Periodic moving waves on 2D lattices with nearest-neighbor interactions
We study the existence of periodic moving waves on two-dimensional periodically forced lattices with linear coupling between nearest particles and with periodic nonlinear substrate potentials. Such discrete systems can model molecules adsorbed on a substrate crystal surface.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509181365190656 |
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| author | Fečkan, M. Фечкан, М. |
| author_facet | Fečkan, M. Фечкан, М. |
| author_sort | Fečkan, M. |
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| description | We study the existence of periodic moving waves on two-dimensional periodically forced lattices with linear coupling between nearest particles and with periodic nonlinear substrate potentials. Such discrete systems can model molecules adsorbed on a substrate crystal surface. |
| first_indexed | 2026-03-24T02:37:01Z |
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UDC 517. 9
M. Fečkan (Comenius Univ., Slovakia)
PERIODIC MOVING WAVES ON 2 D LATTICES
WITH NEAREST NEIGHBOR INTERACTIONS*
PERIODYÇNI RUXOMI XVYLI NA DVOVYMIRNYX ÌRATKAX
IZ VZA{MODIQMY NAJBLYÛÇYX SUSIDIV
We study the existence of periodic moving waves on two-dimensional periodically forced lattices with
linear coupling between nearest particles and with periodic nonlinear substrate potentials. Such discrete
systems can model molecules adsorbed on a substrate crystal surface.
Vyvçeno pytannq isnuvannq periodyçnyx ruxomyx xvyl\ na dvovymirnyx periodyçno zburenyx
©ratkax iz linijnym zçeplennqm miΩ najblyΩçymy çastynkamy ta z periodyçnymy nelinijnymy
potencialamy pidkladynky. Taki dyskretni systemy moΩut\ modelgvaty molekuly, wo adsor-
bugt\sq na krystaliçnu poverxng pidkladynky.
1. Introduction. Recently, several papers have been devoted to the dynamics of
structures on two-dimensional (2d) lattice systems. For instance [1, 2], 2d Frenkel –
Kontorova type models are used to study either coherent localized and extended defects
such as dislocations, domain walls, vortices, grain boundaries, etc., which play an
important role in the dynamical properties of materials with applications to the problem
of adsorbates deposited on crystal surfaces; or in superlattices of ultrathin layers; or in
large-area Josephson junctions. On the other hand [3], the existence of longitudinal
solitary waves is shown for 2d cubic Hamiltonian lattices of particles interacting via
harmonic springs between nearest and next nearest neighborhoods which appear in
elastostatic investigation modeling a particle interaction via interatomic potentials,
which is a natural 2d analogy of the 1d Fermi – Pasta – Ulam lattice.
In this paper, we focus on forced 2d Frenkel – Kontorova models and their
generalizations. Motivated by [4], we consider an isotropic two-dimensional planar
model where rigid molecules rotate in the plane of a square lattice. At site ( n, m ) the
angle of rotation is un m, , each molecule interacts linearly with its first nearest
neighbors and with a nonlinear periodic substrate potential. If χ is the linear coupling
coefficient, ω2 is the strength of the potential barrier or square of the frequency of
small oscillations in the bottom of the potential wells and γ cos µ t is the forcing then
the equation of motion of the rotator at site ( n, m ) is (see Figure)
˙̇ , , , , , ,u u u u u un m n m n m n m n m n m= [ + + + − ]+ − + −χ 1 1 1 1 4 – ω γ µ2 sin cos,u tn m + .
(1.1)
J. M. Tamga et al. [4] studied how a weak initial uniform perturbation can evolve
spontaneously into nonlinear localized modes with large amplitudes and investigated
the solitary-wave and particle-like properties of these robust nonlinear entities.
More general countable systems of nonlinear ordinary differential equations like
(1.1) are investigated in the book [5] focusing on the existence and stability of invariant
tories. We also refer the reader for more motivations to study equations on lattices
to [6].
Our paper has the following structure: Section 2 discusses the mathematical
formulation of the periodic moving wave solutions in two-dimensional lattices and its
connection to a small divisor problem. The existence of weak periodic moving waves
in equations like (1.1) is given in Section 3. More regular and classical periodic moving
waves are shown in Section 4. Final Section 5 is devoted to damped and periodically
*
Partially supported by the Grant VEGA-MS 1/2001/05.
