Periodic Waves in Microstructured Solids and Inverse Problems

Abstract

Conditions for the existence of periodic and solitary waves in 1D microstructured solid of Mindlin type are deduced. Inverse problems to determine material properties are solved.

Full text

Mathematical Modelling and Analysis Publisher: Taylor&Francis and VGTU
Volume 17 Number 5, November 2012, 599–617 http://www.tandfonline.com/TMMA
http://dx.doi.org/10.3846/13926292.2012.732619 Print ISSN: 1392-6292
c
Vilnius Gediminas Technical University, 2012 Online ISSN: 1648-3510
Periodic Waves in Microstructured Solids and
Inverse Problems∗
Ivan Sertakov and Jaan Janno
Tallinn University of Technology
Akadeemia tee 21, 12618 Tallinn, Estonia
E-mail(corresp.): usualis@hotmail.com
E-mail: janno@ioc.ee
Received January 24, 2012; revised September 10, 2012; published online November 1, 2012
Abstract. Conditions for the existence of periodic and solitary waves in 1D mi-
crostructured solid of Mindlin type are deduced. Inverse problems to determine ma-
terial properties are solved.
Keywords: microstructure, inverse problem, dispersion.
AMS Subject Classification: 74J30; 74J35; 74E99; 74J25; 35R30.
1 Introduction
Microstructured materials like alloys, crystallites, ceramics, functionally graded
materials, etc. are gaining a growing importance in contemporary high tech-
nology [4, 9, 13, 16, 20]. This brings along a growing necessity for modelling
mechanical processes in these materials and non-destructive evaluation of phys-
ical parameters.
There are several models of microstructure (see e.g. [5]), but in this paper we
follow the model that was posed by Mindlin [15]. This model was later adjusted
to Rayleigh waves [7] (see also related works [6] and [3]) and approximated by
a Boussinesq-type equation [4]. In the linear case the Rayleigh waves as well as
packets of harmonic waves are informative in the sense of the inverse problems
to determine physical parameters [7, 9, 10, 11].
In the nonlinear case, under a certain balance of nonlinearity and disper-
sion, solitons may emerge. Existence of solitary waves in the one-dimensional
case was proved both for the Boussinesq-type approximation and an original
Mindlin’s coupled system [12, 14]. Numerical simulations [2, 17, 21] and phys-
ical observations [18, 19] support the theoretical results. Solitary waves can be
used to reconstruct material parameters [8, 12].
∗Supported by Estonian Science Foundation Grant 7728 and Estonian Ministry of Educa-
tion and Science Target Financed Theme SF0140083s08.

600 I. Sertakov and J. Janno
In this paper, for the first time periodic waves in nonlinear microstructured
materials of Mindlin’s type are studied from the theoretical viewpoint. The
study is limited to the 1D case. The existence conditions of solitary waves,
deduced formerly in [12], follow as certain limits of the existence conditions
of periodic waves. In the last part of the paper we solve inverse problems to
reconstruct material parameters by means of measurements of periodic and
solitary waves.
2 Description of Mathematical Model and Derivation of
Equations
The mathematical model of the microstructured solid of Mindlin type is based
on assumptions that the microstructure can be interpreted as deformable cells
like “a molecule of polymer or a crystallite of a polycrystal”. It leads to the
necessity to consider deformations on two scales, on macro - and microscopic
scales [15]. The macrodisplacement uis defined as a usual displacement of a
material particle by its components ui=xi−Xi, where xiand Xiare the
components of the spatial and material position vectors, respectively. Ana-
logically the microdisplacement is defined by u0
i=x0
i−X0
i, where the origin
of coordinates x0
iis inside the cell and moves with the displacement u. The
microdisplacement is assumed to be linearly dependent on microcoordinates,
i.e. u0
i=x0
jψji (xi, t), where ψji is the microdeformation tensor [4, 17]. In
this paper we will consider the 1D case so the indices i,jwill be omitted, i.e.
u1=uand ψ11 =ψ. We follow the technique introduced in [4]. The governing
equations of 1D model are







ρ∂2u
∂t2=∂σ
∂x ,
I∂2ψ
∂t2=∂τ
∂x −κ,
where
σ:= ∂W
∂ux
, τ := ∂W
∂ψx
, κ := ∂W
∂ψ ,
ρis the macro-density, Iis the micro-inertia and Wis the free energy function
that is assumed to have the form
W=Au2
x
2+Bψ2
x
2+Cψ2
2+Duxψ+Eu3
x
6+F ψ3
x
6.
Here A, . . . , F are constants and subscripts here and below denote differentia-
tion. Then the governing equations take the form



ρutt =Auxx +Dψx+E
2u2
xx,
Iψtt =Bψxx +F
2ψ2
xx−Dux−Cψ.
Now we introduce nondimensional variables X,T,Uand the quantities I∗,B∗
and F∗by means of the following relations:
x=LX, u =LU, t =LT
c0
, I =I∗l2, F =F∗l3, B =B∗l2,

