The form of moving kink from
figure 3. Time step is equal to 0.022 T0.
approximation [4], and C0 = N if all the
energy is concentrated on one site only.
Parameter C0 indicates energy concen-
tration in all the solitons existing simul-
taneously, but speaks nothing about the
number of solitons and their individual
contributions. In figure 5, the C0 via the
time is given for the chain after thermal-
ization. For the time interval under con-
sideration, C0 oscillates around its av-
erage value in the frames 1.7–3.3, and
it does not exceed 1.5–4.2 during whole
simulation time (t = 106 T0). It follows
from this oscillating behavior of the C0
that the probabilities of kink’s growing
and decay are equal.
Different periodic components of
C0(t) are separated by fast Fourier
transform which is performed taking
8192 time points into account. To sup-
press weak high-frequency fluctuations
of the C0 caused by thermal motion of
112
Kinks in nonlinear acoustic chain
Figure 7. Fourier spectrum amplitude C0(t). Curve 1 is the spectrum smoothed
using 5 points; curve 2 is the spectrum smoothed using 50 points.
atoms, initial time dependence of C0 was smoothed by adjacent averaging using
25 time points. Amplitude Fourier spectrum smoothed using 5 frequency points is
plotted in figure 7 (curve 1). A number of considerable peaks are observed at the
frequencies 10−5 − 10−3 (T0)
−1. However, the width of these peaks is compatible
with the distance between them, and each peak by itself is seen to be meaningless.
Thus, one can evaluate maximal frequency (minimal period) of long-time oscilla-
tions of C0(t) only. For convenience we smooth a frequency dependence of Fourier
amplitudes using a larger number of points (50 points, the curve 2 in figure 7).
We define maximal frequency as a frequency at which Fourier amplitudes drop two
times. Then, minimal period of C0(t) oscillations is evaluated to be ∼ 1000 sT0,
which agrees with the lifetime of most intensive kinks defined directly using maps
of the tracks.
Numerical simulations show a number of kinks existing simultaneously in the
chain. Some of them persist at the stage of growth and the others appear to be
at the stage of decay. Moreover, there are long-time fluctuations of a number of
kinks which correlate with the fluctuations of energy concentration parameter C0:
the larger is the magnitude of C0 the smaller is the number of kinks2. For example,
for a strong concentration of energy (C0 = 3.7) corresponding to figure 3 a single
kink runs along the chain. The number of kinks practically remains unchanged when
enlarged amounts of energy of 300 K/atom and 1000 K/atom are initially fed into
the chain, while the energy concentrated in each kink enhances.
2Similar correlation was observed on the acoustic chain with Lennard-Jones potential [23].
113
L.S.Metlov, Yu.V.Eremeichenkova
In conclusion, there is an hierarchy of excitations in the chain after thermaliza-
tion, starting with the strongest well-localized and long-lifetime kinks and finishing
with phonons. Weak excitations appear to be the background with which strong
kinks interact. Time after time strong kinks emerge spontaneously from the back-
ground. For certain time each kink exhibits systematic tendency to growth, which
changes by stable tendency to decay. The behavior of an ensemble of such kinks
leads to an oscillating time dependence of the energy concentration parameter, and
all the properties averaged during a long enough time appear to be steady.
3.2. White noise
Although there is a stage of chaotic motion of the atoms after a decay of the
initial mode, the question remains about the effect of initial conditions on the form
of the created solitons. To exclude this effect we use white noise as initial conditions.
The system will choose the most appropriate solitons.
At the beginning of the simulations, the atoms move chaotically, but the distri-
bution E(ε) differs a bit from the Gibb’s one. The process of energy concentration
starts immediately. After ≈ 80 T0 the E(ε) agrees with the Gibb’s one. Well local-
ized kinks are created since ≈ 400 T0. Their form and properties are identical to the
ones for the kinks arising after π-mode decay.
Thus, whatever initial conditions, the acoustic chain with realistic Exp-6 po-
tential comes to the same state characterized by a finite number of long-lifetime,
high-energy supersonic moving kinks.
3.3. Shock waves
The form of spontaneously emerging moving kinks is similar to the form of
shock waves. We adopt shock waves as initial conditions, which permit to set the
amplitudes of the kinks in a controlled manner.
Each initial shock wave including three atoms instantly splits into three kink-like
excitations moving with the velocities of 4.5–5 vs; C0 = 17.4. At each moment of
time they are localized on two sites only. Kink-like excitations slowly lose the energy.
Up to ≈ 1100 T0 they become slightly delocalized in space: two, three or more atoms
moving in the same direction are contained in the kinks. The parameter C0 drops
to 3.1, kink velocities decrease to 2.3–2.8 vs. So, quantitative characteristics of shock
waves become compatible with the ones for spontaneously created kinks.
Numerical experiments on the shock waves with different initial amplitudes show
that the more energy is stored in the wave the longer its lifetime is.
We consider the propagation of the same shock waves in the integrable Toda
chain where non-topological moving kinks are exact solutions. Each initial shock
wave splits into three kinks with the form, velocities and parameter C0 similar to
the ones for Exp-6 potential. Afterwards these kinks transfer a bit of the energy
to the background atoms, and, further on, neither the form nor the energies of the
kinks are altered to the end of simulation (t = 28000 T0).