© M. FEČKAN, 2008
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1 127
128 M. FEČKAN
forced differential equations on two-dimensional lattices.
The two-dimensional lattice model of rigid rotation molecules with orientation un m, at site ( n, m ).
2. Periodic moving waves. In this section, we consider the infinite system of
ODEs
˙̇ , , ,u u f u h tn m n m n m= ( ) − ( ) + ( )χ µ∆ , ( n, m ) ∈ Z2, (2.1)
on the two-dimensional integer lattice Z2 for f ∈ C1
( R, R ), h ∈ C ( R, R ), χ > 0,
µ > 0 under the following conditions:
(H1 ) f is odd and 2π-periodic, i.e., f ( – x ) = – f ( x ) and f ( x + 2π ) = f ( x ) ∀x ∈
∈ R;
(H2 ) h ≠ 0 is π-antiperiodic, i.e., h ( x + π ) = – h ( x ) for all x ∈ R.
∆ denotes the discrete Laplacian defined as
( ) = + + + −+ − + −∆u u u u u un m n m n m n m n m n m, , , , , ,1 1 1 1 4 .
For f ( u ) = ω2 sin u and χ = 1, we get the 2d discrete sine-Gordon lattice equation
(1.1). Clearly f is globally Lipschitz continuous on R, i. e., | f ( x ) – f ( y ) | ≤ L | x – y |
∀x, y ∈ R with L : = maxR | f ′ ( x ) |.
We are interested in the existence of periodic moving wave solutions of the form
u t U n m t tn m, cos sin ,( ) = ( + − )θ θ ν µ (2.2)
for some ν, θ ∈ R and U ∈ C2
( R
2, R ) which is 2π-periodic in the both variables.
We may consider solutions of (2.2) to be moving waves on the lattice Z2, in the
direction eiθ. Substitution of (2.2) into (2.1) leads to the equation
ν µν µ2 22U z U z U zzz z( ) − ( ) + ( ), , ,v v vv vv = χ θ θ( ( + ) + ( − )U z U zcos , cos ,v v +
+ U z U z U z f U z h( + ) + ( − ) − ( ) − ( ) + ( )) ( )sin , sin , , ,θ θv v v v v4 (2.3)
with z = n cos θ + m sin θ – ν t and v = µ t.
The linear part of (2.3) has the form
LU : = − ( ) + ( ) − ( )ν µν µ2 22U z U z U zzz z, , ,v v vv vv +
+ χ θ θ( ( + ) + ( − )U z U zcos , cos ,v v +
+ U z U z U z( + ) + ( − ) − ( ))sin , sin , ,θ θv v v4 . (2.4)
Taking en, m : =
1
π
( + )ei nz mv we derive
Len m, = λn, m en m,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
PERIODIC MOVING WAVES ON 2D LATTICES … 129
with
λn, m : = ( − ) − +
n m
n nν µ χ θ θ2 2 24
2 2
sin
cos
sin
cos
.
We see that in general L is not invertible, since we are led to a problem of small
divisors [7 – 9]. To avoid this difficulty, we use the symmetry of f and h in the next
sections.
Finally, the unforced case of (2.1) with the form
˙̇ , , ,u u f un m n m n m= ( ) − ( )∆ , ( n, m ) ∈ Z2, (2.5)
is investigated in [10] by looking for traveling waves of (2.5) of the form
u t U n m tn m, cos sin( ) = ( + − )θ θ ν , (2.6)
when f satisfies assumption (H1 ) in (2.5). Conditions are found in [10] to show the
existence of uniform sliding states and periodic traveling waves of (2.5). Comparing
formulas (2.2) and (2.6), this paper is a natural continuation of [10] to the periodically
forced case (2.1) of (2.5). Next, we are also motivated to study periodic moving waves
by the paper [11] where 1d undamped and periodically forced Frenkel – Kontorova
model is investigated.
3. Weak periodic moving waves. A function U : R2 → R is π-antiperiodic if
U ( z + π, v ) = U ( z, v + π ) = – U ( z, v ) ∀( z, v ) ∈ R2. (3.1)
Note that any such U satisfying (3.1) is also 2π-periodic in the both variables.
