VOLUME
71,
NUMBER 13 PHYSICAL REVIEW
LETTERS
27 SEPTEMBER
1993
Nonlinear
Sinusoidal
Waves
and Their
Superposition
in
Anharmonic
Lattices
Yuriy
A. Kosevich
Department
of
Physics,
Bilkent
University, 06533
Bilkent,
Ankara,
Turkey
and
lV
ational
Surface
and Vacuum
Research
Centrel, l 7334
Moscow,
Russia
(Received
16
3une 1993)
A
new
type
of finite-amplitude
traveling or
standing
wave with
an
exact sinusoidal
form
and
a short
commensurate
wavelength is
predicted to exist in
lattices with
cubic
and/or
quartic
anharmonic
potential
between
any
arbitrary number of nearest
and
non-nearest
neighbors.
Fast
traveling nonlinear
sinusoidal
waves
(NSW) can
generate
sinusoidal
lattice
solitons.
Superposition
of two
NSW
or
sinusoidal solitons
propagating
in
opposite
directions
can
result in
the formation
of an extended
or
a localized
standing-
wave
eigenmode.
New
exact solutions
for localized
standing-wave
structures
are
found within
a
rigorous
discrete-lattice
approach.
PACS
numbers:
63.70.
+h,
63.20.
Pw,
63.20.
Ry
The
only
known
exact solution of nonlinear
discrete-
lattice classical
potential equations
of motion concerns
the
case of the lattice
with
exponential
nearest-neighbor
anharmonic
potential
(Toda
lattice)
[1].
The
exact
periodic
solution
in
the Toda
lattice
has the
form
of
a
cnoidal wave with
arbitrary wavelength,
which
trans-
forms into
a
sinusoidal wave
only
in the small-amplitude
limit. Sinusoidal
waves,
being
the fundamental
low-lying
excitations
(phonon
eigenmodes)
of the
lattice,
are par-
ticularly important
in its
dynamics.
In this
Letter we
present
an exact solution
of nonlinear
discrete-lattice
equations
of motion
of
the
one-dimensional
(1D)
anhar-
monic
lattice in the
form of finite-amplitude
traveling
or
standing
sinusoidal waves with
a
short and commensurate
wavelength.
The lattice is characterized
by
harmonic and
cubic
and/or
quartic
anharmonic
interparticle
potential.
The case
of
the
lattice
with
nearest-neighbor harmonic
and
cubic or
quartic
anharmonic
potential corresponds
to
the
famous I
ermi-Pasta-Ulam
model,
for which the
re-
currence
phenomenon
in the nonlinear
motion was
re-
vealed for the
first time in
computer
simulations
[2].
Be-
cause of the
commensurability of the characteristic
wave-
length
with
the lattice
period,
nonlinear
sinusoidal waves
(NSW) are exact
eigenmodes
of a
1D
lattice with
cubic
and/or
quartic
anharmonic
potential
between
any
arbi-
trary
number
of nearest and
non-nearest
neighbors.
In
anharmonic
lattices
of
higher dimensionality,
plane
NSW
can
propagate
in
certain
directions
(of
high
symmetry).
Contrary
to the linear
sinusoidal
waves,
the
frequency
of
NSW
depends
on the
amplitude,
while
the wave number
is
determined
(fixed)
by
the
anharmonic interactions.
Therefore
large-amplitude
vibrational
eigenstates of the
anharmonic
lattice can be classified with
the
help
of
NSW with
amplitude-dependent
frequency
(and
velocity)
and
a
given wavelength,
and
hence NSW
can
contribute
to the
specific
heat
and
energy
transport
in
the
system.
NSW with
velocity
larger
than the harmonic
wave
veloci-
ty
can
generate
sinusoidal
lattice
solitons of
substantially
difrerent
form
than that
of solitons
of
the Korteweg-de
Vries
(KdV)
(or
modified
KdV
equation),
which
de-
scribes
the
weakly
nonlinear
long-wavelength
dynamics
of
the lattice.
Strongly
localized
solitary
"peaks"
propaga-
ting
at
speeds larger
than the
harmonic wave
speed
were
revealed
in
computer
simulations
of anharmonic
vibra-
tions
of
one-
and two-dimensional
lattices
[3],
while
re-
cent
analytical
and
numerical studies
[4,
5]
concern
only
the discrete-lattice
solitons
moving
with
speeds
much
lower
than the harmonic
wave
speed.
