solitons in elastic solids
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Solitons in elastic solidsTRANSCRIPT
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342 G.A.Maugin / Mechanics Research Communications38 (2011) 341349
without sensible alteration in its essential properties, amplitude
and speed. As a matter of fact, an appropriate relationship between
these last two due to a strict compensation between the effects
of nonlinearity (steepening the wave) and dispersion (spreading
the signal), was shown to be responsible for the effect, that was
to remain a scientific curiosity for more than a half century. Two
now celebrated equations were involved, the Boussinesq (BO)
equation and the Kortewegde Vries (KdV) equation. Now back to
Princeton in the 1960s. A group of scientists prominently amongthem, Martin D. Kruskal paid an almost undue attention to some
wavelike phenomena in conjunction with their plasma studies
and pioneering numerical experiments carried in Los Alamos
in the 1950s (the famous FermiPastaUlam experiment). They
noticed soon with the KdV equation (a one-directional equation
or evolution equation associated with the true wave equation
of Boussinesq) that the strange propagating shape observed
by Scott-Russell shared a property in common with particles in
elastic interactions: during such an interaction head-on collision,
overtaking another fellow they conserved their individuality
but for a change of phase. This particle-like property led to the
coinage of solitons by the facetious M.D. Kruskal and friends
in 1965 (Zabusky and Kruskal, 1965). In a few years they found
the way to generate analytically these muliple-soliton solutions
(the inverse-scattering method; Gardner et al., 1967), and other
remarkable nonlinear and dispersive partial differential equations
were found to exhibit solutions of the same kind. It was no longer
possible to be satisfied with pure hyperbolic systems. Recom-
mended books on the physics and mathematics of soliton theory
are by Drazin and Johnson (1989), Infeld and Rowlands (1990),
Newell (1985), and Ablowitz and Segur (1981).
[It is in these circumstances that the author participated in his
first international conference at Princeton in October 1968, sitting
next to Gerald B. Whitham (from Caltech) who was to publish
soon an influential book on hyperbolic and dispersive systems
(Whitham, 1974).] (I had in fact worked on a paper of this sci-
entist for my memoir of D.E.A. at the University of Paris 6 in the
midst of the somewhat agitated May 1968, and I had been admit-
ted to Whithams Department at Caltech, but I preferred Princeton,a choice I never regretted.) One always recollects with some nos-
talgia the famous Applied Mathematics Colloquium organized by
Martin Kruskal every Friday afternoon, followed by a table-tennis
tournament in the Astronomy building, and working on the solu-
tions of Martin Gardners problems from the Scientific American,
just to finish the working week in good spirits, while Mrs Kruskal
was busy with her origami. I was later to lecture two times at this
Colloquium. Anyway, most research on solitons at that period was
devoted to thefieldsof fluid mechanics, optics, and plasmaphysics,
and to mathematical methods. I became at the time a specialist in
relativistic continuum mechanics (anotherinfluence of the Prince-
tonatmosphere) and a specialistin solidmechanics endowed with
a physical microstructure (the influence of A.C. Eringen) such as in
certain electromagnetic bodies. Using this bias, I was to return tosolitons some ten years later, with one of my Ph.D. students, Jol
Pouget.
The fiery FrenchmanGerard
Reacts to name-slights with en garde
For no insults more drastic
Than an onomastic
Crystal-elastic canard!
MartinD. Kruskal, co-inventor of the soliton andamateur of
limericks; Princeton, October 22, 1984
2. The early introduction of solitons in deformable solids
The mechanics of deformable solid bodies is always more com-
plicated, if not more difficult, to deal with than that of fluids. The
reason for this is multi-fold. But that may explain why nonlinear
waves in general, and solitons in particular, entered that field after
some delay. One of the reasons was that in contrast with fluid
mechanics many specialists of solid mechanics, although dealing
with difficult boundary-value problems, do not deal with nonlin-
earities. It is only with the consideration of physical nonlinearity
in crystals and the phenomenon of plasticity (due ultimately to
the presence of structural defects such as dislocations), that true
nonlinear problems started to appear. The crystalline aspectis tan-tamount to looking at a discrete description. But discreteness is
synonymous with dispersion since a characteristic length then is
necessarily involved. Along this line of thought one must recall
