CONTENTS &
ABSTRACTS

In
English. Summaries in Estonian

Proceedings of the Estonian Academy of Sciences.

Physics * Mathematics

** **

Volume 52 No. 1
March 2003

Special
issue on nonlinear waves in microstructured solids

Preface;
3–4

Nonlinear
wave mechanics of complex material systems; 5–11

Gérard A. Maugin

**Abstract.** Inspired by the original ideas of L. de Broglie on
wave mechanics, which were at the basis of the wave-like interpretation of
quantum mechanics, we tentatively develop possible fruitful analogies between
the conservation equations of nonlinear continuum mechanics in the canonical
framework as expressed on the material manifold and the dispersive kinematic
wave theory of Whitham in order to construct a *nonlinear wave* *mechanics* of structured solid continua*.* The final purpose of this approach
obviously is *not* quantization but the
relationship between dynamical localized concentrations of continuous fields,
such as solitary waves of the envelope type, and the notion of quasi-particles.

**Key words:** waves, action, hyperelasticity, wave mechanics, kinematics,
complex materials, nonlinear waves, solitons.

Waves
in microstructured solids with nonlinearities in microscale;
12–20

Jüri Engelbrecht and Franco Pastrone

**Abstract.** Consistent modelling of wave motion in microstructured solids
is discussed. Based on the Mindlin model, the simplest model equation of motion
is derived. The fundamental properties of such a model are
(i) hierarchical structure distinguishing the macro- and microbalance,
(ii) changes in the wave speed, and (iii) definite influence of
dispersion. The nonlinearity in the microlevel, although presented by a simple
energy density function, leads to a complicated nonlinear term in the equation
of motion as well as in the corresponding evolution equation. Such consistent
modelling opens up direct ways for determining material constants
characterizing the microstructure.

**Key words:** microstructured solids, wave hierarchy, nonlinearity,
evolution equation.

Waves
in solids with vectorial microstructure; 21–29

Franco Pastrone

**Abstract.** A general model of solids with vectorial microstructures is
introduced. Field equations are obtained via a variational principle, with
natural boundary conditions. It is proved that scalar unidimensional bodies and
Cosserat solids are included in this model. Waves and stability problems are
briefly discussed.

**Key words: **microstructured solids, nonlinear waves, stability.

Numerical
simulation of waves and fronts in structured materials: a thermodynamic
approach; 30–42

Arkadi Berezovski, Jüri Engelbrecht, and Gérard A. Maugin

**Abstract.** A thermodynamically consistent form for the finite-volume
numerical method for thermoelastic wave and front propagation is proposed. Such
reformulation provides applicability of the Godunov-type numerical schemes
based on averages of field variables to the description of nonequilibrium
situations. The nonequilibrium description uses contact quantities instead of
numerical fluxes. These quantities satisfy the thermodynamic consistency
conditions which generalize the classical equilibrium conditions.

**Key words:** thermoelastic waves, phase transition fronts, finite-volume
methods, thermodynamics of discrete systems.

Nonlinear
waves guided at a liquid–solid interface; 43–62

Andreas P. Mayer and Alexander S. Kovalev

**Abstract.** Nonlinear acoustic waves propagating at the interface between
a solid and a fluid with a compressibility much higher than that of the solid
are considered. It is shown that their waveform evolution in the fluid is
governed by the two-dimensional Zabolotskaya–Khokhlov (ZK) equation, with a
linear boundary condition determined by the acoustic mismatch between fluid and
solid. Two evolution equations used for the interpretation of recent
experiments are derived as two different limiting cases of the ZK equation,
with the corresponding boundary condition at the interface. The possibility of
the formation of solitary waves is discussed for the case of Scholte waves
becoming dispersive due to inhomogeneity of the solid.

**Key words:** Scholte waves, interface waves, solitons.

Lattice
modelling of nonlinear waves in a bi-layer with delamination;
63–75

Karima R. Khusnutdinova and Vadim V. Silberschmidt

**Abstract.** A lattice model consisting of two one-dimensional periodic
chains with linear links between elements and nonlinear interaction between the
chains is suggested to study nonlinear dynamics of a bi-layer. The properties
of the model are discussed, and the influence of a delamination zone on the
propagation of solitary waves is studied numerically.

**Key words:** bi-layer, delamination, Frenkel–Kontorova model, coupled
Klein–Gordon equations, nonlinear waves.

Exact
discrete breather solutions and conservation laws of lattice equations;
76–84

Mikhail M. Bogdan and Gérard A. Maugin

**Abstract.** Exactly-solvable differential-difference equations describing
the nonlinear dynamics of one-dimensional lattices and electrical
transmission lines are investigated. The equations considered are discrete modified
Korteweg–de Vries equation, nonlinear self-dual network equations, and the
Hirota lattice equation. Explicit expressions for principal integrals of motion
of the equations are presented and discussed. All the above-mentioned
differential-difference equations have exact discrete breather solutions.
Conservation laws and values of the basic integrals, energy, energy flow, and
total momentum for main types of solitons, as well as the adiabatic invariant
for the discrete breather, are found. Quasi-classical quantization of the
discrete breather oscillation is performed and the breather energy spectrum
found.