Superposition
of
two NSW
propagating
in
opposite
directions
with
equal
amplitudes can result in
the formation of
a
stable
standing-wave
extended
eigenmode,
while
superposition
(during
resonant
head-on
collision)
of
two
sinusoidal
soli-
tary
waves
can result
in the
formation of
a localized
non-
linear model.
Such
a
phenomenon,
which is
quite
unusu-
al for
the behavior
of
solitary
waves
in
completely
inte-
grable systems
where
solitons
restore their
coherent
shapes
after
colliding
[6,7],
was
also revealed in
computer
simulations
[3].
New
exact solutions
for
strongly
local-
ized dark-profile
standing-wave
lattice
structures are
found. We claim
that
the existence
of
NSVV
in
anhar-
monic
lattices is
not occasional
since localized
short-
wavelength
lattice
vibrational
modes with
sinusoidal
en-
velope
and
small
amplitude-independent
spatial
width
are
exact
solutions
of the
continuous envelope-function
equa-
tion with
the nonlinear
gradient
terms
[8].
Solitary
waves
with
compact
support
(compactons)
and sinusoidal
profile
were
recently
obtained
as the
solutions
of the
con-
tinuous
KdV-like
equations
with
nonlinear
dispersion [9].
[Continuous
KdV-like
equations
with
nonlinear disper-
sion,
which
represent
the
corresponding
long-wavelength
limit
of the
below
discrete-lattice
equations
(2),
also have
exact solutions
in the form of
NSW
[10]1.
We
start
from the
model of
a monatomic
periodic
chain with
anharmonic
potential
of the
order r
=3
and/or
r
=4
between 6
~
l
nearest and
non-nearest
neighbors,
U=g
g
g
—
It,
,
(u„+,
—
u.
)',
l
n
y=2b=1
where
u„
is the (real
scalar)
displacement of
the nth
par-
ticle
from
its
equilibrium
position
and
Kyq
are harmonic
(y=2)
and
anharmonic
force
constants.
The lowest
or-
der
potential
describing the
anharmonicity of the longitu-
dinal
or
pure
transverse
motion
in
a
centrosymmetric lo
lattice
corresponds to
r=3
or
r=4,
respectively.
From
0031-9007/93/71
(13)/2058
(4)$06.00
1993
The
American
Physical
Society
VOLUME
71,
NUMBER
13
PH
YSICAL REVIEW
LETTERS
27
SEPTEMBER
1993
mu„=
g
g
K„~[(u„+,
u—
„)
'
'
—
(u„—
u„g)
"
'l
.
y
2b
1
(2)
We
are
looking
for the solution
of
Eqs.
(2)
for the
dis-
placements
u„(or
relative
displacements
r„—=
u„+i
—
u„)
in
the
form of
traveling (shifted)
sinusoidal
waves:
=2
cos(kna
—
cot
+
p)
+
8
.
r„
(3)
In the lattice with
quartic
anharmonic
nearest-
neighbor
interaction
(ANNI),
NSW with
u„pattern
(3)
are exact solutions
of
Eqs.
(2)
with
the
amplitude-
dependent
frequency
t0 and
definite
wave
number k:
Eq.
(I)
we
obtain the nonlinear discrete-lattice
equations
of motion:
Using
trigonometric
identities, it
can
be
easily
shown
that
linear
superposition
of
two
NSW
with
u„patterns (3)
propagating
in the
same
direction
with
B=O,
equal
am-
plitudes
2
and
different
initial
phases
p~ 2
is also an
exact
eigenmode
with
the
same
amplitude
8,
if
the
phase
difference
p~
—
p2
is
equal
to
2tt(n
~
—,
'
tt)
with
an
integer
n.
;
i.e.
,
if it
coincides
with
one of
the
allowed
values
of ka
given
by
Eq.
(5). Then,
using
the
identity
u„+q
—
u„
~8
—
I
=~y=or„+y
and
the above
property
of
superposition
of
two
NSW,
it
can
be shown
that NSW
with
r,
pattern
(3)
are exact
eigenmodes of
Eqs. (2)
with
~ka~
=
—,
'
tr
(or
~ka~
=x)
in the
lattice
with
cubic+quartic
(or
cubic)
anharmonic
interaction
between
any
arbitrary
number
6~
1
of nearest
and
non-nearest
neighbors.