two remarkable works. One is by Frenkel and Kontorova (1938) in
Leningrad, when these authors conceived of a dislocation motion
as the strongly localized solution exhibited by a chain of mass
particles so-called atoms in lattice dynamics connected by
linear springs but placed in a periodically varying external field
(a substrate or a foundation representing the action of neigh-
bouring parallel chains). With appropriate normalization, and in a
continuum long-wave length limit, the relevant partial differen-
tial equation for an elastic displacement noted reads (here thecharacteristic speed is normalized to one)
2t2
2x2
sin = 0, (2.1)
where both nonlinearity and dispersion are contained in the sin
term.This apparently innocuousequation wasto have a remarkable
destiny. It is only now that we call it the sine-Gordon (SG) equation
by imitation with the KleinGordon (KG) linear equation of atomic
physics. Eq. (2.1) presents kink solutions that have a topological
nature: theamplitude of thesolution(jump betweenthe twovalues
onthe two sides ofthe kink) isfixedandthespeed ofpropagation is
not analytically related to this amplitude. As a matter of fact, such
a (subsonic) solution exists even at rest (the relationship of(2.1)
with the nonlinear pendulum equation is obvious)! Later on most
of the popularity gained by an equation such as (2.1) was in the
domains of magnetism, superconductivity and so-called Joseph-
son junctions, and the dynamics of molecular chains such as DNAchains(Baroneetal.,1971 ). However, remaining in defectivecrystal
physics, another fundamental step was taken due to the ingenuity
of Alfred Seeger in Stuttgart. In his very Diploma (Seeger, 1949)
and then in further papers (Seeger, 1955, 1979), the formidably
well-educated Seeger recognized in (2.1) an equivalent equation
from the differential geometrical theory of two-dimensional sur-
facesof constantnegative curvature, the so-calledEnneperequation
obtained by introducing characteristic coordinates , =x tas
2
= sin . (2.2)
What is remarkable with Seegers remark is that he knew that
Bcklund (1882) had devised analytical means to generate other
solutions if one knew one solution to (2.2), hence to (2.1). Accord-ingly, in modern jargon, we can generate multiple-soliton solutions
to (2.1) if we know a one-soliton solution! Not enough credit is
granted to Seeger for this beautiful uncovery. It was while visit-
ing Seeger in Stuttgart that Wesolowski has shown that Eq. (2.1)
was also governing the torsion of elastic bars with a rectangular
cross section (Seeger and Wesolowski, 1981; Wesolowski, 1983).
Having devoted much work to the construction of contin-
uum models of magneto-elastic and electro-elastic media such
as magnetostrictive elastic ferromagnets and piezoelectric and
electrostrictive elastic materials with the accompanying thorough
study of coupled linear bulk and surface waves with an empha-
sis on effects such as the resonance between modes of different
nature, I naturally became involved in the study of the dynamics
of domain walls. Through these walls one witnesses finite rota-
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tion/precession of magneticor electric dipoles. This, in the absence
of couplings withmechanicaldegrees of freedom, naturallyleads to
theintroduction of thesine-Gordon equationto describethe special
casesof transition through a wall, e.g., Blochtype (out of plane rota-
tion around a direction orthogonal to the wall) and Nel types (in
plane rotation effected in a plane containing the directionorthogo-
nal to the wall). The coupling of these orientational effects of walls
with the elastic properties of the relevant crystal was to bring me
back to soliton theory but in a field not as much cultivated as fluidmechanics.
Here we must distinguish between three different kinds of sit-
uations:
(i) the sine-GordondAlembert system and themechanics of fer-
roc states;
(ii) the BoussinesqKdV paradigm for purely mechanical effects;
(iii) the generalized nonlinear Schrdingermodel (small amplitude
surface waves on solid structures).
These are going to be examined successively.
3. Going to solitons in deformable solids via the
sine-Gordon equation: ferroc states
Ferroc states are, by analogy with well known ferromagnetism,
states of matter where the usually considered effect may be present
without a cause. For example, a local magnetization can exist in a
small region of a ferromagnet while there is no applied magnetic
field. Ferroelectricity, named by analogy withferromagnetism, also
exhibits local permanent electric dipoles in the absence of applied
electric field. Furthermore, there even exists ferro-elasticity in
materials exhibiting local spontaneous strains (Aizu, 1970). Here
we consider the first two cases.
