Nonlinear Rayleigh waves in a medium with a monatomic nonlinear coating
A. S. Kovalev* and E. S. Sokolova
B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sciences
of Ukraine, pr. Lenina, 47, 61103Khar’kov, Ukraine
A. P. Mayer
Institut fu ¨r Theoretische Physik, Universita ¨t Regensburg, D-93040, Regensburg, Germany
G. A. Maugin
Laboratoire de Mode ´lisation en Me `canique, Universite ´ Pierre et Marie Curie, 75252, Paris, France
Submitted March 3, 2003
Fiz. Nizk. Temp. 29, 530–538 May 2003
The nonlinear dynamics of surface acoustic waves at the surface of a linear elastic half-space
coated with a monolayer of a nonlinear material is investigated. A one-dimensional
nonlinear integrodifferential equation describing the dynamics of such a system is derived. The
model proposed is used to study Rayleigh solitons with a stationary profile. The possible
phenomenological generalizations of the equations derived and their exact soliton solutions are
discussed. © 2003 American Institute of Physics. DOI: 10.1063/1.1542503
The theory of nonlinear acoustic waves in one-
dimensional atomic chains has been developed in detail.
1,2
The problem of nonlinear acoustic surface waves at the sur-
face of an anharmonic half-space is much more complicated
because it is two-dimensional.
3–12
However, this problem be-
came especially topical immediately after a series of experi-
ments were performed on the propagation of nonlinear sur-
face waves
13,14
and high-intensity acoustic surface
pulses.
15–17
Most experiments were performed on samples
consisting of a substrate coated with a film of a different
material. The presence of a film coating is very important
because it results in the appearance of an additional strong
dispersion of linear waves and the competition between this
dispersion and the nonlinearity gives rise to stationary non-
linear surface waves and surface solitons with a stationary
profile. The analytic study of these nonlinear waves simpli-
fies somewhat when the substrate can be treated in the linear
approximation.
18
The two-dimensional problem for the sub-
strate can be solved exactly. It is important to underscore that
such a formulation of the problem a linear half-space with a
nonlinear coating is fully realizable experimentally. For ex-
ample, in Ref. 19 experimental data are presented for anhar-
monic surface phonon effects in systems with a metallic sub-
strate coated with a monolayer of inert-gas atoms Ar or Xe.
A similar situation arises for an inert-gas monolayer on a
graphite surface.
1. FORMULATION OF THE MODEL
We shall consider the propagation of a nonlinear surface
acoustic wave in the X direction along the surface of the
half-space Z 0 coated with a monatomic layer. The dis-
placements are independent of the coordinate Y and the
problem is effectively two-dimensional. We shall confine
our attention to the particular case where the nonlinear inter-
action of the atoms in the surface monolayer with one an-
other and with atoms of the substrate surface is central. Each
atom of the monolayer interacts with its nearest neighbors in
the direction of the X axis in the monolayer and with the
nearest neighbor and next two neighboring atoms in the sub-
strate surface see Fig. 1. Keeping only the first nonlinear
terms in the expression for the interaction of the atoms in the
surface film with one another and with substrate atoms we
obtain the following expression for the total energy of the
monatomic coating:
E =
n
M
2
dU
n
dt
2
+
dV
n
dt
2
+
n
2
n , n +1
2
-
3
n , n +1
3
+
2
n
2
-
3
n
3
+
2
n , n +1
2
-
3
n , n +1
3
+
2
n , n -1
2
-
3
n , n -1
3
, 1
where U
n
and V
n
are the displacements of the n th atom in
the monolayer in the X and Z directions;
n , n +1
=
U
n
-U
n +1
-a
2
+ V
n
-V
n +1
2
-a =
˜
n , n +1
-a
is the deviation of the interneighbor distances in the surface
film from their values a in equilibrium; M is the mass of an
atom in the monolayer;
FIG. 1. Geometry of the problem.
LOW TEMPERATURE PHYSICS VOLUME 29, NUMBER 5 MAY 2003
394 1063-777X/2003/29(5)/7/$24.00 © 2003 American Institute of Physics