Soliton trains in dispersive media
In this paper two Boussinesq-type mathematical models are described which lead to solitonic solutions. One case corresponds to microstructured solids, another case to biomembranes. The emergence of soliton trains in both cases is demonstrated by using numerical simulation. The pseudospectral method...
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irk-123456789-1762022021-02-05T01:30:06Z Soliton trains in dispersive media Engelbrecht, J. Peets, T. Tamm, K. Свойства нелинейных биофизических систем со сложной внутренней структурой In this paper two Boussinesq-type mathematical models are described which lead to solitonic solutions. One case corresponds to microstructured solids, another case to biomembranes. The emergence of soliton trains in both cases is demonstrated by using numerical simulation. The pseudospectral method guarantees the high accuracy in computing. The significance of the nonlinearities — either deformation-type or displacement-type, is demonstrated. 2018 Article Soliton trains in dispersive media / J. Engelbrecht, T. Peets, and K. Tamm // Физика низких температур. — 2018. — Т. 44, № 7. — С. 887-892. — Бібліогр.: 34 назв. — англ. 0132-6414 PACS: 46.40.Cd, 47.35.Fg, 47.54.Fj http://dspace.nbuv.gov.ua/handle/123456789/176202 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свойства нелинейных биофизических систем со сложной внутренней структурой Свойства нелинейных биофизических систем со сложной внутренней структурой |
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In this paper two Boussinesq-type mathematical models are described which lead to solitonic solutions. One case corresponds to microstructured solids, another case to biomembranes. The emergence of soliton trains in both cases is demonstrated by using numerical simulation. The pseudospectral method guarantees the high accuracy in computing. The significance of the nonlinearities — either deformation-type or displacement-type, is demonstrated. |
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Soliton trains in dispersive media |
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Soliton trains in dispersive media |
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Soliton trains in dispersive media |
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soliton trains in dispersive media |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Свойства нелинейных биофизических систем со сложной внутренней структурой |
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Soliton trains in dispersive media / J. Engelbrecht, T. Peets, and K. Tamm // Физика низких температур. — 2018. — Т. 44, № 7. — С. 887-892. — Бібліогр.: 34 назв. — англ. |
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Физика низких температур |
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AT engelbrechtj solitontrainsindispersivemedia AT peetst solitontrainsindispersivemedia AT tammk solitontrainsindispersivemedia |
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Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7, pp. 887–892
Soliton trains in dispersive media
Jüri Engelbrecht, Tanel Peets, and Kert Tamm
Laboratory of Solid Mechanics, Department of Cybernetics, School of Science, Tallinn University of Technology
Akadeemia tee 21, Tallinn 12618, Estonia
E-mail: je@ioc.ee, tanelp@ioc.ee, kert@ioc.ee
Received December 19, 2017, published online May 28, 2018
In this paper two Boussinesq-type mathematical models are described which lead to solitonic solutions. One
case corresponds to microstructured solids, another case to biomembranes. The emergence of soliton trains in
both cases is demonstrated by using numerical simulation. The pseudospectral method guarantees the high accu-
racy in computing. The significance of the nonlinearities — either deformation-type or displacement-type, is
demonstrated.
PACS: 46.40.Cd Mechanical wave propagation (including diffraction, scattering, and dispersion);
47.35.Fg Solitary waves;
47.54.Fj Chemical and biological applications.
Keywords: dispersion, nonlinearities, microstructure, biomembranes, solitons.