There is a natural tendency among applied mathematicians to
passover or ignore thederivationfrom firstprinciplesof basicequa-
tions to be considered as mathematical objects, their primary
interest.This last task may be a difficultone in continuum mechan-ics. Fortunately, W.F. Brown Jr, H.F. Tiersten and the author have
constructed in the 1960s1970s a sound phenomenological theory
of finitely deformable ferromagnets, from which one can deduce
in confidence easily exploitable equations for nonlinear dispersive
wave propagation processes (see,e.g., Maugin, 1988). A typical sys-
tem of partial differential equations obtained in the configuration
of so-calledNel walls in magnetoelasticferromagnets readsas fol-
lows in an obvious notation for partial space and time derivatives
(Maugin and Miled, 1986a):
tt xx sin = ux cos , utt c2Tuxx = (sin )x. (3.1)
Thiscouples,via magnetostriction (of coefficient ), the in-planerotation angle of magnetic spins with the transverse elastic dis-
placement u. Just like for Eq. (2.1) the essential nonlinearity anddispersion are contained in the spin equation while the elastic
equation remains linear although coupled to the second equa-
tion. This has a drawback. While the uncoupled equation for is exactly integrable and exhibits true multiple soliton solutions,
the displacement equation has the pure standard wave nature and
will in fact destroy the exact integrability of the system, which
becomes then one example of such nonexactly integrable systems
justified by real physics. Kivshar and Malomed (1989) coined the
name ofsine-GordondAlembert(SGdA) systems for systems such
as (3.1) and established a durable contact, and later co-operation
with the author. Pougetand the author introduced a similar system
in the physics of elastic ferroelectrics (Pouget and Maugin, 1984),
studied the multiple soliton solutions and the accompanying
wave radiation generated by the coupling with the displacement
wave equation (Pouget and Maugin, 1985a), and also the tran
sient motion of such a wavelike phenomenon under the action o
an applied external field (Pouget and Maugin, 1985b) by mean
of a perturbation method applied to the canonical conservation
laws associated with that system. Later on, it was shown tha
the mechanics of deformable bodies endowed with an interna
degree of freedom of the rotational type (so-called micropolar, ori
ented,or Cosserat continua)are likelyto yield systems such as (3.1
prone to developing close to soliton solutions, but still with somgenerated radiations in the intercourse between several signal
(cf. Maugin and Miled, 1986b; Pouget and Maugin, 1989). Mor
problems including the effect of the application of an externall
alternating field (Sayadi and Pouget, 1991) and the transition t
chaos (Sayadi and Pouget, 1992) have been expertly treated b
Sayadi and Pouget. Much more later, in collaboration with th
Kosevich group originally the Landau Institute from Kharkov
(Ukraine)who had already dealt at length with many aspects of th
sine-Gordon equation in ferromagnetism (see the book by thes
authors; Kosevich et al., 1988), it was possible to show that on
can improve on both dispersion and nonlinearity of the system
based on the sine-Gordon equation (by adding appropriate new
terms) andstill keep the essential solitonic properties, andcreat
ing thus new soliton complexes (Bogdan et al., 1999, 2001). It i
impossible to cite here the extremely rich bibliography about th
sine-Gordon equation and its generalizations (see Chapter 7 and
more particularly Section 7.8 in our book; Maugin, 1999).
4. TheBoussinesqKdV paradigm
Eventually (see Section 6 below), system (3.1) could be viewed
as a two-degree of freedom elasticity system placed on a foun
dation (external force field) affecting only the degree. But thiis just for the commodity of some computations. What we wan
to consider now are purely elastic systems. Nonlinearity can b
introduced via a potential of interactions (physical nonlinearitie
in crystals). As to dispersion, the other necessary ingredient fo
the existence of solitary waves, it can be introduced through different paths, all introducing one or several characteristic lengths
e.g., discrete description (like in a lattice), object of finite thicknes
(thin film glued on top of a body, finite transverse size of a wav
guide) introduced in the system. We focus here on the first case o
which the theory can be traced back to Boussinesq who obtained
the relevant bi-directional wave equation, not only for fluids, bu
also for elastic solids (Boussinesq, 1870). Then Korteweg and d
Vries (1895) derived the unidirectional version of the Boussinesq
equation which now bears their name (KdV equation). In modern
terms, the KdV equation is deduced by the method ofreductiveper
turbations. All equations that are extensions of the BO and KdV
equations are said to belong to the Boussinesq paradigm of wav
propagation (Christov et al., 1996, 2007). The standard derivation
of the crystal Boussinesq equation from a discrete lattice is given in
manybooks (e.g., Kosevich,1999;Maugin,1999). In theappropriat
non-dimensionalization, it reads:
utt uxx uxuxx 2uxxxx= 0, (4.1
where is a characteristic length and is a parameter of nonlinearity. A balance between these two effects favours the existenc
of solitary-wave solutions. In the starting lattice the fourth-orde
space derivative follows fromthe considerationof next-neighbour
interactions andnot onlythe immediate neighbours responsible fo
classic elasticity. Consideration of farther neighbours such as next
next ones would yield a stiffer equation with higher order spac
derivatives, e.g.,
utt uxx [F
(u) 2
uxx + uxxxx]xx = 0, (4.2
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Fig. 1. Examples of Kawahara solitons(after Christov and Maugin).
where F(u) is at least a cubic in u. In some cases we can deduce an
equation of the type
utt uxx [F(u) (1utt 2uxx)]xx = 0, (4.3)
where a mixed space-time fourth order derivative can be present
yielding two types of dispersion, in the same way as the equation
for transverse vibrations of elastic beams does.The regularizedlong-
wave BO equation (Benjamin et al., 1972)
utt uxx [F(u)]xx uttxx = 0, (4.4)
belongs in the same class. We have also introduced what we called
the MaxwellRayleigh equation (Maugin, 1995)
utt uxx [F(u)]xx (uxx utt)tt= 0. (4.5)
This is an example of so-called double-dispersion equation with
two wave operators as obtained by Samsonov et al. in quasi-one-
dimensional elastic rods (Samsonov, 2001):
utt c21uxx [F
(u)]xx + (utt c22uxx)xx = 0. (4.6)
Here the characteristic length comes from the finite cross section
of the rods.