We
can consider
a
"small-amplitude
modulation"
of
NSW
(3)
in the
form
mco
=3K2+
(27/4)K43
(4)
=
2
cos(kna
—
cot+&)
rn
K4A
sin
(2
ka)[sin
(2
ka)
—
3cos
(2
ka)1=0,
(5)
+gb;
cos(k;na
—
co;t+P;)+8, (7)
where
K„—
=
K„
l.
Equation
(5)
is obtained from
requiring
the absence of
third (and
correspondingly
higher)
har-
monic
contribution
to
NSW
[3].
In
the
reduced-
Brillouin-zone
picture,
we
find
the
"allowed"
wave
nurn-
bers:
ka
=
~
3
x.
In
a 10 lattice
with harmonic and
quartic
anharmonic
interaction between 6
~
2
non-
nearest
neighbors,
parameter
ka
in
Eq.
(5)
should
be
re-
placed
by
the
parameter
ka6,
and
the
corresponding
equation
is also satisfied
by
the above
values
of
the wave
number k. Therefore
NSW
(3)
with
a
commensurate
wavelength k
=3a
are exact
eigenmodes
of 1D
lattice
(I)
with
harmonic and
quartic
anharmonic interaction
be-
tween
any
arbitrary
number
of
nearest and
non-nearest
(8)
1) neighbors.
For wavelengths
different
from
the
above
one,
the
term
proportional
to the
left-hand
side of
Eq.
(5)
determines the
generation
of
the
higher
harmon-
ics
of
NSW
(3).
Therefore
the
generation
is
strongly
suppressed
for
NSW with wave numbers close
to
the
al-
lowed ones.
NSW with
u„pattern
(3), edge
Brillouin-zone
wave-
length
ka
=
~
tt,
and (amplitude-independent)
upper
cutoff
frequency
of
harmonic oscillations
(mai
=4K'
for
6
=
I
)
are exact solutions of
Eqs.
(2)
in a 1 D
lattice with
cubic
anharmonic interaction between
any
arbitrary
num-
ber of nearest and
non-nearest
neighbors.
NSW with
r„pattern
(3)
and
frequency
where
b;
«A,
k;,
and
co;
are
(small)
amplitudes,
wave
numbers,
and
frequencies of modulation.
Then from
Eqs.
(2)
we
can find
that
"linear"
sinusoidal waves
(7)
(with
amplitudes
b;)
are lattice
modes
with
the Goldstone
harmonic-wave
dispersion
relation with
the
force constant
renormalized
by
NSW
(3).
For
instance, in the lattice
with
quartic ANNI,
the ith linear
sinusoidal wave
with
u„pattern
(7)
and
k,
=
—,
'
tr
has
spectrum
mco;
=4sin
(
2
k;a) [K2+ (9/4)K4A ].
In
the
lattice with
cubic+quartic
ANNI,
the
ith
linear
sinusoidal wave with
r„pattern (7)
has
a
spectrum
of the form
similar
to
Eq.
(6)
(and
with the
same
notation):
mai;
=4
sin
(
&
k;a ) [K2+2KiB
+K4[38
+A (1+
~
cos(ka))]}.
Therefore finite-amplitude
NSW
(3),
(4),
and
(6)
with
&
0
are stable
eigenmodes
of an anharmonic
chain
(1). Moreover,
they
represent a new
basis
for the
classification
of the vibrational
eigenstates
of the
system.
In the lattice
solitary
wave,
relative
displacements
of
neighboring particles
r„(as
well
as
displacements
u„or
values h, u„=u
—
u„,
where
u is
the
finite
displacement
at
infinity;
see below) vanish
at both
infinities
according
to
the
exponential
law,
the inverse
decay length
rc in
which
is
determined
by
the
linear
part
of
Eqs.
(2):
mai
=
4sin
(ka/2)
&&
[K2+2K38+K4[38
+A
(I+
~
cos(ka))]]
r„~e
p[x~ x(na
ct)],
—
4K2sinh
(xa/2)
=mc
x
(9)
(6)
are
exact
eigenmodes of
Eqs.