To the memory of A.M. Kosevich
1. Introduction
The celebrated wave equation is one of the classical
equations of mathematical physics and describes the mo-
tion of a wave with a constant speed. For many practical
applications this model must be generalised. One of such a
generalisation in conservative media is called after
Boussinesq, who derived this model for surface waves on a
fluid layer [1,2]. Nowadays the Boussinesq-type models
are widely used also in solid mechanics [3]. In brief, such
models are (i) bi-directional (including the d’Alembert
operator); and in addition (ii) include nonlinear terms (of
any order); (iii) include higher order terms (the presence of
space and time derivatives of the fourth order or higher)
describing the dispersive effects [3]. In general terms, the
Boussinesq equation may be presented in a following form:
2
0 ( ) = ( ),tt xxu c u N u D u− + (1)
where u is the displacement, 0c is the velocity and indices
,x t here and further denote differentiation. Operator ( )N u
expresses nonlinear effects
2 3( ) = ( , , , , )xN u N u u uu (2)
and ( )D u describes dispersive effects
4 4 2 2 6( ) = ( , , , , ).x t x t xD u D u u u u (3)
There are many studies of this type of equations derived
using various physical assumptions [3–7]. Attention is paid
to the mathematical correctness of models in the sense of
Hadamard, i.e., establishing whether the initial value prob-
lem is well-posed or ill-posed [3,5]. In physical terms, the
Boussinesq-type models describe waves in crystals [5,8],
longitudinal waves in rods [6,9], waves in microstructured
solids [10,7], waves in biomembranes [11], etc.
The most remarkable phenomenon resulting from using
models where nonlinearity and dispersion are both taken
into account, is the possibility of existence and/or emerg-
ing of solitons. Many studies are devoted to the special
type of dispersion when ( ) = xxxxD u u . This is typical for
cases when the governing equation is derived on the basis
of lattice dynamics [5]. The “well-posedness” of such a
model is analysed in detail [12,13].
The Boussinesq-type equations which actually model
weak dispersion, are not the only ones able to describe
solitons, the sine-Gordon equation, for example, is able to
model solitons and bound-soliton complexes [14] emerging
in ferromagnets. The combination of the sine-Gordon and
the Boussinesq-type equations permits to analyse the dislo-
cation (crowdion) motion in crystals [15]. This model is
2sin 0,tt xx x xx xxxxu u u u u u− + − γ −β = (4)
where γ and β are the physical parameters. Equation (4) is
nowadays called the Kosevich–Kovalev equation describing
© Jüri Engelbrecht, Tanel Peets, and Kert Tamm, 2018
Jüri Engelbrecht, Tanel Peets, and Kert Tamm
motion in a strongly dispersive medium. Other generalisations
are possible demonstrating the richness of the model [16]. The
striking duality of solitons and quasi-particles is noted [17].
Based on this brief overview, it is clear that Bous-
sinesq-type equations govern complicated dynamics.
However, one should clearly describe the physics behind
the mathematical models. In what follows, the focus is on
the emergence of soliton trains modelled by the Bous-
sinesq-type equations. The attention will be paid to the
model equations focusing on the influence of various
nonlinearities on the emergence process together with the
fourth-order dispersive terms. The physical background
of models is related to the microstructured media:
microstructured Mindlin-type solids and biomembranes
which possess internal structure. The latter is qualified
also as a microstructure. In Sec. 2 types of nonlinearities
(deformation-dependent and displacement-dependent) are
presented and analysed. Section 3 is devoted to mathe-
matical models with dispersive terms. In this case the
inertia of a microstructure is taken into account which
leads to dispersion operator ( ) = ( , )xxxx xxttD u D u u . This
means that in mechanics of microstructured solids disper-
sion is more complicated than proposed in lattice dynam-
ics [5]. The emergence of soliton trains for two model
cases is analysed in Sec. 4. Finally, the discussion and
conclusions are presented in Sec. 5. The results obtained
earlier are here analysed from a unified viewpoint.
2. Nonlinearities
In mechanics of solids the nonlinearities are caused by
physical and geometrical effects (see [18]). According to
the conventional continuum theory [19], the physical non-
linearity means the nonlinearity of the stress-strain rela-
tions and the geometrical nonlinearity is related to the fi-
nite deformations, i.e., to the strain tensor. The free energy
function (potential) in terms of the strain tensor is then
presented in the form including beside quadratic terms also
the higher order terms. Such a form reflects better the
shape of the potential in terms of forces between the atoms
in the discrete lattice. Note that as far as the stress tensors
are determined by the derivatives of the potential with re-
spect to the components of the deformation tensor, the
quadratic terms lead to the conventional linear theory [19].