The unidirectional version of the original Boussinesq equation
(4.1), i.e., the KdV equation, is deduced by the method ofreductive
perturbations as:
vt+ vvx + dvxxx= 0, (4.7)
where d is a dispersion parameter.
The same mathematical method applied to Eqs. (4.2)(4.6) will
yield generalizations of theKdV equations. However, only the orig-
inal Eqs. (4.1) and (4.7) are exactly integrable in the sense of the
mathematical theory of solitons (as shown originally by Kruskal
and his co-workers), the others providing solitary wave solutions
whichdo produce some radiationin thecourse of interactions. This
shows how rare are the more physically based equations that are
true solitonic in their interaction behaviour.
Before concluding this section, several remarks are in order:
(i) Among the most interesting ones from the point of view of
physics is Eq. (4.2), based on a nonconvex elasticity potential
presenting three minima (one austenite and two marten-
sites of opposite shear angle). This allows the reproduction of
the various phase transitions observed between the phases
of such materials as martensitic alloys (Maugin and Cadet,
1991). This followed the static considerations ofFalk (1983)
and the work ofPouget (1988).
(ii) A faithful numerical simulation of stiff partial differential
equations such as (4.2) requires special attention in devising
an appropriate finite-difference scheme. This question was
pondered by Christov and Maugin (1995a).
(iii) Generalized KdV equations and the evolution of soliton sys-tems therein are analysed by various numerical techniques
in a number of papers by Salupere et al. (1994, 1996, 2001,
1997).
(iv) Inclusion of cubic terms in the elastic energyprovides drastic
alterations as shown by Porubov and Maugin (2005, 2006,
2008). In particular, cubic nonlinearity is responsible for the
formation of so-called fat solitary waves.
(v) Two-dimensional (in space) problems of course become rel-
atively complex. Equations for plates are obtained from
discrete equations for two-dimensional lattices (Collet, 1993;
Potapov et al., 2001) but exhibit a strong phenomenon
of localization, and a strong amplification accompanied by
depressions (Porubov et al., 2004; Porubov, 2003) (see also
Porubovs books (Porubov, 2003, 2009)). Localization andinstability of patterns were also studied by Pouget (1991) in
his 2D modeling of martensitic alloys.
(vi) In recentworks, it was shown in co-operation with Eron Aero
(a pioneer in the generalized continuum mechanics of the
Cosserat type; Aero and Kuvshinskii, 1961) that media with
such an internal structure and elastic models issued from
a purely elastic theory with higher-order nonlinearity may
present localized solutions so that experiments may decide
on the really existing microstructure (Porubov et al., 2009).
(vii) Other generalizations include linear atomic chains account-
ing for both longitudinal and transverse elastic displacement
Fig. 2. Interaction of two counter propagating Kawahara solitons (after Christov et al., 2007).
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Fig. 3. Examples of pterons and nanopterons (afterChristov and Maugin).
(cf. Cadet, 1989; Kosevich, 1999) and diatomic chains
(Pnevmatikos et al., 1986); also truly nonlocal interactions
with an integral over space.
(viii) From the point of view of shapes, the most original onesobtained are the self-similar propagating shapes solutions
of the system mentioned at (i) above (Christov and Maugin,
1995a), theKawahara type of subsonic solutions for system
(4.2) see Fig. 1 of which the nonperfectly elastic interac-
tion of twooppositelytravellingshapes is shownin Fig.2; and
supersonic weakly nonlocal oscillatory shapes with wings
therefore called pterons (Christov et al., 1996) and affec-
tionately nanopterons when the wings are small enough
of which examples are shown in Fig. 3.
5. Surface solitons on deformable structures
We have been concerned by surface waves on solids for quite
a long time. It was therefore natural that we came to ponder theproblem of the existence of solitonic surface waves, in more vivid
words, surface solitons, on solids. Technologically, this was envis-
aged as soon as 1981 (Ewen et al., 1981, 1982). But there was no
deduction from first principles of continuum mechanics or physics.