(2)
for
ka
=+'
x and
arbi-
trary
8
in
the
chain with
cubic
ANNI
(K4
=0),
and
for
ka
=
~
—'
,
tt,
8
=
—
K3/3K4
in the chain
with
cubic
+quartic
ANNI
(or
with
quartic ANNI,
K3=0,
8
=0).
where
c
is
a
velocity
of the lattice
solitary
wave.
Equa-
tion (9) has a solution for the
real
x.
only
in the
case
when
velocity
c
is
larger
than the harmonic wave
velocity:
c)
JK2a
/m.
So
far
as the
(phase) velocity
c
=—
ro/k
of
NSW
(3)
depends,
in
the
general
case,
on the
amplitude
2059
VOLUME
71,
NUMBER 13 PHYSICAL REVIEW
LETTERS
27 SEPTEMBER
1993
A (or
8),
the form
of
the large-amplitude
lattice solitons
can be
obtained
by
matching
the
sinusoidal
r„pattern (3)
of
"supersonic"
NSW (near
corresponding
zeros)
with
exponential
tails
(8).
The
matching
can be
performed
by
solving
Eq.
(2)
for the
particles
(with
u„«A
or
Au„«A)
at the
moving
"border
points"
between the sinusoidal
r„
pattern (3)
and (small-amplitude)
tails
(8).
For
this
con-
struction it is essential that fast
lattice
solitary
waves
(with
mc
)&K2a
)
have
very
short
exponential
tails:
tea&)1
[see
Eq.
(9)].
Therefore
large-amplitude
solitary
waves
in
the
anharmonic
lattice
(
I
)
actually
have
sinusoidal form.
In
the lattice
with
hard
quartic
ANN!
(K4&
0),
any
half
period
(between
two
successive
zeros) of
the
cosine
r„pattern
(3)
with
ka
=
—'
,
tr,
mcus
=3K2+
4
K4A,
and
8=0
corresponds to a half
period
(between
zero
and
suc-
cessive
maximum)
of
the
cosine
u„pattern (3)
with the
amplitudes
A'=A/v3
and
~8'~
~ A'.
Thus a lattice
soli-
tary
wave,
which is
described
by
a half
period (or
by
a
period)
of the
cosine
r„pattern (3)
with
8=0,
—
tr/2
&
kna
—
cot
+
p
&
+
tc/2 (or
—
tr/2
&
kna
—
cot
+
P
&
3tr/
2),
corresponds,
e.
g.
,
to a
steplike
with
u
=0
and
u~
=2A/J3
(or
a
pulselike
with
u
—
=u+
=0)
cosine
u„pattern (3):
u„=u
~
cos
[(kna
—
cot+/')/2],
—
tr
&
kna
—
kct+p'
&
0
(or
—
tr
&
kna
—
cot+&'
&
+tr).
The
former
(steplike) solitary
wave
represents
the gen-
eralization
of the soliton
of
the
(weakly
nonlinear)
modified KdV
equation
in the limit of
large
amplitudes,
when
mc
»K22
and the soliton
has small amplitude-
independent
spatial
width
A =3a/2.
The
latter (pulse-
like) lattice
solitary
wave
has
no
counterpart
in
the
weak-
ly
nonlinear limit.
These
most
strongly
localized
fast
soli-
tary
waves
can be
generalized
to the
traveling
localized
modes with
three
and more half
periods
of the
cosine
r„
pattern
(3).
In
the lattice
with
cubic
ANNI, large-amplitude
soli-
tary
waves
can be obtained
by
matching
the
cosine
r„pattern (3)
of
supersonic
NSW (with
ka
=tr,
mt'
=4K2+8K38,
K38»K2
and
A
~
~B~)
with
short
exponential
tails
(8).
Then the
one-pulse
r„pattern
[r„=A
cos
[(kna
—
kct
+
p)/2],
—
tr
&
kna
—
kct
+
p
&
tr,
B=A]
corresponds
to a one
step
with
u
—
=0
and
u+
=A
cosine
u„pattern:
u„=A
cos
[(kna
—
kct
+
P')/2],
—
tr
&
kna
—
kct
+
P'
&
0.