**Key words: **lattice equations, exact soliton solutions, discrete breathers,
integrals of motion.

Selection
of localized nonlinear seismic waves; 85–93

Alexey V. Porubov, Vitaly V. Gurski, and Gérard A. Maugin

**Abstract.** The asymptotic solution is obtained for the nonlinear
evolution equation governing seismic wave propagation in the Earth’s crust. The
conditions are found under which the amplitude and velocity of an initial
solitary wave tend to the finite values prescribed by the equation
coefficients. Numerical simulations demonstrate validity of these predictions
in case of an arbitrary localized pulse evolution, and in the presence of the
solitary wave interactions.

**Key words:** solitary wave, nonlinearity, dispersion, dissipation.

Nonlinear
excitations of incommensurate surface structures; 94–102

Alexander S. Kovalev, Igor V. Gerasimchuk, and Gérard A. Maugin

**Abstract. **Nonlinear dynamics of the
incommensurate surface layer with a spatially periodical structure is investigated
analytically. In the framework of the Frenkel–Kontorova model the nonlinear
excitations of the periodic soliton lattice, such as moving additional kinks and gap
solitons, are discussed.

**Key words: **incommensurate
structure, kink, gap soliton.

Wave
propagation in dissipative microstructured materials;
103–114

Tarvo Sillat and Jüri Engelbrecht

**Abstract.** One-dimensional deformation waves in microstructured materials
are described by an hierarchical evolution equation that clearly distinguishes
macro- and microstructural behaviour. The pseudospectral method is used for
numerical simulation supported by the analytical solution for the linear case.
It is shown how dissipative effects on various scales affect the harmonic wave.
The shock wave formation on macroscale is strongly influenced by the
microstructure. The results of this study can be used for material processing.

**Key words: **wave hierarchy, dissipation, microstructure.

On the
existence of bulk solitary waves in plexiglas; 115–124

Alexander M. Samsonov, Galina V. Dreiden, and Irina V. Semenova

**Abstract.** The study of the nonlinear waves in plexiglas was aimed to
prove the nonexistence of positive strain (compression) solitary waves in this
polymer. The estimation and calculation were based on the only, relatively old
data available on plexiglas’ elasticity. We succeeded to generate and observe
for the first time a compression solitary wave in plexiglas and prove that this
polymer is a transparent material suitable for observation of the *compression
soliton* in an elastic solid wave guide, and may be of interest for
applications in fracture or nondestructive testing.

**Key words: **soliton, nonlinear elasticity, solids, plexiglas, holographic
interferometry.

Solitons
in hierarchical Korteweg–de Vries type systems;
125–134

Lauri Ilison and Andrus Salupere

**Abstract. **Wave propagation in dilatant granular materials is studied by using
a hierarchical Korteweg–de Vries type evolution equation. The model
equation is solved numerically under harmonic initial conditions. The behaviour
of the solution is described and analysed over a wide range of material
parameters (two dispersion parameters and one microstructure parameter). Two
main solution types with different subtypes are introduced. The character of
the both solution types is found to be solitonic.

**Key words:** dilatant granular materials, solitons, wave hierarchies,
Korteweg–de Vries type evolution equations.

On the
formation of solitons in media with higher-order dispersive effects;
135–144

Olari Ilison and Andrus Salupere

**Abstract.** Wave propagation in microstructured materials is strongly
influenced by dispersive effects. In the present paper two
Korteweg–de Vries type model equations, with the third- and fifth-order
dispersion, are studied. Both model equations are solved numerically, under
harmonic initial and periodic boundary conditions, by making use of the
pseudospectral method. The character of the solution is found to be solitonic
in both cases. The number of visible and hidden solitons in the emerging train
is detected. Phenomena of recurrence and super-recurrence are examined.

**Key words:** microstructure, dispersion, nonlinearity,
Korteweg–de Vries type evolution equations, solitons, pseudospectral
method.

Periodically
forced solitonic structures in dispersive media; 145–156

Andrus Salupere and Martti Kukk

**Abstract. **The influence of the amplitude-dependent periodic driven field on
the formation and propagation of solitary waves in nonlinear dispersive media
is studied. The model equation – the forced Korteweg–de Vries equation –
is integrated numerically under harmonic initial and periodic boundary
conditions by using the pseudospectral method. Main attention is paid to
solitonic solutions. The driven field is classified as weak, moderate, strong
or dominating, according to the character of the solution. The solution is
found to be solitonic in the case of weak, moderate, and strong fields.

**Key words:** periodic driven field, forced Korteweg–de Vries equation,
solitary waves, solitons, pseudospectral method.

Nonlinear
interaction of waves with material inhomogeneity; 157–168

Andres Braunbrück and Arvi Ravasoo

**Abstract.** A relatively simple method for nondestructive evaluation of
weak and smooth variation of the physical properties of the material from their
constant values is proposed. The method is based on the analysis of nonlinear
effects of simultaneous propagation, reflection, and interaction of two
ultrasonic waves in the material. The results of the analysis enable one to
solve several problems of material parameter evaluation provided some
preliminary information about the material is available.

**Key words: **longitudinal waves, nonlinear interaction, inhomogeneity, material
characterization.