Some authors were satisfied with a directapplication of a plausible
equation (Boussinesq or KdV) (Bataille and Lund, 1982; Cho and
Miyagawa, 1993) and this provided not even the beginning of a
proof. It seems that the first good sign of a proof was given at a
Euromech Colloquiumthat we co-organized in Nottingham (UK)in
1987 (Parker and Maugin, 1988) in a contribution by Maradudin
(1988), but while providing the incentive for going further only
the part on the production of harmonics was correct. It was due
to the author and his Ph.D. student Hadouaj to give a mathemati-
cally correct proof of the existence of envelope mechanical solitonspropagating on top of a structure made of a nonlinear elastic sub-
strate and of a glued superimposed slow-velocity thin film. Their
proof was based on the following influences and arguments:
the discovery of a new type of transverse surface shear wave by
Mozhaev (1989) due to the nonlinearity; the concept of a material boundary having its own inertia and
elasticity (Murdoch, 1976); the exploitation of the wave-kinematics formalism ofBenney and
Newell (1967) in the study of localized waves (as a matter of fact
the first, and almost unique, application of such a formalism in
2D problems of solid mechanics (Maugin and Hadouaj, 1989)). the combination of a nonlinear elastic substrate (half space) and
an a slow linear-elastic superimposed layer of infinitesimal
Fig. 4. Interaction of two counter propagating surface envelope solitons (figur
shows the square of the envelope at different depths with decreasing amplitud
with depth in the substrate (numericalsimulation by Hadouaj and Maugin, 1992)
thickness (usually allowing for the existence of the Love-type osurface waves).
The result was announced in 1989 (Maugin and Hadouaj, 1989
and details of the proof given in 1991 (Maugin and Hadouaj
1991). What was really proved was the existence of stable prop
agating bright (envelope-type) solitons, providing a mechanica
equivalent to the known optical solitons in optical fibers; henc
the qualification of bright. These have complex small ampli
tude ultimately governed by a cubic Schrdinger (NLS) equation
It is a true surface wave guided by the superimposed film a
the amplitude decreases with depth in the substrate. This, a
well as the quasi-solitonic interactions of two counter propagat
ing such waves were checked numerically (Hadouaj and Maugin
1992) see Fig. 4. Furthermore, the weak coupling of this transverse horizontal shear wave with a Rayleigh wave componen
(in the so-called sagittal plane) was established (Hadouaj et al
1992) yielding a system that we baptized Generalized Zakharo
system (GZS=coupling a NLS equation and a wave equation)
Such systems present interesting peculiarities: existence of a for
bidden window in the range of speeds, new inelastic solitoni
process called perestroika of the solution in a dissipation
induced evolution, collision-induced fusion of subsonic soliton
(Hadouaj et al., 1991a,b).
After the just-mentioned works, a long series of works started
to appear being devoted to Rayleigh solitary waves and exploit
ing different mathematical techniques, but still for a configuration
involving a substrate and a glued lid. Among these works w
may mention those of Kovalev et al. (Eckl et al., 2001; Kovalevet al., 2002a,b, 2003a) including the possibility of spin surfac
waves (Kovalev et al., 2003b) and the consideration of incommen
surate surfaces (Kovalev et al., 2004), and of Gorentsveig et a
(1990) and Porubov and Maugin (2009). The latter have shown
that the obtained exact solutions for longitudinal motion allow
one to describe simultaneous propagation of tensile and compres
sive localized strain waves. Interactions between these waves giv
rise to both the multi-hump and Mexican hat (central peak wit
side depressions) localized wave structures. Recent reviews and
lecture notes on surface elastic solitons are those of the autho
(Maugin, 2005; Maugin, 2007) seeotherreferences therein. Olde
ones devoted generally to nonlinear surface waves but not nec
essarily solitons are by Maradudin and Mayer (1991) and Maye
(1995).
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6. Strange quasi-particles
6.1. Generalities
The notion of soliton itself calls for a more general introduc-
tion of the notion ofquasi-particle. Indeed, if the elastic interaction
between solitons is so true then we must be able to associate a
particle in inertial motion with each of these strongly localized
objects. To do this one must establish a mechanics say of mas-sivepoints thatreproduces thesame characteristics.A convenient
tool exists forthat,at least for exactly integrable equationslikelyto
produce solitons. It happens that such systems of equations admit
an infinite number ofconservation laws (not to be mistaken for the
physical balance equations of continuum mechanics we started
with). These conservation laws in field theory are related to the
existence of an infinity of symmetries, so that Noethers celebrated
theorem (one conservation law for each symmetry) holds good
(cf. Fokas (Fokas, 1979)). In our dynamics these reflect the exis-
tence ofconstants of motion. It happens that the construct yielding
such constants is greatly facilitated for us because all wave equa-
tions considered here are issued from elasticity (with or without a
microstructure), and elasticity is the paragon of field theory when
written in the proper formalism (essentially finite strains where
we distinguish clearly between placement and spatial parameteri-
zation).We have discussedat length the relevantconservationlaws
in that framework (Maugin, 1992; Maugin, 1993).
As a preliminary remark, we note that for a massive point
particle in 1D space, a point mechanics is defined by definite
relationships between four quantities: the velocity c, the linear
momentum p, the (kinetic) energy E and the mass m such that,
in Newtonian mechanics for an inertial motion:
d
dtp = 0, p = mc, E=
p2
2m, (6.1)
where the first is the equation of motionper se. This one remains
valid in standard (LorentzEinstein) relativistic physics while the
last two in (6.1) are replaced by the equations
p = m(c)c, E2 = p2 + m20; m(c) = m0(c),
(c) (1 c2)1/2
, (6.2)
where m0 is the rest mass (independent ofc) and we have set the
limit velocity (usually the light velocity in vacuum) equal to one.