This
solitary
wave
represents
the
generalization of
the soliton
of the KdV
equation
in the
large-amplitude
limit,
when
mc
»K2a
and
a
steplike distribution of
the lattice
displacements
u„
is located
on
an
interparticle
spacing
a.
The two-pulse
r„
pattern corresponds to
a
two-step
u„distribution
(be-
tween
u
—
=0
and
u+
=2A),
etc.
In the lattice with
cubic+hard
quartic
ANN
I,
large-amplitude
solitary
waves with
similar
u„patterns
can
be
obtained
by
match-
ing
the cosine
r„pattern (3)
of
supersonic
NSW
[with
ka
=
—
',
tr,
mco
=3K2+
(9/4)K4A
—
K3/K4,
8
=
—
K3/
3K4
and
arbitrary
amplitude
A
~
~8~]
with
short
ex-
ponential
tails
(8).
We can
consider a sinusoidal
standing-wave
mode
as a
superposition
of
two
NSW
(3)
propagating
in
the
oppo-
site
directions
with
equal
amplitudes
2:
ug
=
A(t)cos(kna
+
a)
+
8
.
r„
(10)
In the lattice with
cubI'c
anharmonicity,
the mode with
u„
or
r„pattern (10)
is
an
exact solution
of
Eqs.
(2)
with
~ka
~
=tr,
arbitrary
a,
and
A(t)
=A
cos(cot)
(where, e.
g.
,
mao
=4K2
or men =4K2+8K3B
in the
lattice with
the
nearest-neighbor
interaction).
In
the
lattice with
quartic
(or
cubic+quartic)
anharmonicity,
sinusoidal
modes
(10)
are exact solutions
of
Eqs.
(2)
with
~ka
~
=
—
',
tr and
arbi-
trary
a,
and also
with
~ka
~
=tr
or
~ka
~
=
2
tc and definite
initial
phase
a,
which
is
determined from
the
requirement
cos[3(kna+a)]
=
+'cos(kna+a)
(e.
g.
,
a
=0).
Time
dependence
of
the
amplitude
A(t)
of
these
modes
is
governed
by
the
eA'ective
equation
of motion
of
a
decou-
pled
(single)
anharmonic
oscillator.
Thus we find
the
equations
m A
=
—
2K2A
—
2K4A,
mA
=
—
3K2A
—
(27/4)K4A,
and
mA=
—
4K2A
—
16K4A
for the
modes with
u„pattern (10)
and
ka
=
2
tr,
ka
=
—
',
tr,
and
ka
=z
in
the lattice
with
quartic
ANNI. For
the mode
with
r„pattern (10)
and
ka
=
3
tc,
8
=
—
K3/3K4,
we
find
the
equation
mA
=
—
[3K2
—
(K3/K4)]A
—
(9/4)K4A
in the
lattice with
cubic and
quartic
ANNI
(or
quartic
ANNI,
K3
=0,
8
=0).
From
these
equations we can
ex-
actly
determine the
period
of
oscillation T
of nonlinear
sinusoidal
mode
(10)
as a
function
of
amplitude Am.
„(or
energy
per
single
oscillator)
(see,
e.
g.
,
[11]).These
equa-
tions can
also
be
easily
solved
within
a
"rotating
wave ap-
proximation"
(RWA) when
A(t)
=Acos(cut)
and
only
a
single
frequency component
is
included
in
the time depen-
dence.
For the
considered
short-wavelength
optical-like
oscillations, this
approximation
holds
due
to
the
weakness
of
nonresonant
interaction
between
the modes with
fun-
damental
frequency
and
its third
harmonic.
In
the
lattice
standing-wave
localized
mode with
bright
profile,
the
envelope
of
particle displacements
f„=(
—
I)"u„decays
at
both infinities
according
to the
exponential
law
[cf.
Eqs.
(8)
and
(9)]:
f„cx:exp(~
qan)cos(cot),
4K2cosh
(qa/2)
=mco
(12)
Equation
(12)
has
a
solution
for
real
q
only
for
the
local-
ized
mode with
frequency higher
than
the
upper
cutoA
frequency
of
harmonic
oscillations
(mco
&
4K2),
and for
the
high frequency
anharmonic
mode the
decay length
of
the
exponential tail
(11)
is much
shorter than
interparti-
cle
spacing: qa))1 for mco
))4K2.