Periodic Waves in Microstructured Solids 601
where Land L are fixed lengths (possibly a length and an amplitude of a
wave), c0is a fixed velocity and lis a size of the microstructure. By using
these relations we obtain the following nondimensional equations:
ρc2
0L
L2UT T =AL
L2UXX +D
LψX+E2L2
2L3U2
XX,
I∗l2c2
0
L2ψT T =B∗l2
L2ψXX +F∗l3
2L3ψ2
XX−DL
LUX−Cψ.
Let us divide the first equation by ρc2
0L
L2and the second by I∗c2
0. Also we
differentiate the first equation with respect to Xand introduce a new variable
v=UXand a scale coefficient δ=l2
L2. Having done that, we reach the following
equations:
vT T =A
ρc2
0
vXX +D
ρc2
0ψXX +E
2ρc2
0v2XX ,
δψT T =δB∗
I∗c2
0
ψXX +δ3/2F∗
2I∗c2
0ψ2
XX−D
I∗c2
0
v−C
I∗c2
0
ψ.
For simplicity we use lowercase letters xand tinstead of Xand Tand define
new coefficients:
a0=A
ρc2
0
, α =D
ρc2
0, µ =E
ρc2
0
, a1=B∗
I∗c2
0
, λ =F∗
I∗c2
0
, γ =C
I∗c2
0
, β =D
I∗c2
0
.
Then the system of the equations is transformed to
(vtt =a0vxx +αψxx +µ
2v2xx,
δψtt =δa1ψxx +δ3/2λψxψxx −γψ −βv. (2.1)
Here, according to the physical background, the coefficients satisfy the following
conditions:
a0, a1, δ, γ, αβ > 0.(2.2)
The system is supposed to be hyperbolic, i.e. the following condition must be
fulfilled by the physical parameters:
γa0−αβ > 0.(2.3)
We emphasize that in the linear case the type of dispersion is related to the
sign of γa0−γa1−αβ. Namely, it holds (see [10, 12])
1. for normal dispersion: γa0−γa1−αβ > 0,
2. for dispersionless case: γa0−γa1−αβ = 0,
3. for anomalous dispersion: γa0−γa1−αβ < 0.
Math. Model. Anal., 17(5):599–617, 2012.

602 I. Sertakov and J. Janno
From now on, let us consider the travelling wave solutions. This means that
the solution of the system (2.1) has the form
(v(x, t) = w(x−c1t),
ψ(x, t) = ϕ(x−c2t),(2.4)
where wand ϕare arbitrary functions and c1and c2are velocities of the compo-
nents of the wave. In physically reasonable cases it holds c:= c1=c2. Indeed,
if either wor ϕis a constant, then we may set c1=c2without restriction of
generality. But if both wand ϕare non-constant, then we plug (2.4) into the
first equation of (2.1) to obtain the relation f1(x−c1t) = f2(x−c2t), where
the functions f1= (c2
1−a0)w00 −µ
2(w2)00 and f2=αϕ00 are not constant in
physically relevant cases. From this relation we infer c1=c2.
Thus, let us rewrite the system (2.1) in a new variable η=x−c t:
(c2w00(η) = a0w00(η) + µ
2w(η)200 +αϕ00(η),
δc2ϕ00(η) = δa1ϕ00(η) + δ3/2λϕ0(η)ϕ00(η)−γϕ(η)−βw(η),(2.5)
where the first equation is twice integrable. After integrating we have
ϕ(η) = 1
αhc2−a0w(η)−µ
2w(η)2i+C1η+C0,(2.6)
where C0and C1are constants.
We are interested in periodic and solitary wave solutions. This immediately
implies C1= 0. Moreover, for the sake of simplicity we will be limited to the
case C0= 0. The latter condition is always satisfied for the solitary wave
solutions. But in the case of periodic w,ϕand µ= 0 the condition C0= 0
implies that the integrals of w,ϕthat are the macro- and microdeformation,
respectively, are also periodic. Consequently, (2.6) takes the form
ϕ(η) = 1
αhc2−a0w(η)−µ
2w(η)2i.(2.7)
After replacing ϕ(η) in the second equation in (2.5) by (2.7) the following
equation for unknown function w(η) is deduced:
w00 =µ(w0)2−δ1/2λµ(c2−a0)(w0)3
α(c2−a1)+δ1/2λµ2(w0)3w
α(c2−a1)+γµw2
2δ(c2−a1)−γ(c2−a0)+αβ
δ(c2−a1)w
(c2−a0−µw)1−δ1/2λw0
α(c2−a1)(c2−a0−µw).
(2.8)
Doing the substitution y(w) = (c2−a0−µw(η))w0(η), the equation (2.8) is
transformed into the more simple and integrable one:
δc2−a1−δ3/2λ
αy(w)y(w)y0(w)
=µγ
2w2−γc2−a0+αβwc2−a0−µw.(2.9)

Periodic Waves in Microstructured Solids 603
3 Periodic Waves
For the further analysis it is more comfortable to rewrite the equation (2.8)
into the following system for the pair (w(η), z(η)):







w0=z,
z0=µz2−δ1/2λµ(c2−a0)z3
α(c2−a1)+δ1/2λµ2z3w
α(c2−a1)+γµw2
2δ(c2−a1)−γ(c2−a0)+αβ
δ(c2−a1)w
(c2−a0−µw)1−δ1/2λz
α(c2−a1)(c2−a0−µw).
(3.1)
Depending on µ, this system has a different number of equilibrium points, i.e.
the points where z0(η) = w0(η) = 0. If µ6= 0 then there are two equilibrium
points and if µ= 0 then there is a single equilibrium point. Combining two
parameters of nonlinearity three different cases can be coped with:
1) λ= 0 and µ= 0,2) λ6= 0 and µ= 0,3) µ6= 0.
The first case is purely linear and involves sinusoidal periodic travelling wave
solutions [12]. We will present a more detailed treatment of the second case
λ6= 0, µ= 0 in the next subsection where we deduce conditions for the velocity
cthat are necessary and sufficient for the existence of periodic travelling wave
solutions. Results in the third case µ6= 0 can be obtained in a similar manner
and will be given more shortly. Moreover, in the latter case we can present a
common treatment for the subcases λ= 0 and λ6= 0.
3.1 Case λ6= 0 and µ= 0
In this case we have the following system of nonlinear differential equations:





w0=z,
z0=−γ(c2−a0) + αβ
δ(c2−a1)(c2−a0)w.1−δ1/2λ(c2−a0)
α(c2−a1)z.(3.2)
The system linearized near the equilibrium point (w;z) = (0; 0) is





w0=z,
z0=−γ(c2−a0) + αβ
δ(c2−a1)(c2−a0)w.
The characteristic equation of the latter system is
k2+γ(c2−a0) + αβ
δ(c2−a1)(c2−a0)= 0.
Periodic waves are related to complex roots. Therefore, the inequality
γ(c2−a0) + αβ
(c2−a1)(c2−a0)<0
must hold. Taking this into account we deduce the following conditions for the
velocity c:
Math. Model. Anal., 17(5):599–617, 2012.

604 I. Sertakov and J. Janno
1. for normal dispersion, i.e., γa0−γa1−αβ > 0 and
either a)c2> a0> a0−αβ
γ> a1,or b)a0−αβ
γ> c2> a1; (3.3)
2. for dispersionless case, i.e., γa0−γa1−αβ = 0,
c2> a0; (3.4)
3. for anomalous dispersion, i.e., γa0−γa1−αβ < 0 and
either a)c2> a0, a1> a0−αβ
γ,or b)a0, a1> c2> a0−αβ
γ.(3.5)
Observing the second equation of the system (3.2) we see that there exists
a singular line z=zswith zs=α(c2−a1)
δ1/2λ(c2−a0)on the phase plane. Therefore, the
first derivative of the function w(η) is bounded by zs. More precisely, w0< zs
in the case of positive zsand w0> zsin the case of negative zs. Using such
a restriction for w0we can deduce certain bounds for the extrema wmin and
wmax of the periodic wave, too. Let us do that.
Firstly, we mention that wmin =−wmax . This follows from the symmetry
with respect to the z-axis of the phase curves of (3.2). Further, we consider
the equation (2.9) that was deduced by means of the substitution y(w) = (c2−
a0)w0(η) (recall that µ= 0 in the present case). Since the phase curve z=z(w)
has two branches, passing through the upper and lower half-planes, respectively,
the function y(w) also has two branches: a positive and a negative one. Let us
choose such a branch of y(w) that satisfies the condition sign y
c2−a0= sign zs.
Integrating (2.9) we get
δ(c2−a1)
2−δ3/2λ
3αyy2
y2
y1
=−(c2−a0)(γ(c2−a0) + αβ)
2w2
w2
w1
,(3.6)
where y1=y(w1) and y2=y(w2). Setting w2=wext, where wext is either
wmax or wmin , we have y2=y(wext) = 0, because w0= 0 in the extremum
point. Moreover, let w1= 0. Then (3.6) yields
δ(c2−a1)
2−δ3/2λ
3αy1y2
1=(c2−a0)(γ(c2−a0) + αβ)
2w2
ext
with y1=y(0). This in turn implies w2
ext =f(z1) with
f(z) = 2(c2−a0)
γ(c2−a0) + αβ δ(c2−a1)
2−δ3/2λ(c2−a0)
3αzz2,
where z1=y1/(c2−a0). Due to the choice of the branch of y, the numbers z1
and zshave the same signs. Moreover, 0 <|z1|<|zs|. One can immediately
check that the maximum of the cubic function f(z) between 0 and zsis achieved

Periodic Waves in Microstructured Solids 605
at z=zs. Therefore, w2
ext < f(zs). Computing f(zs) we reach the following
bound for the extrema:
w2
max =w2
min <α2(c2−a1)3
3λ2(c2−a0)(γ(c2−a0) + αβ).(3.7)
Summing up, the conditions (3.3)–(3.5) give ranges of the velocity when
periodic wave may exist. The inequality (3.7) shows that the amplitude of
the periodic wave is restricted. In this connection, the crucial role has the
microstructure nonlinearity parameter λ. The bigger λ, the smaller range of
the amplitude.
3.2 Case µ6= 0
As it was mentioned, in the case µ6= 0 two equilibrium points occur. Let us
consider separately these two cases.
1. Waves related to the equilibrium point (w;z) = (0; 0).
Using the same technique of linearisation near the equilibrium points as in
Subsection 3.1 we deduce the following restrictions for the velocity c:
1. for normal dispersion, i.e., γa0−γa1−αβ > 0 and
either a)c2> a0> a0−αβ
γ> a1or b)a0−αβ
γ> c2> a1; (3.8)
2. for dispersionless case, i.e., γa0−γa1−αβ = 0,
c2> a0; (3.9)
3. for anomalous dispersion, i.e., γa0−γa1−αβ < 0 and
either a)c2> a0, a1> a0−αβ
γor b)a0, a1> c2> a0−αβ
γ.(3.10)
Observing the second factor of the denominator of the second equation in
the system (3.1) we see that the variable y(w) = (c2−a0−µw(η))w0(η) has
the singular value ys=α(c2−a1)
δ1/2λ. Therefore, the value of yis located between
0 and ys. This enables to deduce estimates for the extrema. Let us integrate
the equation (2.9):
δ(c2−a1)
2−δ3/2λ
3αyy2
y2
y1
(3.11)
=−µ2γ
8w4+µ(3γ(c2−a0)+2αβ)
6w3−(c2−a0)(γ(c2−a0) + αβ)
2w2
w2
w1
,
where y1=y(w1) and y2=y(w2). The present situation is more complicated
than in the case µ= 0, because the right- hand side of (3.11) contains a quartic
Math. Model. Anal., 17(5):599–617, 2012.