With this convention, we check that E=m0 at rest, the celebrated
Einsteinequationof equivalence between massand energy (usually
written with a conversion factor equal to the squared velocity of
light).
However, it is only our limited imagination and the present
knowledge of physics whichconstrain us to the two examples (6.1)
and (6.2). As we shall see, other point mechanics can be designed
(think ofa rocketseenas a massivepointconsumingitsfuel(mass)).
Now, to proceed with the present problem, the most relevantconservation laws for our systems are the canonical conservation
equations of linear momentum and energy (symmetries related
to translational invariance with respect to the space-time param-
eterization (Maugin, 1993)). These are constructed easily locally,
and then they are integrated along the whole real line R (for 1D
motion). The specific analytic solutions of the soliton type found
for the various equations are then carried in these integral formu-
las, the evaluation of which yields the global equation of inertial
motion at constant energy in the form
d
dtP= 0,
d
dtH= 0, (6.3)
where Pand Hare the global values obtained by integration along
the real line R. With some real luck we have then the expression
ofPand Hwhich compare more or less favourably with those ofp
and E in (6.1) or (6.2) the velocity being known and fixed since
the motion described by the first of(6.3) is inertial.
6.2. Sine-Gordon systems and their generalizations
For instance, for the sine-Gordon equation (2.1) with solutions
of the subsonic kink-like type
(x, t) = 4 tanh1{exp[(x ct)]}, (6.4)
one obtains thus
P(c) = M0c, H 2(c) = P2(c) + M20 , (6.5)
with defined as in the last of(6.2) and M0 = 8 =H(0), so that thequasi-particle associated wit the exactly integrable sine-Gordon
equation satisfies the LorentzianEinsteinian mechanics (6.2) of
point particles. This should not come as a surprise since Eq. (2.1)
itself is Lorentz invariant.
Nowwe easily imagine thatterms added to (2.1) case of the so-
called double sine-Gordon equation and of perturbed sine-Gordon
equations or any further coupling such as in the sine-Gordon
system thatdestroys the exactintegrability, willdrasticallycompli-
cate the matter. Nonetheless, it must be realized that all quantitiesconsidered in the canonical formulation of conservation involve a
summationover contributions of alldegreesof freedom (cf. Maugin,
1993; Maugin and Christov, 2002). Accordingly, as already men-
tioned, system (3.1) is formally viewed as a system for two elastic
components (,u) and the corresponding P is defined canonicallyas
P=
R
(utux + xt) dx, (6.6)
in which must be carried the found soliton-like solution for the
two functions. Since the true displacement component u is only
secondary and generated by the one via magnetostriction, weknown before hand, without producing analytical results, that the
linear momentum of the quasi-particle associated with (3.1) has,
for small magnetostriction coupling the general perturbed form
P= PSG + G(M0, cT; c), (6.7)
where PSG is the LorentzEinstein value for the pure sine-Gordon
equation. We obtain thus a point mechanics that deviates from
the LorentzianEinsteinian one.
6.3. Generalized Zakharov systems
Another example of the same deviation property from a known
classical point mechanics is given by the generalized Zakharov
model obtained in the surface wave problem of Section 5. Indeed,
a remarkable property of the pure NLS (cubic) equation is that itsassociated quasi-particle point mechanics is none other than the
pure Newtonian one (6.1). Accordingly, for the GZ model and its
soliton-like solutions, the above procedure yields a general expres-
sion for Pas
P= PNLS + F(M0, cT; c); PNLS = M0c, (6.8)
in which the summation property over various degrees of freedom
in the canonical momentum has been exploited, and an exact ana-
lytical expression is known for Fon account of the exact known
one-soliton solution to the GZ system for these see (Hadouaj
et al., 1991a). In the present surface-wave problem, the rest
mass M0 is physically interpreted as the total number of sur-
face phonons. The relationship P(c) in (6.8) is not bi-univoque:
the point-mechanics is Newtonian for smallcs, becomes Lorentzian
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from below at a characteristic speed equal to one, is again Newto-
nian at very large speeds; but there exists in between a forbidden
window between value one and another characteristic speed.
Equations such as (6.7) and (6.8), established for inertial
motions, are useful in the perturbation of these motions by exter-
nally applied fields or additional effects, an applied electric field of
magnetic field in the first case, the influence of viscosity in the sec-
ond case of interest, for which one deduces the non-zero value of
dP/dt. The acceleration of soliton solutionsof thesine-Gordon equa-tion was studied by this method by Pouget and Maugin (1985b)
also Sayadi and Pouget (1990). The dissipation (viscosity) induced
evolution of soliton-like solutions of the GZ system was studied by
Hadouaj et al. (1991a) exhibiting interesting drastic phenomena of
reconstruction (perestroika) in the course of propagation.