Therefore the
f„
pattern
of
the localized
standing-wave
mode can
be
ob-
tained
by
matching (near
corresponding
minima)
the
en-
velope
function
f„=(
—
1)
"u„of
sinusoidal mode
(10)
(with oscillation
period
T)
with
short
exponential
tails
(11)of the
same
(high)
frequency
co=2rc/T.
In
this
way
2060
VOLUME
71,
NUMBER
13 PHYSICAL REVIEW
LETTERS
27 SEPTEMBER
1993
we
can
establish
that
in
the lattice
with
hard
quartic
anharmonicity,
two
symmetric
most
strongly
localized
large-amplitude
sinusoidal
modes of
odd
and
even
parity
exist,
which
correspond to
a half
period
of the cosine
en-
velope
function
f„=A(t)cos(
—,
'
trna+a) with
a
=0
or
a=
—,
'
tr
and
have
(approximate)
displacement
patterns
u„=A(t)(.
. .
,
0,
—
—,
',
I,
—
—,
',
0,
. . .
)
or
u„=A(t)(. . .
,
O,
—
1, 1,0,
. . .
).
Intrinsic
localized modes
with
such
dis-
placement
patterns
were
previously
revealed
in
asymptot-
ic
analytical
and numerical studies
of
the
dynamics
of
monatomic lattice
with
hard
quartic
ANNI
[12,
13]
(see
also
[8]
for
the
continuous envelope-function
description
of these and other nonlinear localized
modes).
A
slowly
moving
large-amplitude
localized
sinusoidal mode can
also
exist in
the
lattice,
which
has
in the RWA the
en-
velope
f„=A
cos
[k
(na
—
Vst
)
]cos(hk
—
cot
)
[for
~
k
(na
—
Vgt)~
(
—,
'
n,
ka=
—,
'
tr,
mco
=3K2+(81/16)K4A
],
where
Vg(hk,
A)
«
QK2a
/m
and hk «k
are the
(small)
group
velocity
and
"reduced"
wave
number
of
the
mode.
In
the
lattice with cubic+hard
quartic
anharmoni-
city,
a
standing-wave localized mode
with
the
one-period
cosine
r„pattern
(10)
exists [with
ka=
—,
'
tr,
8
=
—
K3/
3K4,
A )&
~8
~,
and
mco
=3K2+
(27/16)K4A
—
K3/K4
in
the
RWA],
which has
(slightly)
asymmetric
u„and
f„
patterns.
Essentially
the
formation of
any
of the above
nonlinear
sinusoidal standing-wave
(or
slowly
moving)
lo-
calized
modes can occur
in
consequence
of
superposition
(during
resonant
head-on
collision)
of
two
sinusoidal
soli-
tary
waves
[with
the one-period
cosine
r„patterns (3)]
propagating
in
opposite
directions
with
equal
(or
close)
amplitudes
A,
similar
to
the
formation
of
the extended
sinusoidal standing-wave mode
(10).
Matching
sinusoidal
eigenmodes (10)
with
ka
=
z
tr,
ka
=
3
n,
or
ka
=
x,
we can
find new exact solutions
for
strongly
localized transition
regions
between
two
extend-
ed
standing-wave
vibrational domains of
definite
wave
number
in the lattice
with
soft
quartic
ANNI
(K4&0).
Indeed,
we reveal
in
the
RWA that
(at
least)
three
ex-
tended
eigenmodes
[with
the
patterns
u„=Apcos(
—,
'
trn)
&icos(copt),
u„=Ap(44/3)cos(
—,
'
trn+
—,
'
tr)cos(copt),
and
u„=Apcos(trn)cos(copt)]
exist in the
lattice,
in
which
particles
oscillate with the
same
frequency coo
=
jI2K2/7m
and
amplitude
Ap=
j4K2/21~K4~.
With
the
help
of
these
eigenmodes,
we
easily
obtain
the exact
form
of
(most
strongly
localized)
domain
walls between
two wavelength-four modes [u„=Ap(. . .
,
—
1,0,
1,0,
—
1,
1,0,
—
1,0,
1,
. . .
)cos(copt
)],
between two
upper
cutoff
modes [u„=Ap(.
. .
,
1,
—
1, 1,
0,
—
1, 1,
—
1,
.