Let
Hr : = U W Ur∈ ( ) π{ }loc
, is antiperiodic2 2
R -
be Hilbert spaces for r ∈ Z+ : = N ∪ { 0 } with scalar products
( u, w ) r : = (∂ ∂ ) + (∂ ∂ )z
r
z
r r ru w u w, ,0 0v v
for r ∈ N and ( u, w ) 0 : =
u z w z dz d( ) ( )∫ , ,v v v
Ω
with Ω : = ( 0, π ) × ( 0, π ) (see cf.
[7, 12]). The corresponding norms are denoted by || ⋅ || r .
In the first part of this section, we are interested in the existence of weak π -anti-
periodic solutions U of (2.3), i.e., U ∈ H
0 satisfying
− ( ) ( ) + ( ) ( ) − ( ) ( ){∫ ν µν µ2 22U z w z U z w z U z w zzz z, , , , , ,v v v v v vv vv
Ω
+
+ χ θ θ θ( ( + ) + ( − ) + ( + )( U z U z U zcos , cos , sin ,v v v +
+ U z U z f U z h w z dz d( − ) − ( ) − ( ) + ( )) ( )}) ( )sin , , , ,θ v v v v v v4 = 0 (3.2)
for all w ∈ H
0 ∩ C
2
( R
2, R ). Since the integration by parts formula holds for π-
antiperiodic functions, if U is π-antiperiodic and C
2-smooth solving (2.3) then U is
also a weak π-antiperiodic solution of (2.3). Clearly, (3.2) has the form
LU + N ( U ) + h = 0, (3.3)
where L : D ( L ) ⊂ H
0 → H
0 is defined by (2.4) and N : H
0 → H
0 is a Nemytskij
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
130 M. FEČKAN
operator N ( U ) : = – f ( U ). Note that assumptions (H1 ) and (H2 ) imply that really N
maps H
0 to itself and h ∈ H
0. Now we are ready to prove the following result.
Theorem 3.1. Suppose (H1 ) and (H2 ) hold. If µ > L + 8χ and one of the
following conditions holds:
i) ν = µ 2 1
2
p
k
+
for some p ∈ Z and k ∈ N such that 2k <
µ
χL + 8
,
ii) ν = µ 2
2 1
k
p +
for some k ∈ Z and p ∈ Z+ such that 2p + 1 <
µ
χL + 8
,
then for any θ ∈ R, (2.3) has a unique weak π-antiperiodic solution.
Proof. We expand u ∈ H
0 in the Fourier series
u ( z, v ) = c en m n m
n m
, ,
,
2 1 2 1− −
∈
∑
Z
, c cn m n m, ,= − + − +1 1.
Then u 0
2 =
cn mn m ,,
2
∈∑ Z
and L u =
c en m n m n mn m , , ,,
λ2 1 2 1 2 1 2 1− − − −∈∑ Z
. If i)
holds, then we have
λ ν µ χ2 1 2 1
22 1 2 1 8n m n m− − ≥ ( − ) − ( − ) −( ), =
= ( − ) + − ( − )
−2 1
2 1
2
2 1 8
2
n
p
k
mµ µ χ =
=
µ χ µ χ
2
2
2
2
24
2 1 2 1 2 2 1 8
4
8
k
n p k m
k
( )( − )( + ) − ( − ) − ≥ − > 0.
If ii) holds, then we have
λ ν µ χ2 1 2 1
22 1 2 1 8n m n m− − ≥ ( − ) − ( − ) −( ), =
= ( − )
+
− ( − )
−2 1
2
2 1
2 1 8
2
n
k
p
mµ µ χ =
=
µ χ µ χ
2
2
2
2
22 1
2 1 2 2 1 2 1 8
2 1
8
( + )
( − ) − ( + )( − ) − ≥
( + )
−( )
p
n k p m
p
> 0.
Consequently, L– 1
: H
0 → H
0 satisfies
|| L– 1
|| ≤
4
32
2
2 2
k
kµ χ−
under condition i),
|| L– 1
|| ≤
( + )
− ( + )
2 1
8 2 1
2
2 2
p
pµ χ
under condition ii).
Note that N : H
0 → H
0 is Lipschitz continuous with a constant L. Next, we rewrite
(3.3) as a fixed point problem
U = F ( U ) : = – L– 1 N ( U ) – L– 1 h. (3.4)
Clearly, F : H
0 → H
0 is Lipschitz continuous with a constant || L– 1
|| L. The
assumptions of Theorem 3.1 ensure that || L– 1
|| L < 1. So the Banach fixed point
theorem gives a unique solution U of (3.4) in H
0. This is a unique weak π-
antiperiodic solution of (2.3).
