Solitons in elastic solids

5/18/2018 Solitons in elastic solids

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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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G.A.Maugin / Mechanics Research Communications38 (2011) 341349 34

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.

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