. .
)
x
cos(copt
)],
between
the
upper
cutoff and
wavelength-
four
modes
[u„=Ap(.
.
.
,
—
1,1,
—
1,0, 1,0,
—
1,
0, 1,
.
.
.
)
x
cos(copt
)
],
etc. A
link with the
pattern
u„
=Ap(0,
—
1,
1,
0)cos(copt)
can
play
the
role
of
a
kink in
the
upper
cutoff
(or
wavelength-four)
mode,
while
a
link
with
the
pattern
u„=Ap(0,
1,
0)cos(copt)
can
play
the role
of
a
kink
in
the
ka
=
—,
tr
(or
upper
cutoff)
mode.
All of
these dark-profile localized
structures
in the lattice with
interparticle
anharmonic
potential
(1)
substantially
differ
from
similar structures in
the
lattice with
on-site anhar-
monic
potential,
which have
been
recently
observed in the
lattice of
coupled pendulums suspended
in
a
gravitational
field [14].
In
conclusion,
we have shown that finite-amplitude
traveling
or
standing
sinusoidal
waves
with
a
short and
commensurate
wavelength
and,
in
general,
amplitude-
dependent
frequency,
are
exact
eigenmodes
of
a
1D
lat-
tice
with
cubic and/or
quartic
anharmonic
potential
be-
tween
any
arbitrary
number of nearest and
non-nearest
neighbors.
New
dark-profile localized standing-wave
structures are
predicted
within
a
rigorous
discrete-lattice
approach.
The existence of NSW influences the
classi-
fication
of
the large-amplitude
vibrational
eigenstates,
and therefore NSW must
be
considered in
a
complete
thermodynamic description
of
the
anharmonic
lattices.
Supersonic traveling
NSW
can
generate
sinusoidal lattice
solitons and contribute to the
energy
transport
in
the sys-
tem. The
processes
of the formation
of
standing-wave
(or
slowly moving)
nonlinear sinusoidal
modes, including
lo-
calized
ones,
as a
consequence
of the
superposition
of
two
NSW
traveling
in
opposite
directions,
can influence
An-
derson
localization of vibrational states in
disordered
anharmonic solids.
I am
grateful
to
M.
J.
Ablowitz for the useful
discus-
sion. I would like
to
thank Bilkent
University
for
the
hos-
pitality
and TUBITAK for the
support
of
the
work.
[I]
M.
Toda,
in
Theory
of
Nonlinear
Lattices, edited
by
M.
Toda,
Solid State Sciences Vol.
20
(Springer,
Berlin,
1981).
[2]
E. Fermi,
J. R. Pasta,
and
S.
M.
Ulam,
in Collected
Works
of
F.. Fermi,
edited
by
E.
Segre
(University
of
Chicago
Press,
Chicago,
1965).
[3]
R.
Bourbonnais and R.
Maynard,
Phys.
Rev.
Lett.
64,
1397
(1990).
[4] S.
R. Bickham,
A. J.
Sievers,
and
S. Takeno,
Phys.
Rev.
B
45,
10344 (1992).
[5]
K.
W.
Sandusky,
J. B.
Page,
and K. E.
Schmidt,
Phys.
Rev.
B
46,
6161 (1992).
[6]
N. J.
Zabusky
and M.
D. Kruskal,
Phys.
Rev.
Lett.
15,
240
(1965).
[7]
M.
J.
Ablowitz
and
H.
Segur,
Solitons and the
Inverse
Transform
Method
(SIAM,
Philadelphia,
1981).
[8]
Yu. A.
Kosevich,
Phys.
Rev.
B
47,
3138 (1993).
[91
P. Rpsenau
and
J.
M.
Hyman,
Phys.
Rev.
Lett.
70,
564
(1993).
[10]
Yu.
A. Kosevich
(unpublished).
[11]
L. D.
Landau and E.
M.
Lifshitz, Mechanics
(Pergamon,
New
York,
1976).
[12]
A. J.
Sievers
and S.
Takeno,
Phys.
Rev.
Lett.
61,
970
(1988).
[13]
J.
B.
Page,
Phys.
Rev.
B
41,
7835 (1990).
[14]
B. Denardo
et
al.
,
Phys.
Rev.
Lett.
68,
1730 (1992).
2061