The theorem is proved.
Remark 3.1. We prove in [10] that if ν > L + 8 then (2.5) has a unique uniform
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
PERIODIC MOVING WAVES ON 2D LATTICES … 131
sliding state, i.e., a solution of the form (2.6) satisfying U ( z + 2π ) = U ( z ) + 2π
∀z ∈ R.
Now we look for solutions of (2.3) satisfying
U ( z + π, v ) = U ( z, v + π ) = – U ( z, v ) + 2π ∀( z, v ) ∈ R2. (3.5)
Note that any U satisfying (3.5) is 2π-periodic in the both variables. So we change
U ( z, v ) = u ( z, v ) + π, where u is π-antiperiodic. Substituting this into (2.3), we get
ν µν µ2 22u z u z u zzz z( ) − ( ) + ( ), , ,v v vv vv = χ θ θ( ( + ) + ( − )u z u zcos , cos ,v v +
+ u z u z u z f u z h( + ) + ( − ) − ( ) − ( ) + π + ( )) ( )sin , sin , , ,θ θv v v v v4 . (3.6)
Assumption (H1 ) implies that
f ( π – x ) = – f ( π + x ) ∀x ∈ R. (3.7)
Using (3.7) we easily check that if u is π-antiperiodic, then also f ( u ( z, v ) + π ) is
π-antiperiodic. So the Nemytskij operator
˜ ,N u z( )( )v : = f ( u ( z, v ) + π ) maps H
0 to
H
0. Consequently, by repeating the proof of Theorem 3.1, we obtain the following
result.
Theorem 3.2. Under the assumptions of Theorem 3.1, for any θ ∈ R , (2.3) has
a unique weak solution satisfying (3.5), i.e., (3.6) possesses a weak solution u ∈
H
0.
Hence under the assumptions of Theorem 3.1, for fixed involved parameters, we
have at least two 2π-periodic weak solutions of (2.3): one satisfying (3.1) and other
satisfying (3.5).
We can further utilize the symmetries of f and h as follows. We look for
solutions of (2.3) satisfying
U ( z + π, v ) = U ( z, v ), U ( z, v + π ) = – U ( z, v ) ∀( z, v ) ∈ R2. (3.8)
Again, any U satisfying (3.8) is 2π-periodic in the both variables.
Instead of Hilbert spaces Hr, we consider similar ones defined by
Xr : = U W Ur∈ ( ){ }loc
, satisfies (3.8)2 2
R
keeping the scalar products ( ⋅, ⋅ ) r.
Theorem 3.3. Suppose (H1 ) and (H2 ) hold. If
µ > L + 8χ and ν = µ k
p2 1+
for some p ∈ Z+ and k ∈ Z such that 2p + 1 <
µ
χL + 8
, then for any θ ∈ R ,
(2.3) has a unique weak solution satisfying (3.8), i.e., a solution U ∈ X
0 satisfying
(3.2) for all w ∈ X
0 ∩ C2
( R
2, R ).
Proof. We expand u ∈ X
0 in the Fourier series
u ( z, v ) = c en m n m
n m
, ,
,
2 2 1−
∈
∑
Z
, c cn m n m, ,= − − +1.
Then u 0
2 = cn m
n m
,
,
2
∈
∑
Z
and Lu = c en m n m n m
n m
, , ,
,
λ2 2 1 2 2 1− −
∈
∑
Z
. From our
assumptions, we derive
λ ν µ χ2 2 1
22 2 1 8n m n m, − ≥ − ( − ) −( ) = 2
2 1
2 1 8
2
n
k
p
mµ µ χ
+
− ( − )
− =
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
132 M. FEČKAN
=
µ χ µ χ
2
2
2
2
22 1
2 2 1 2 1 8
2 1
8
( + )
− ( + )( − ) − ≥
( + )
−( )
p
nk p m
p
> 0.
The rest of the proof is similar to one of Theorem 3.1, so we omit it.
Similarly to (3.5), we look for solutions of (2.3) satisfying
U ( z + π, v ) = U ( z, v ), U ( z, v + π ) = – U ( z, v ) + 2π ∀( z, v ) ∈ R2. (3.9)
Clearly, any U satisfying (3.9) is 2π-periodic in the both variables. Repeating the
proof of Theorem 3.3, we obtain the following result.