606 I. Sertakov and J. Janno
polynomial instead of the simple square of w. Therefore, in the first stage we
will get results for such a polynomial of wext , not directly for wext .
Let us set w2=wext , where wext is again either wmax or wmin , and w1= 0,
y1=y(0) in (3.11) to get
δ(c2−a1)
2−δ3/2λ
3αy1y2
1=R1w4
ext +R2w3
ext +R3w2
ext ,(3.12)
where
R1=γµ2
8, R2=−3µγ(c2−a0)+2µαβ
6, R3=(c2−a0)(γ(c2−a0) + αβ)
2.
Arguing similarly as in Subsection 3.1 we deduce the following estimate for the
extrema in an implicit form:
λ2<R0
R1w4
max +R2w3
max +R3w2
max
=R0
R1w4
min +R2w3
min +R3w2
min
,(3.13)
where
R0=α2(c2−a1)3/6.(3.14)
Moreover, from (3.12) another useful equation follows:
R1w4
max −w4
min +R2w3
max −w3
min +R3w2
max −w2
min = 0.(3.15)
To deduce explicit bounds for the extrema of w, we make use of the second
singular value of wc2−a0
µof the system (3.1) and the circumstance that w
cannot reach the second equilibrium point 2
µ(c2−a0+αβ
γ) in the case of periodic
wave. We can argue as follows. Due to the Cauchy’s theorem, the solution
of (3.1) is unique for a given initial condition. This implies that the phase
curves related to different periodic waves cannot intersect and they form a
family of closed curves inserted into each-other. Consequently, decreasing wmin
results in the increase of the corresponding value of wmax and vice versa. If
we approach with wmin or wmax any of the bounds c2−a0
µor 2
µ(c2−a0+αβ
γ),
the corresponding bound for the opposite extremum can be found solving the
equation (3.15).
Let us consider in detail the particular case 2
µ(c2−a0+αβ
γ)≤c2−a0
µ<0
and µ > 0. Then 0 < c2≤a0−2αβ
γand the lower bound of wmin is c2−a0
µ. We
plug this bound for wmin into (3.15) and deduce the following equation for the
limit of the amplitude A=wmax −wmin :
A2
24 3γµ2A2−8µαβA −6c2−a02αβ −γa0+γc2= 0.
The positive nontrivial solution is
Aext =1
6γµ 8αβ +s72γ2c2−a0+4αβ
γc2−a0+2αβ
3γ.

Periodic Waves in Microstructured Solids 607
Therefore, we obtain c2−a0
µ< wmin < wmax <c2−a0
µ+Aext .
Having studied all cases of location of c2−a0
µand 2
µ(c2−a0+αβ
γ) and the
sign of µwe can summarize the bounds for the extrema in the following form:
(i) min(g1, g2)< wmin < wmax <max(g1, g2) for 0 < c2≤a0−2αβ
γ,
(ii) min(g3, g4)< wmin < wmax <max(g3, g4) for a0−2αβ
γ< c2< a0−αβ
γ,
(iii) min(g3, g4)< wmin < wmax <max(g3, g4) for a0−αβ
γ< c2≤a0−2αβ
3γ,
(iv) min(g1, g5)< wmin < wmax <max(g1, g5) for a0−2αβ
3γ< c2< a0,
(v) min(g1, g5)< wmin < wmax <max(g1, g5) for a0< c2,
where g1, . . . , g5are defined as follows:
g1=c2−a0
µ, g2=c2−a0
µ+
8αβ +q72γ2(c2−a0+4αβ
3γ)(c2−a0+2αβ
3γ)
6γµ ,
g3= 2c2−a0
µ+αβ
µγ , g4=2(pαβ(3γ(a0−c2)−2αβ)−αβ)
3γµ ,
g5=c2−a0
µ+
8αβ −q72γ2(c2−a0+4αβ
3γ)(c2−a0+2αβ
3γ)
6γµ .(3.16)
2. Waves related to the equilibrium point (w;z)=(2
µ(c2−a0+αβ
γ); 0).
By means of the linearisation technique the following conditions for the
velocity are obtained:
1. for normal dispersion, i.e., γa0−γa1−αβ > 0 and
either a)c2≥a0> a0−αβ
γ> a1or b)a0> c2> a0−αβ
γ> a1,
or c)a0−αβ
γ> a1> c2> a0−2αβ
γ≥0,or d)a0−2αβ
γ> c2> a1;
(3.17)
2. for dispersionless case, i.e., γa0−γa1−αβ = 0,
c2> a0−2αβ/γ; (3.18)
3. for anomalous dispersion, i.e., γa0−γa1−αβ < 0 and
either a) c2=a0> a1> a0−αβ/γ, or b) c2> a0, a1> a0−αβ/γ ,
or c) a0> c2> a1> a0−αβ/γ,
or d) a0, a1> a0−αβ/γ > c2> a0−2αβ/γ ≥0.(3.19)
Math. Model. Anal., 17(5):599–617, 2012.