The first approximation of a nonlinear stress tensor in-
cludes the quadratic polynomial of displacement gradients.
For example, in the one-dimensional case the Kirchhoff
stress tensor is [18]
2
11 1 2 3
1= ( 2 ) 3 3 3 ,
2x xK u u λ + µ + λ +µ + ν + ν + ν +
(5)
where λ, µ are the elastic constants of the second order
(the Lamé parameters) and 1ν , 2ν , 3ν are the elastic con-
stants of the third order. It means that the velocity is de-
termined by the relation
2 2
0= (1 ),xc c ku+ (6)
where 0c is the velocity in the unperturbed state and k is
the constant of nonlinearity which according to expression
(5) is 0= 3(1 )k m+ , 1
0 1 2 3= 2( )( 2 )m −ν + ν + ν λ + µ .
Such a situation may lead to the formation of shock
waves, i.e., the discontinuities of the solution [18,20].
More complicated free energy potentials lead certainly to
more complicated mathematical models [18] but the situa-
tion described briefly above is the fundamental case of the
nonlinear wave motion in solids.
In biological tissues and cells, Exp. (6) might be differ-
ent, especially when the discreteness of structures is taken
into account. Such a situation is the case of biomembranes
which are built by bilayers of lipid molecules. Based on
experimental observations, Heimburg and Jackson [21]
have proposed for longitudinal waves in biomembranes
2 2 2
0= ,c c pu qu+ + (7)
where = Au ∆ρ , Aρ is the density and p, q are experimen-
tally determined constants.
It means that contrary to solids with deformation-
dependent (ux) nonlinearities, for biomembranes the gov-
erning wave equation based on expression (7) includes
nonlinearities in terms not ux but simply u, i.e., the nonlin-
earities are of displacement-type (see below).
3. Mathematical models for dispersive waves
In general, dispersion of waves is the separation of
waves into constituents of different wave-lengths and may
be caused by either geometrical or physical effects. The
geometrical dispersion takes place in waveguides due to
the influence of the existence of lateral surfaces [9]. The
physical dispersion in solids is caused by the existence of
the microstructure of the material [5,7]. In the first case
dispersion depends on the transverse dimensions of wave-
guides, in the second case — on the scale effects.
Here we present two nonlinear mathematical models
where the physical dispersion is of importance. Both mod-
els are of the Boussinesq-type like Eq. (1).
Microstructured solids. In the theory of microstructured
solids [22,23] the behavior of the macro- and microconti-
nuum is described by the separate balance laws. In terms of
macro-displacement u and microdeformation ϕ , the sim-
plest free energy W is governed by a cubic function
2 2 2 3 31 1 1 1= ,
2 2 6 6x x x x xW u A u B C Nu Mα + ϕ + ϕ + ϕ + + ϕ (8)
where α, A , B , C , N , M denote material parameters [10].
The balance laws are derived then from the Euler–
Lagrange equations. Introducing dimensionless variables
0= /X x L , 0 0= /T c t L , 0= /U u U where 2
0 = /c α ρ and 0U
and 0L are certain constants, along with geometrical pa-
888 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7
Soliton trains in dispersive media
rameters 2
0 0= ( / )l Lδ and 0 0= /U Lε , where 0l is the char-
acteristic scale of the microstructure, the governing
Boussinesq-type equation of motion is derived. By using
the slaving principle (see [23]) this governing equation in
terms of deformation ( = XV U ) is [24]
( )
( )
2
2
=
2
= ,
2
TT XX XX
TT XX X X
XX
V bV V
V V V
µ
− −
λ δ
δ β − ν
(9)
where 2= 1 /( )b A B− α , 0 0= /( )NU Lµ α , 2 2 2
0= / ( ),IA l Bβ ρ
2 2 2
0= /( )CA B lν α and 3 3 3
0 0 0= /( )A MU B l Lλ α are con-
stants. Here I denotes microinertia.