114
Kinks in nonlinear acoustic chain
3.4. Exact breathers
It was found that in monoatomic acoustic chains with realistic Lennard-Jones
and Morse potentials breathers are not created [6]. We have performed numerical
simulations on the chain with Exp-6 potential with initial displacements of atoms
in the form of single odd-parity breather placed in the center of the chain. For the
time less than T0, the breather decays into two kinks with equal energies moving in
opposite directions. Afterwards, shock wave scenario is realized. Kink-like excitations
are slightly delocalized in space, and at up to 5000 T0, the parameter C0 and kink
velocities decrease and become compatible with the characteristics of spontaneously
created kinks. Further, each kink grows and decays as it was described above. The
simulations with the even parity breather give similar results.
Instability of breathers in the chain with Exp-6 potential is caused by strong
asymmetry of the potential. Inside the breather, one of interatomic bonds is com-
pressed and its potential energy increases. If the potential were symmetrical, the
increase of the potential energy would be equal when the atoms both approached
each other and removed. So, if the slope of potential curve were large enough, the
pair of atoms would vibrate almost like the isolated bound state as it does in FPU
chain. In Exp-6 potential, the attractive part is much weaker than the repulsive
one, and the atoms, if approached, are removed almost freely. They collide with the
neighboring atoms and transfer a whole momentum to them. As a result, two kinks
run along the chain in opposite directions.
3.5. Power potentials
The form and some properties of moving kinks in the chain with Exp-6 poten-
tial are similar to the ones of the solutions of Bussinesk or KDV equation. These
equations can be obtained from equation 1 in continual approximation if only cu-
bic anharmonism is taken into account. This similarity would suggest that high
order anharmonisms could be neglected. To clarify the role of different order an-
harmonisms we perform simulations for the chains with the potentials obtained as
power expansion of Exp-6 potential (equation 3).
The following cases are considered (initial conditions – π-mode).
• Harmonic and cubic terms are only taken into account in equation 3 (2–3
potential). π-mode decays, the solitons appear and collapse immediately.
• 2–4 and 2–3–4 potentials. π-mode decays by modulational instability, and the
breathers occur in agreement with [1–6]. Maximal velocity of the breather is
∼ 0.1 vs, the lifetime is ≈ 120 T0 (2–4 potential) and ≈ 50 T0 (2–3–4 poten-
tial). Unlike the kinks, breathers strongly interact with each other. After the
breathers decay, the movement of atoms becomes chaotic, energy distribution
agrees with the Gibb’s one. Afterwards, kink-like solitons are spontaneously
created in the chain3. However, lifetime of these kinks is very short ( ≈ 50 T0),
3Numerical simulation shows kink-like solitons to arise in the chains with 2–4 and 2–3–4 poten-
tials regardless of initial conditions.
115
L.S.Metlov, Yu.V.Eremeichenkova
indicating that both kinks and breathers strongly interact in the chains with
2–4 and 2–3–4 potentials.
• 2–3–4–5–6 and 2–3–4–5–6–7–8 potentials. π-mode is stable.
• 2–3–4–5–6–7–8–9–10 potential. π-mode decays through the period-doubling
instability. Solitons identical to the kinks in the chain with Exp-6 potential
are created (with long lifetime and weak interaction).
Thus, in the chain with 2–3 potential, the nonlinear term is not balanced by the
dispersion term (at least from the energy 90 K/atom), and the solitons collapse.
Cubic anharmonism added to 2–4 potential give quantitative changes only in the
picture of soliton formation. The 5–6 and 7–8 powers stabilize π-mode. Higher (9–
10) powers give the change of the type of instability, and the properties of the
kinks coincide with the ones in realistic Exp-6 potential. So, quantitatively small
corrections give a qualitative change in the behavior of the atoms, and realistic
interatomic potentials should be used to avoid such difficulties in real crystals.
4. Summary
The behavior of nonlinear acoustic chain with realistic Exp-6 potential is inves-
tigated. Regardless of the type of initial conditions, moving non-topological kink-
shaped solitons occur spontaneously after thermalization of the system. These kinks
are one-parameter solitons with the velocity as a parameter. They weakly inter-
act with each other and pass through each other without loss of their individuality
like solitons in integrable systems. Travelling along the chain, each kink, firstly, col-
lects the energy from the background atoms, and, afterwards, transfers it to small-
amplitude phonons. Dynamic equilibrium between the processes of kink growth and
decay takes place. So, there is a finite number of high-energy excitations in the sys-
tem at each moment of time. Time intervals at which the most part of the energy
is concentrated in the kinks are repeated periodically. Average lifetime of a kink is
estimated to be ∼ 1000 T0.
Acoustic chains with power interatomic potentials are investigated adding higher
powers of atom displacements consequently up to ten powers. In the chains with 2–
4 and 2–3–4 potentials, discrete breathers are observed in agreement with [1–6].
After the breathers decay, thermal equilibrium is achieved and kink-like solitons
also appear in these chains. However, these kinks strongly interact with each other
and have short lifetime (50–120 T0). Thus, unlike the breathers resided only in
power potentials, the kinks can be created both in power and in realistic potentials
whatever initial conditions may be.
Molecular dynamics study of π-mode stability in the chains with power potentials
shows that the type of instability is changed (from modulational to period-doubling)
when high (9 and 10) powers are added. It indicates that an extreme caution is
required if power expansions of interatomic potentials are used in real crystals.
116
Kinks in nonlinear acoustic chain
Thus, in thermal equilibrium, long-lifetime high-energy supersonic moving kinks
appear in an acoustic chain with Exp-6 potential at high temperatures. One can
expect that similar excitations can exist in the three-dimensional case and produce
a considerable enhancement of thermal conductivity at high temperatures as well as
effect the long-time strength of solids.
References
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117
L.S.Metlov, Yu.V.Eremeichenkova
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