608 I. Sertakov and J. Janno
Taking the singular value of ys=α(c2−a1)
δ1/2λof yinto account, we again deduce
restrictions for the extrema of w. To this end, let us set in (3.11) w2=wext ,
w1=2
µ(c2−a0+αβ
γ) and y1=y(2
µ(c2−a0+αβ
γ)). Then we obtain
δ(c2−a1)
2−δ3/2λ
3αy1y2
1=R1w4
ext +R2w3
ext +R3w2
ext +R4,(3.20)
where R1,R2,R3are defined by (3.12) and
R4=2αβ(γ(c2−a0) + αβ)3
3γ3µ2.
From this relation we derive the following implicit estimate:
λ2<R0
R1w4
max +R2w3
max +R3w2
max +R4
=R0
R1w4
min +R2w3
min +R3w2
min +R4
,(3.21)
where R0is given by (3.14).
Furthermore, similarly as in the previous case, we obtain the following
bounds:
(vi) min(g1, g5)< wmin < wmax <max(g1, g5) for 0 < c2< a0−2αβ
γ,
(vii) min(g1, g5)< wmin < wmax <max(g1, g5) for a0−2αβ
γ< c2< a0−4αβ
3γ,
(vii) min(g6, g7)< wmin < wmax <max(g6, g7) for a0−4αβ
3γ≤c2< a0−αβ
γ,
(viii) min(g6, g7)< wmin < wmax <max(g6, g7) for a0−αβ
γ< c2≤a0,
(ix) min(g1, g2)< wmin < wmax <max(g1, g2) for a0< c2,
where g1,g2,g5are defined by (3.16), g6= 0 and
g7=2(p4α2β2+ 3αβγ(c2−a0)+2αβ + 3γ(c2−a0))
3γµ .
To conclude this subsection, we point out that the conditions (3.8)–(3.10)
(resp. (3.17)–(3.19)) give ranges of the velocity when periodic wave may ex-
ist. The restrictions for the extrema (i.e. the minima and maxima) of the
wave depend on the nonlinearity parameters µand λ. In addition to the condi-
tion (3.13) (resp. (3.21)), the maxima and minima must satisfy the inequalities
(i)–(v) (resp. (vi)–(ix)). The bigger λand µ, the smaller the range of the ex-
trema. In case λ= 0 the inequality (3.13) (resp. (3.21)) drops.

Periodic Waves in Microstructured Solids 609
4 Solitary and Related Waves
4.1 Solitary waves
As it was mentioned the system (2.5) is supposed to have a solitary wave
solution, i.e. a solution that satisfies w→0 as |η|→∞. A solitary wave
solution can exist only if parameter µdoes not equal zero. This condition can
be easily verified by integrating the equation (2.9) with the lower and upper
limits y(wmax ) = y(wmin ) = 0.
In the mathematical sense, the solitary wave is a limit case of the periodic
wave related to the equilibrium point w= ( 2
µ(c2−a0+αβ
γ); 0) whose phase
curve approaches the origin (0;0). Let us consider the relation (3.20) deduced
for such a periodic wave. By subtraction we immediately get (3.15). Taking
either the limit wmin →0 or wmax →0 there, we reach the following quartic
equation
w4
amp −3µγ(c2−a0)+2µαβ
6w3
amp +(c2−a0)(γ(c2−a0) + αβ)
2w2
amp = 0,
where wamp =wmax in the case of the positive wave and wamp =wmin in
the case of the negative wave. The nontrivial roots of this equation give the
possible values for the amplitude:
wamp12 =2(±p4α2β2+ 3αβγ(c2−a0)+2αβ + 3γ(c2−a0))
3γµ .(4.1)
In order wamp12 to be real, the velocity must satisfy the inequality c2≥a0−4αβ
3γ.
A detailed analysis of the behavior of the system (3.1) near the equilibrium
point (w;z) = 2
µ(c2−a0+αβ
γ); 0enables to extract the right formula for
amplitude of the wave:
wamp =2(p4α2β2+ 3αβγ(c2−a0)+2αβ + 3γ(c2−a0))
3γµ .(4.2)
Inserting this formula to (3.21), a more simple condition for λfollows:
|λ|<sα2µ2γ3(c2−a1)3
4αβ(αβ +γ(c2−a0))3.(4.3)
To deduce conditions for the velocity c, we begin with the consideration that
the singular value c2−a0
µof wcannot be located between 0 and the equilibrium
value 2
µ(c2−a0+αβ
γ). This implies c2< a0. Furthermore, since the phase
curve of the solitary wave approaches the origin point (0,0) as |η|→∞, we can
again linearize the system (3.1) at this point. The vanishing solution occurs
only in case γ(c2−a0)+αβ
(c2−a1)(c2−a0)<0. Combining this condition with the inequalities
c2< a0and c2≥a0−4αβ
3γ, obtained before, we derive the following restrictions
for the velocity:
Math. Model. Anal., 17(5):599–617, 2012.