This equation has a hierarchical structure [20] — two
nonlinear wave operators (one at the l.h.s., another at the
r.h.s.) describe motion in macro- and microstructure, re-
spectively. In such a way, this is an explicit description of
mechanical waves in microstructured media [7,20] which
takes into account the leading effects. The accuracy of this
approximation is established by the analysis of dispersion
relations of the original and approximated equations and
depends on the ratio of velocities in macro- and micro-
structure [25]. Note also that even without the operator of
the wave motion in the microstructure, the velocity of
waves in the macrostucture is affected (see the structure of
the coefficient b). It is possible to solve the inverse prob-
lem for Eq. (9) in order to determine the values of its coef-
ficients with a suitable accuracy [26].
Equation (9) has soliton-type solutions if the following
condition
32 2
2 2
4>c
c b
β − ν λ
− µ
(10)
is satisfied [26]. Here the velocity of a soliton c is a free
parameter. It is interesting to note that for solutions of the
highly dispersive Kosevich–Kovalev equation (4) there are
also restrictions which govern the existence of soliton
complexes [16]. Such conditions seem to be characteristic
to generalised models.
Biomembranes. These important building blocks for
cells and nerves are built by lipid molecules which have
hydrophobic tails directed inwards [25]. It has been demon-
strated experimentally that such bilayers are able to carry me-
chanical waves [26,27]. The molecules of a bilayer can be
treated as a microstructure and similarly to solids, the inertia
of the microstructure must be taken into account.
The mathematical model for longitudinal waves in
biomembranes including nonlinearity of the biomembrane
expressed by Eq. (7) was derived by Heimburg and Jack-
son [21] and improved by Engelbrecht et al. [28] including
also inertial effects. The governing equation in the dimen-
sionless form is the following:
2
2
1 2
= (1 )
( 2 ) ,
TT XX
X XXXX XXTT
U PU QU U
P QU U H U H U
+ + +
+ + − + (11)
where = /X x l , 0= /T c t l , = / AU u ρ and 2
0= /AP p cρ ,
2 2
0= /AQ q cρ , 2 2
1 1 0= /( )H h c l , 2
2 2= /H h l . Note that
= Au ∆ρ and l is a certain length (in case of a nerve fibre it
is the fibre diameter). The constants 1h and 2h are disper-
sion parameters reflecting the elasticity and inertia of the
structure, respectively.
The accounting of inertia (term XXTTU ) means that the
propagation velocity is bounded for higher frequency har-
monics [28]. Moreover, neglecting this term, i.e., the pres-
ence only the spatial fourth order derivatives in Eq. (10)
can lead to instabilities of the solution [5].
4. The emergence of soliton trains
Both mathematical models presented in Sec. 3 are spe-
cific cases of the Boussinesq-type Eq. (1) including non-
linear and dispersive terms. As well it is known, under the
certain conditions the nonlinear and dispersive effects
could be balanced resulting in solitons. Since the pioneer-
ing studies of Zabusky and Kruskal [29] much attention is
paid to the emergence of solitons and soliton trains from an
Fig. 1. (a) 2sech -type displacement U (solid line) and defor-
mation XU (dashed line), (b) 2sech -type deformation XU
(dashed line) and displacement U (solid line).
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 889
Jüri Engelbrecht, Tanel Peets, and Kert Tamm
arbitrary input. Here we demonstrate main features of the
emergence processes for both governing equations presented
in Sec. 3. Note that there is a fundamental difference in solu-
tions of these equations. Equation (9) describes the defor-
mation while Eq. (10) describes the displacement. If we con-
sider soliton-type solutions for both equations then there is a
significant difference in displacements and deformations. A
single pulse of a displacement means actually a sign-changing
(bipolar) profile of a deformation (Fig. 1. (a)) and a single
pulse of a deformation means a change of the displacement
from one niveau to another (Fig. 1. (b)) [30].
The following results are obtained by numerical simula-
tion using the pseudospectral method (see [31]) which
gives high accuracy in computing. As far as the Bous-
sinesq-type equations are bi-directional, from a localised
initial input two soliton trains will emerge, one to the right,
another to the left. Such a solution for the Eq. (9), i.e., the
case of a solid with a microstructure is shown in Fig. 2
[24]. The initial input is taken
2
0 0 0( ,0) = sech ( ).V X A B X X− (12)
where A0 is the amplitude, B0 is related to the width of the
pulse and X0 defines the spacial shift of the input.