6.4. BoussinesqKdV systems
It remains to considerthe case of the BoussinesqKdV paradigm
for which invariants of the motion were found quite early (Kruskal
and Zabusky, 1966) for the pure KdV case. In order to include all
cases mentioned in Section 4, we note that the total canonical
momentum in 1D should be defined by
P=
R
ux Lut
dx, (6.9)
where L/utis a functional (EulerLagrange) derivative of the rel-evant Lagrangian density. This allows one to account for strange
cases of inertia such as in Eqs. (4.3)(4.6). In the case of the KdV
systems that have only first-order derivatives in time (cf. Eq. (4.7)),
the canonical definition ofPis recovered by introducing a potential
u such that = uxand Preads (Maugin and Christov, 2002)
P=
R
utuxdx =
R
v
1
2v
2 + dvxx
dx. (6.10)
In thegeneralcase where (6.9) applies, while exactspecialsolutions
of the soliton type can exist, it is not possible to establish analyti-
cal point-mechanics type relations between M, Pand H. However,the wavicle dynamics of Eq. (4.2) is dominated by its pseudo-
Lorentzian (in fact anti-Lorentzian) character. The localized wave
solutions of such an equation have momenta and energies that
finally decrease with an increase in their speed andthey eventually
decay to zero at the characteristic speed. It was possible to estab-
lishby bestnumerical fitting a possiblerelationshipbetweena fixed
rest mass (for which the wave solution exists) and P. For instance
(Christov and Maugin, 1995b)
P= [M0(1 c2)
13/8]c, (6.11)
for monotone sech4-like shapes, and
P= [M0(1 c2)
3/2]c, (6.12)
for Kawahara solitons with oscillatory tails. It would have beenalmost impossible to imagine such point mechanics before hand
(see the graphs for M, P, andHin Maugin (1999), p. 182).
7. Conclusions
The above given developments somewhat summarize most of
the works on solitons in elastic solids done in the 19702010
period, of which many by the author and co-workers. They essen-
tially emphasize an evolution from the standard exactly integrable
partial differential equations and the corresponding exact soli-
ton solutions obtained by the creators of soliton theory, to the
physically more realistic, but simultaneously more complex and
no longer exactly integrable systems considered here. All basic
equations exploited were based on first principles. The physical
Fig. 5. A typical inhomogeneous 2D shape in the elastic-plate problem: Mexi
can hat with side depletions along the propagation directionx, and monotonou
decrease on both sides in theorthogonaly direction (after Porubov et al., 2004).
landscape thus is widely enlarged, notto speak of the strange poin
mechanics that the quasi-particles associated with the found solu
tions enjoy. Nothing was said of experiments that are not in ou
field of expertise. Suffice it to record the experimental proof o
the existence of solitons in elastic polystyrene rods by the group
of Samsonov in St Petersburg (this is documented at length i
Samsonovs book (Samsonov, 2001), Chapter 4; see more partic
ularly Samsonov et al. (1996)) and the evidence of the existencof surface solitons by Nayanov (1986). Concerning applications, in
addition to the case of crystal structures and/or structural mem
bers, we note the recently emphasized applications to geophysica
situations (Ostrovsky and Johnson, 2001).
To conclude, a word on two-dimensional problems is in order
It does notescape the reader that analytical difficulties met in such
problems may be insuperable. Thatis whyso-calledinhomogeneou
waves onlyhaveoftenbeenconsidered. Bythiswe mean waves tha
are essentially propagating in a prescribed direction (propagatio
space) but are not spatially uniform in a lateral direction (orthogo
nal space). The surface solitons illustrated in Fig. 5 belong in thi
class (orthogonal space then is the depth in the substrate). This i
also the case of the 2D problems mentioned at point (v) at the end
of Section 4. A nice illustration of this is provided in Fig. 5 (fromPorubov et al., 2004)). Here, considering an elastic plate as a 2D
object and both longitudinal and shear deformations, we observ
inhomogeneous soliton-like solutions in the form of humps tha
typically exhibit a Mexican-hat shape (also obtained in Porubov
and Maugin, 2009) in the propagation x-direction say at fixed
lateral position y=0 and monotonous decrease on both side
orthogonally to that direction.
Acknowledgements
Most of the works andresults reported above have been carried
out or obtained in a small number of places: Paris, St Peters
burg, Nizhny-Novgorod, Tallinn, and Kharkov with the Laboratoir
de Modlisation en Mcanique, now integrated in the Institut JeaLe Rond dAlembert, as the central pivoting point of these co
operations. The author thus expresses his immense debt to hi
former and present co-workers over the last thirty years: B. Col
let, J. Pouget, A. Miled, H. Hadouaj, B.A. Malomed, C.I. Christov, A
Salupere, J. Engelbrecht, A.V. Porubov, A.S. Kovalev, M.M. Bogdan
and the late S. Cadet, A.M. Kosevich and A.I. Potapov.