610 I. Sertakov and J. Janno
1. for normal dispersion, i.e., γa0−γa1−αβ > 0,
either a) a0> c2> a0−αβ/γ > a1
or b) a0−αβ/γ > a1> c2≥a0−4αβ/(3γ); (4.4)
2. for dispersionless case, i.e., γa0−γa1−αβ = 0,
a0> c2≥a0−4αβ(3γ); (4.5)
3. for anomalous dispersion, i.e., γa0−γa1−αβ < 0,
either a) a0> c2> a1> a0−αβ/γ
or b) a1> a0−αβ/γ > c2≥a0−4αβ/(3γ).(4.6)
We mention that the same conditions for the velocity and the parameter λ
were deduced also in [12], but by means of different techniques.
Moreover, we underline that the numerical solitary wave solution is unstable
near the point (w;z) = (0; 0), so instead of the pure numerical solution in
practice the following (piecewise analytical-numerical) approximation is used:
w(η) = 






w−∞(η) = w(−ˆη1)e√κ(η+ˆη1)η≤ −ˆη1,
w(η)−ˆη1≤η≤ˆη2,
w+∞(η) = w(ˆη2)e−√κ(η−ˆη2)ˆη2≤η,
(4.7)
where κ=−γ(c2−a0)+αβ
δ(c2−a1)(c2−a0)and ˆηj,j= 1,2, are sufficiently large numbers.
4.2 Another wave of infinite length
An interesting wave of infinite length can be obtained from the periodic wave
related to the equilibrium point (0; 0) in case the phase curve approaches w=
(2
µ(c2−a0+αβ
γ); 0). Such a wave approaches a nonzero constant as |η|→∞,
hence it is not a solitary wave in the classical sense. Physically, this wave may
occur in predeformed materials.
Since the phase curve turns around the origin, the wave changes the sign
around η= 0. More precisely, the following inequalities hold for those wave:
γ(c2−a0) + αβ
δ(c2−a1)(c2−a0)>0,γ(γ(c2−a0) + αβ)
δ(c2−a1)(γ(c2−a0)+2αβ)<0
and one of the following conditions must be satisfied:
either µ > 0,c2−a0
µ<0,c2−a0
µ<2c2−a0
µ+αβ
γµ 
or µ < 0,c2−a0
µ>0,c2−a0
µ>2c2−a0
µ+αβ
γµ .

Periodic Waves in Microstructured Solids 611
The solution obeys the following properties:
w(0) = 2pαβ(3γ(a0−c2)−2αβ)−αβ
3γµ , w0(0) = 0,
lim
η→±∞ w(η)=2c2−a0
µ+αβ
γµ ,lim
η→±∞ w(i)(η) = 0.(4.8)
Restrictions for the velocity are
either c2> a1, a0−2αβ/γ < c2< a0−αβ/γ
or c2< a1, a0−αβ/γ < c2< a0−2αβ/(3γ) (4.9)
and the condition for λis
|λ|<sα2µ2γ3(a1−c2)3
4αβ(αβ +γ(c2−a0))3.(4.10)
The stable approximation of this solution is
w(η) = 






w−∞(η) = w(−ˆη1)e√κ(η+ˆη1)+Θ1−e√κ(η+ˆη1)η≤ −ˆη1,
w(η)−ˆη1≤η≤ˆη2,
w+∞(η) = w(ˆη2)e−√κ(η−ˆη2)+Θ1−e−√κ(η−ˆη2)ˆη2≤η.
Here
κ=−γ(γ(c2−a0) + αβ)
δ(c2−a1)(γ(c2−a0)+2αβ), Θ = 2c2−a0
µ+αβ
γµ ,ˆηj, j = 1,2,
are again sufficiently large numbers.
5 Inverse Problems
5.1 Description of method
Solving an inverse problem is understood as finding (extracting) values of pa-
rameters of an equation or a system of them (i.e. in this article finding the
value of a0,a1,γ,α,β,µand λon condition that the parameters cand δ
are supposed to be known with high accuracy and to be controlled) from the
experimental (simulation) data. The number of parameters and their accuracy
depend on how the data is presented. We use the periodic or solitary waves
and assume that the data can contain information either about the macrode-
formation only, or about the micro- and macrodeformations. Also the data can
be given either as wave profiles or as a collection of characteristics, i.e. the
amplitudes of waves, the lengths etc.
From the practical viewpoint, it is important that the traveling waves that
we use, are orbitally stable, because otherwise these waves are very difficult to
observe. Rigorous mathematical proof of the stability is a complicated task,
Math. Model. Anal., 17(5):599–617, 2012.