The coefficients here and further for solutions of Eq. (9)
are taken = 0.7188b , = 1.1395µ , = 0.09δ , = 45β ,
= 9.3867ν , = 1.1470λ .
The number of solitons in a train depends on the energy
of the initial pulse [24]. For example, two cases are shown
in Figs. 3 and 4.
In the case of Eq. (10), i.e., the case of a biomembrane,
the similar situation is observed. Here one should distin-
guish between the normal ( 1 2<H H ) and anomalous
1 2( > )H H ) dispersion. Given the values of < 0P and
> 0Q [21] the following results are obtained [32]. Figure 5
demonstrates the emergence of soliton trains for anoma-
lous dispersion and Fig. 6 — the emergence of soliton
trains for normal dispersion.
The significant difference between the cases demon-
strated above is the structure of a train. For microstructured
solids the trains follow the conventional structure — the
larger the amplitude of a soliton, the faster it propagates
[29]. For biomembranes, however, given the values of non-
linear coefficients ( < 0P and > 0Q ) and anomalous dis-
persion, the outcome is different. In a train smaller solitons
move ahead and larger solitons follow with a smaller speed.
The comparison of two cases is demonstrated in Fig. 7.
5. Summary
The emergence of soliton trains is demonstrated for two
cases of nonlinear and dispersive operators in the governing
equations of the Boussinesq type for weakly dispersive
media: microstructured solids and biomembranes. The
Fig. 4. Solutions of Eq. (9) in case of 0 = 1A , 0 = 0.05B at dimen-
sionless times = 2230T (dashed line) and = 7210T (solid line).
Fig. 5. Solutions of Eq. (10) in case of = 0.2186P − , = 0.0043Q ,
1 = 0.072144H , 2 = 0.001H at dimensionless times = 15042T
(dashed line) and = 98001T (solid line).
Fig. 3. Solutions of Eq. (9) in case of 0 = 1A , 0 = 0.01B at
dimensionless times = 2230T (dashed line) and = 7000T
(solid line).
Fig. 2. Formation of trains of solitons from pulse-type initial
condition for Eq. (9). Right- and left-going structures are plotted
at every = 2000T∆ . For details see [24].
890 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7
Soliton trains in dispersive media
dispersive effects in both cases are caused by the
embedded microstructures while the competing nonli-
nearities are different. For microstructured solids, the non-
linearities are of the conventional deformation-type terms but
for biomembranes due to the structure of bi-layers, the
nonlinearities in terms of the governing wave equation are of
the displacement-type. The corresponding governing
equations allow variation over a wide range of values which
results in the changes of nonlinear dynamics. The analysis
demonstrates that both models permit the emergence of
soliton trains. However, an important problem is to establish
whether the single solitons in trains are ‘pure’ solitons or
not. The studies reveal that the interactions of solitons
obtained in both cases are not fully elastic. Referring to
earlier studies out of scope of this paper [11,24], we note
here that during the interactions the radiation occurs
demonstrating the ‘inelasticity’ of interaction processes. So,
strictly speaking, one should call the observed entities
‘quasi-solitons’ which is characteristic in many solitonic
systems with inelastic interactions [3].
Acknowledgements
This research was supported by the European Union
through the European Regional Development Fund (Esto-
nian Programme TK 124) and by the Estonian Research
Council (projects IUT 33-24, PUT 434).
________
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Fig. 6. Solutions of Eq. (10) in case of = 0.2186P − , = 0.0043,Q
1 = 0.072144H , 2 = 0.1H at dimensionless times = 21207T
(dashed line) and = 98001T (solid line).
Fig. 7. (a) soliton train for Eq. (9); (b) soliton train for Eq. (10);
arrows mark the direction of propagation.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 7 891
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1. Introduction
2. Nonlinearities
3. Mathematical models for dispersive waves
4. The emergence of soliton trains
5. Summary
Acknowledgements
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