References
Ablowitz, M.J.,Segur, H., 1981. Solitons and the Inverse Scattering Transform. SIAMPhiladelphia.
Aero, E.L., Kuvshinskii, E.V., 1961. Fundamental equations of the theory of elastimediawithrotationallyinteractingparticles(Engl.Transl.). Sov.Phys.Solid Stat2, 12721281 (in Russian, 1960).
Aizu, K., 1970. Possible species of ferromagnetic, ferroelectric and ferroelastic crys
tals. Phys. Rev. B2,754772.
-
5/18/2018 Solitons in elastic solids
8/9
-
5/18/2018 Solitons in elastic solids
9/9
G.A.Maugin / Mechanics Research Communications38 (2011) 341349 34
Pouget, J., Maugin, G.A., 1989. Nonlinear dynamics of oriented elastic solids. II.P ro pagation o f s olit on s. J. Elast icit y 22, 157183 (Alon g t he same line o f thought, Potapov, A.I., Pavlov, A.S., Maugin, G.A., 1999. Nonlinear wave prop-agation in 1D crystals with complex (dumbbell) lattice. Wave Motion, 29,297312).
Salupere, A., Maugin, G.A., Engelbrecht, J., 1994. KdV soliton detection from a har-monic input. Phys. Lett. A 192,58.
Salupere,A., Engelbrecht, J.,Kalda, J.,Maugin, G.A., 1996. On theKdV soliton forma-tion anddiscrete spectral analysis. Wave Motion23, 4966.
Salupere, A., Engelbrecht, J., Maugin, G.A., 1997. Solitons in systems with a quar-tic potential and higher-order dispersion (Proc. EUROMECH 348, Tallinn, May
1996). Proc. Est. Acad. Sci. Math. Phys. 46, 118127.Salupere, A., Engelbrecht, J., Maugin, G.A., 2001. Solitonic structures in KdV-based
higherordersystems.Wave Motion 34, 5161.Samsonov, A.M., 2001. Strain Solitons in Solids, and How to Construct Them. Chap-
man & Hall/CRC, Boca Raton, Florida (This book contains a large bibliographyon nonlinear waves in structural members, especially rods and plates; see thepioneering works by Nariboli, G.A., Sedov, A., 1970. BurgersKdV equation forviscoelastic rods and plates. J. Math. Anal. Appl., 32, 661677; Ostrovsky, L.A.,Sutin, A.M., 1977. Nonlinear elastic waves in rods. Priklad. Matem. i Mekhan.,41/3, 531537 (in Russian); Soerensen, M.P., Christiansen, P.L., Lomdahl, P.S.,1984. So litary waves in n on linear elas tic r ods I. J. Acous. S oc. Ame r. , 76,871879).
Samsonov, A.M., Dreiden, G.V., Porubov, A.V., Semenova, I.V., 1996. Generatioand observation of longitudinal strain soliton in a plate. Tech. Phys. Lett. 226168.
Sayadi, M.K., Pouget, J., 1990. Propagation dexcitations acoustiques non linairedans les matriaux dots de microstructure.J. Phys. Coll. 51 (C3), 219230.
Sayadi, M., Pouget, J., 1991. Soliton dynamics in a microstucturd lattice model.Phys. (UK) A: Gen. Phys. 24, 21512172.
Sayadi, M., Pouget, J., 1992. Chaos transition of a motionin microstructured latticePhysica D 55, 259268.
Seeger, A., 1949. Diploma Physik, T.U. Stuttgart. (Ph.D., 1951).Seeger, A., 1955. Theorie der Gitterfehlstellen. In: Flgge, S. (Ed.), Handbuch de
Physik Bd. 7. Springer-Verlag, Berlin, pp. 383665.Seeger, A., 1979. Solitons in crystals. In: Continuum Models of Discrete Systems
(Proc. Symp.Freudenstadt,1979), vol.15 ofSolid MechanicsStudiesSeries. UnivofWaterloo Mechanics, Canada.
Seeger, A., Wesolowski, Z., 1981. Standing wave solutions of the Enneper equatio(sine-Gordonequation). Int. J. Eng. Sci. 19, 15351549.
Stoker, J.J., 1957. Water Waves. J. Wiley-Interscience, New York.Whitham, G.B.,1974. Linear and Nonlinear Waves. J. Wiley-Interscience, New YorkWesolowski, Z., 1983. Dynamics of a bar of asymmetric cross section. J. Eng. Math
17, 315322.Zabusky, N.J.,Kruskal, M.D.,1965. Interactions of solitons in a collisionless plasm
and recurrence of initial states. Phys. Rev. Lett. 15, 240243.