Theorem 3.4. Under the assumptions of Theorem 3.3, for any θ ∈ R, (2.3) has
a unique weak solution satisfying (3.9), i.e., (3.6) possesses a weak solution u ∈ X
0.
Corollary 3.1. Suppose (H1 ) and (H2 ) hold. If
µ > L + 8χ and ν = µ 2
2 1
k
p +
for some p ∈ Z+ and k ∈ Z such that 2p + 1 <
µ
χL + 8
, then for any θ ∈ R,
(2.3) has at least four weak 2π-periodic solutions: ones satisfying conditions (3.1),
(3.5), (3.8), and (3.9), respectively.
Clearly, if (H2 ) holds, then h is 2π-periodic. Now we only suppose
(H3 ) h ≠ 0 is 2π-periodic, i.e., h ( x + 2π ) = h ( x ) ∀x ∈ R.
Then we look for solutions of (2.3) satisfying either
U ( z + π, v ) = – U ( z, v ), U ( z, v + 2π ) = U ( z, v ) ∀( z, v ) ∈ R2, (3.10)
or
U ( z + π, v ) = – U ( z, v ) + 2π, U ( z, v + 2π ) = U ( z, v ) ∀( z, v ) ∈ R2. (3.11)
Clearly, any U satisfying either (3.10) or (3.11) is 2π-periodic in the both variables.
Now we consider Hilbert spaces defined by
Y
r : = U W Ur∈ ( ){ }loc
, satisfies (3. )2 2 10R
with scalar products
〈 u, w 〉 r : = 〈∂ ∂ 〉 + 〈∂ ∂ 〉z
r
z
r r ru w u w, ,0 0v v
for r ∈ N and 〈 u, w 〉 0 : =
u z w z dz d( ) ( )∫ , ,˜ v v v
Ω
with Ω̃ : = ( 0, π ) × ( 0, 2π ).
Theorem 3.5. Suppose (H1 ) and (H3 ) hold. If µ > L + 8χ and condition i)
of Theorem 3.1 holds, then for any θ ∈ R , (2.3) has a unique weak solution
satisfying (3.10) and other unique weak solution satisfying (3.11), i.e., there are two
functions U 1, 2 ∈ Y
0 satisfying (3.2) and (3.6) for all w ∈ Y
0 ∩ C 2
( R
2, R ),
respectively.
Proof. We expand u ∈ Y
0 in the Fourier series
u ( z, v ) = c en m n m
n m
, ,
,
1
2 2 1−
∈
∑
Z
, c cn m n m, ,= − + −1 .
Then the rest of the proof is the same as for Theorem 3.3, so we omit it.
Remark 3.2. The function h in (2.1) could be arbitrary satisfying either (H2 ) or
(H3 ). Next, the above results are applied to (1.1) for any γ > 0 and µ > 0, χ > 0, ω >
> 0 satisfying µ > ω χ2 8+ .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
PERIODIC MOVING WAVES ON 2D LATTICES … 133
4. More regular periodic moving waves. Now we study (2.3) in H r for r ∈ N.
First, we note the Sobolev embedding result [7, 12]
H2 ⊂ Cπ ( R
2
) : = H
0 ∩ C ( R
2, R ).
Moreover, we have
U U z c e c
z
n m n m
n m
n m
n m
0 2 1 2 1
1
: max ,
,
, ,
,
,
,
= ( ) ≤ ≤
π∈
− −
∈ ∈
∑ ∑
v
v
R
Z Z
≤
≤
1
2 1 2 1
1
2 1 2 1
2 4 4
4 4π
( − ) + ( − )
( − ) + ( − )
( )
∈ ∈
∑ ∑c n m
n m
n m
n m n m
,
, ,Z Z
≤
≤
1
1 2π
c U
with
c
n m n m n mn m n m n m
1
2
4 4 4 4 2 2 2
1
2 1 2 1
4
1
8
1=
( − ) + ( − )
≤
+
≤
( + )∈ ∈ ∈
∑ ∑ ∑
, , ,Z N N
≤
≤ 8 2
1
2 1 32 2 2
0
3 3
1
dy
n y n
dx
xn n( + )
= π ≤ π +
= π
∞
∈ ∈
∞
∫∑ ∑ ∫
N N
.