612 I. Sertakov and J. Janno
because the system (2.1) is not integrable. The stability can be seen from nu-
merical simulations [17, 21, 22]. Moreover, physical solitary waves are observed
in some microstructured materials [18, 19].
As it was mentioned, analytical solutions of the problems (2.5) and (2.8) are
not realistic to find. Therefore, it remains to solve them numerically. To find
the parameters a0,a1,γ,α,β,µand λwe have to fit the computed solution to
the experimental data, i.e. we have to use methods analogous to the shooting
method and methods of minimization. We must underline that there are some
tricks which allow us to extract a part of parameters analytically, but their
accuracy is too low, so we will not consider them.
As for the method used to find the parameter, we have to construct the
special positive objective functional (it is special because of the dependence
on the presentation of data and the number of parameters and waves) and
numerically minimize it. The minimal value of functional is either zero in case
of the data are not affected by the noise or the smallest (as possible) positive
number in the case of presence of the noise. According to that we obtain either
an exact or a quasi-solution.
Let us start with the case when the data contain information exclusively
about the wave profile of the macrodeformation. Then the basic system (2.8)
contains the three parameters α,βand λonly in the form of the quotient λ
α
and the product αβ. Therefore, the values of these three parameters cannot be
extracted from macro-measurements. The vector of unknown physical parame-
ters contains 6 components: a0,a1,αβ,γ,λ
α,µ. To solve the inverse problem,
we have to minimize the following functional:
F(P) = sX
i,s ˆw(ηi, cs)−w(ηi, cs, P )2+X
k,s
Gk(P, cs),(5.1)
where PT= (p1, . . . , pn) is a vector of n parameters to be determined (below
we see that this may contain Cauchy data in addition to physical parameters),
csis the velocity of the s-th wave (the number of waves in this case must
be greater or equal to three), Gkare penalty functions which are zero if the
physical constraints (2.2) and (2.3) are fulfilled and positive and grow very
fast if not, w(ηi, cs, P ) is the computed wave solution at η=ηicorresponding
to given P,csand ˆw(ηi, cs) is the measured data (in our case generated by
simulation).
The simulated data are as follows: ˆw(ηi, cs) = w(ηi, cs, P †) + iwhere iis
the noise and P†is the prefixed “exact solution” of the inverse problem.
As we mentioned, the Cauchy data for the system (3.1) may also be un-
known. This means that, in general, the vector Pconsists of the following
components:
p1=a0, p2=a1, p3=αβ, p4=γ, p5=λ/α, p6=µ,
p5+2s=ws(η0), p6+2s=w0
s(η0), s = 1, . . . , J, (5.2)
where Jis the number of waves incorporated. In special cases the number of
unknowns can be reduced (if either µor λis known to be zero or both of them

Periodic Waves in Microstructured Solids 613
are known to be zero or Cauchy conditions are either same for all waves or
exactly known).
Secondly, let us consider the case when the wave profiles of both macro- and
microdeformation are given. Then the whole vector of physical data a0,a1,α,
β,γ,λ,µcan be recovered, because the additionally given microdeformation
depends on αseparately from the product αβ (see formula (2.7)). Now the
objective functional reads
F(P) = sX
i,s ˆw(ηi, cs)−w(ηi, cs, P )2+X
i,s ˆϕ(ηi, cs)−ϕ(ηi, cs, P )2
+X
k,s
Gk(P, cs),
where Pagain is a vector of parameters to be determined, csis the velocity,
Gkare the penalty functions, w(ηi, cs, P ), ϕ(ηi, cs, P ) are the macro- and micro-
components of the computed wave and ˆw(ηi, cs), ˆϕ(ηi, cs) are the measured
data. The simulated data are constructed in a manner similar to the previous
case.
Finally, if the data contain a collection of characteristics of waves then the
functional analogical to (5.1) may have the form
F(P) = sX
i,s ˆ
Ki(cs)−Ki(cs, P )2+X
k,s
Gk(P, cs).(5.3)
Here K1and K2are the minimum and the maximum of the periodic wave of
a macrocomponent (in case of both micro- and macrodeformation there are
also K5and K6, i.e. the minimum and the maximum of the periodic wave of
a microcomponent ) and K3and K4are the half-lengths, i.e. w(η0−K3) =
w(η0+K4) = K1(or ϕ(η0−K3) = ϕ(η0+K4) = K5) where η0is some value
of ηwhere the wave-function attains the maximum K2. Note that now the
number of parameters is n= 6 + J(in this case for simplicity we assume that
p6+2s= 0) and can be reduced on some conditions.
5.2 Numerical results
In case of wave profiles the number of points of a wave profile is denoted by N.
Also we define the noise as i=kR mins|w(cs, P †)max −w(cs, P †)min |,where
Ris the uniformly distributed pseudorandom number in the interval [−1,1]
and the coefficient kis specified for every case.
In case of a collection of wave characteristics we define two types of noise:
A=k1R×mins|w(cs, P †)max −w(cs, P †)min |for wave amplitudes and L=
k2Rmins|T(cs, P †)|for wave lengths, where k1and k2are specified for every
case and T(cs, P †) is the length of the s-th wave.
Evidently, a single wave doesn’t contain enough information to recover all
unknown parameters whatever is the number of measured points or character-
istics of this wave. Indeed, the basic equation of the macro-component of the
traveling wave (2.9) has 5 degrees of freedom (coefficients of yy0,y2y0,w,w2
Math. Model. Anal., 17(5):599–617, 2012.

614 I. Sertakov and J. Janno
Table 1. Maximal absolute errors (wave profile, macrodeformation only).
N= 150, T= 14.8491, k= 0.005, max ||= 0.00826, δ= 1
Number of waves 3 4 5 6 7
Exact values Maximal absolute errors
a03 0.03151 0.01648 0.01763 0.01781 0.01769
a11 0.09697 0.05569 0.04943 0.02927 0.04370
αβ 1 0.05932 0.03517 0.03259 0.02390 0.02681
γ1 0.00663 0.00487 0.00454 0.00280 0.00269
µ1 0.01747 0.00888 0.00910 0.00915 0.00808
λ/α 1 0.02497 0.02220 0.02012 0.01665 0.01099
wi1 0.00103 0.00076 0.00065 0.00089 0.00068
w0
i0 0.00130 0.00130 0.00124 0.00075 0.00078
and w3). But the number of unknown parameters contained in this equation
is 6 (i.e. a0,a1,αβ,γ,λ
αand µ). Therefore, at least 2 waves with different
velocities have to be measured. Similar situation occurs if both macro- and
micro-components of the waves are incorporated.
Increasing the number of measured waves reduces the error of the solution
of the inverse problem. The reasons are that that then the amount of infor-
mation used in the problem is bigger and the influence of stochastic errors of
measurements is smaller (the mathematical expectations of the errors of the
measurements are equal to 0).
The 50 simulations are done for every case. The minimization of related ob-
jective functionals was performed using the Nelder–Mead algorithm [1]. Num-
ber Tin the tables denotes the maximal wave-length, i.e. T= maxsT(cs, P †).
In the examples the exact parameters and the velocities are chosen so that
the theoretical existence conditions for the periodic waves are satisfied. More
precisely, the range of velocity of the measured waves is the interval [2.2,2.8],
where the particular velocities are taken by the formula cj= 2.2 + τ(j−1),
j= 1, . . . , M , where τ=2.8−2.2
M−1and Mis the number of measured waves.
Tables 1 and 2 show absolute errors of numerical results that are obtained
solving inverse problems that use measurements of Npoints on the profiles of
different number of measured waves (from 3 to 7). Tables contain maximal ab-
solute errors of the parameters. In addition to the physical parameters, Cauchy
data of all waves are assumed to be unknown (the parameters wiand w0
i).
Tables 3 and 4 contain absolute errors of results obtained by means of mea-
surements of wave characteristics (minima, maxima and half-lengths). Since
the results are worse than in the previous case, the number of measured waves
is increased (from 5 to 30).
The results show that the informativity of measured waves is very different
with respect to different parameters. Wave profiles are much more informative
than the wave characteristics. The biggest error has the parameter a1. In case
only the macrodeformation is measured, the best results are obtained for γ. In
case both macro- and microdeformation are measured, the results are better.
In particular, then the parameters αand βare separated form the product and

Periodic Waves in Microstructured Solids 615
Table 2. Maximal absolute errors (wave profile, macro- and microdeformation).
N= 150, T= 14.8491, k= 0.005, max ||= 0.00826, δ= 1
Number of waves 3 4 5 6 7
Exact values Maximal absolute errors
a03 0.00514 0.00609 0.00384 0.00311 0.00405
a11 0.04250 0.01630 0.04345 0.03570 0.02361
α1 0.00079 0.00105 0.00087 0.00084 0.00153
β1 0.03010 0.01032 0.02625 0.02168 0.01400
γ1 0.00175 0.00114 0.00270 0.00239 0.00160
µ1 0.01150 0.01198 0.00694 0.00500 0.00553
λ1 0.00499 0.00324 0.00598 0.00561 0.00389
wi1 0.00131 0.00143 0.00087 0.00089 0.00107
w0
i0 0.00030 0.00038 0.00034 0.00042 0.00047
Table 3. Maximal absolute errors (wave characteristics, macrodeformation only).
max |A|= 0.00165, max |L|= 0.00975, k1=k2= 0.001, δ= 1
Number of waves 5 10 20 30
Exact values Maximal absolute errors
a03 0.07604 0.05938 0.04131 0.01876
a11 0.56430 0.25994 0.24591 0.19478
αβ 1 0.19849 0.19849 0.18713 0.13654
γ1 0.01390 0.00804 0.00906 0.00661
µ1 0.05486 0.02625 0.02354 0.01452
λ/α 1 0.07038 0.03470 0.03587 0.03434
the best results are obtained for α. We point out that the numerical results are
comparable with the results obtained formerly for inverse problems for solitary
waves [12].
6 Conclusions
The microstructure has a dispersive impact to the wave propagation. Under
a proper balance between the dispersion and the nonlinearity, periodic and
solitary waves may occur. In this paper we deduced existence conditions for
such waves. Here the crucial role play the nonlinearity parameters µand λ,
corresponding to the macro- and micro-levels, respectively. In case µ= 0 only
a single family of periodic waves exist and solitary waves do not occur. In case
µ6= 0 two families of periodic waves exist and the extrema of the periodic
waves have bounds that depend on µ(see conditions (i)–(ix)). Approaching
these bounds the periodic wave attains infinite length and becomes either a
solitary wave or a wave described in Subsection 5.2.
The micro-level parameter λrather disturbs the balance between the non-
linearity and the dispersion. The relations (3.13), (3.21), (4.3) and (4.10) show
Math. Model. Anal., 17(5):599–617, 2012.

616 I. Sertakov and J. Janno
Table 4. Maximal absolute errors (wave characteristics, macro- and microdeformation).
max |A|= 0.00165, max |L|= 0.00975, k1=k2= 0.001, δ= 1
Number of waves 5 10 20 30
Exact values Maximal absolute errors
a03 0.00581 0.00382 0.00356 0.00150
a11 0.13040 0.10708 0.05625 0.03569
α1 0.00140 0.00098 0.00100 0.00044
β1 0.07860 0.06552 0.03979 0.02140
γ1 0.00933 0.00516 0.00383 0.00304
µ1 0.00568 0.00310 0.00452 0.00232
λ1 0.02346 0.01797 0.01102 0.00568
that |λ|must be sufficiently small. Increasing |λ|to a certain critical level, the
balance breaks.
Moreover, we showed that periodic and solitary waves can be used to recon-
struct the physical parameters of the material. The numerical tests insist that
the measurements of the wave profiles contain enough information to recover
the parameters with acceptable accuracy. The measurements of wave charac-
teristics (extrema and half-lengths) are less informative. Some parameters can
be well-recovered, but some other parameters are very sensitive with respect
to measurement errors.
Acknowledgements
Authors thank Prof. J¨uri Engelbrecht for very useful discussions and referees
of the paper for suggestions that lead to improvement of the presentation.
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