Hence we get the Sobolev inequality [7, 12]
U U0 2
3≤
π
∀U ∈ H2. (4.1)
Next, supposing f ∈ C2
( R
2, R ), we compute for U ∈ Cπ
∞ : = H
0 ∩ C
∞
( R
2, R )
f U f U f Uzz( ) ≤ ( ) + ( )2 0 0vv . (4.2)
Furthermore,
f U f U U f U U L U L Uzz zz z zz z( ) ≤ ′( ) + ′′( ) ≤ +
0 0
2
0 0 2
2
0
,
where L2 : = max
x
f x
∈
′′( )
R
. Similarly we derive
f U L U L U( ) ≤ +vv vv v0 0 2
2
0
.
Hence by (4.2)
f U L U U L U Uzz z( ) ≤ +( ) + +( )2 0 0 2
2
0
2
0vv v .
Next, using integration by parts, we derive
U z dz U z U z U z U z U z dzz z z z zz
4
0
3
0
2
0
3( ) = ( ) ( ) − ( ) ( ) ( )
π
=
π
π
∫ ∫, , , , , ,v v v v v v ≤
≤
3 0
4
0
2
0
U U z dz U z dzz zz( ) ( )
π π
∫ ∫, ,v v
which implies
U z dz U U z dzz zz
4
0
0
2 2
0
9( ) ≤ ( )
π π
∫ ∫, ,v v .
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
134 M. FEČKAN
Consequently, we obtain
U U z dz d U Uz z zz
2
0
4
0 0
3= ( ) ≤∫ , v v
Ω
.
Similarly, we get
U U Uv vv
2
0 0 03≤ .
Summarizing, we arrive at
f U L L U U Uzz( ) ≤ +( ) +( )2 2 0 0 03 vv ≤ 2 3 2 0 2L L U U+( ) . (4.3)
Of course, (4.3) is the well-known Moser inequality [7, 12]. Assuming that µ >
> 2 8L + χ and one of the following conditions holds:
I) ν = µ 2 1
2
p
k
+
for some p ∈ Z and k ∈ N such that 2k <
µ
χ2 8L +
,
II) ν = µ 2
2 1
k
p +
for some k ∈ Z and p ∈ Z+ such that 2 1p + <
µ
χ2 8L +
,
and using (2.4), we get
( LU, U ) r ≥ c̃ U r
2 ∀U ∈ Cπ
∞ (4.4)
for any r ≥ 0 and with either c̃ =
µ χ
2
24
8
k
− > 0 or c̃ =
µ χ
2
22 1
8
( + )
−
p
> 0.
Supposing h ∈ C
2
( R, R ), from (4.1), (4.3), and (4.4) we derive
( LU – f ( U ) + h, U ) 2 ≥ c̃ L L U U h U− +
π
−2 3
3
2 2 2
2
2 2 .
If
c̃ L> 2 , (4.5)
then we get
( LU – f ( U ) + h, U ) 2 ≥ A B U U h U−( ) −( )2 2 2 2
with
A : = c̃ L− 2 > 0, B : = 3
6
2L
π
> 0. (4.6)
The quadratic function x → ( A – B x ) x has its maximum
A
B
2
24
at x0 =
A
B2
. So if
h 2 <
A
B
2
24
, then there is κ ∈ 0
2
,
A
B
such that for any U ∈ Cπ
∞ with U 2 = κ,
the following inequality holds:
( LU – f ( U ) + h, U ) 2 > 0. (4.7)
Now we take the finite-dimensional Banach spaces Hk ⊂ Cπ
∞ , k ∈ N, given by
Hk : = c e n m c cn m n m
n m k
k
n m n m, ,
,
, ,, are odd integers and
= −
− −∑ =
.
Next, like in [13], we take a convex set
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PERIODIC MOVING WAVES ON 2D LATTICES … 135
Vk : = U H Uk∈ ≤{ }2 κ .
Then, on the boundary ∂Vk of Vk
, (4.7) holds. Hence we consider the homotopy
H ( λ, U ) : = λ Pk ( LU – f ( U ) + h ) + ( 1 – λ ) U, U ∈ Vk , (4.8)
where Pk : H
2 → Hk ⊂ H
2 is the orthogonal projection. According to (4.7), we see
that
( H ( λ, U ), U ) 2 = λ( LU – f ( U ) + h, U ) 2 + ( 1 – λ ) U 2
2 > 0 (4.9)
for any λ ∈ [ 0, 1 ] and U ∈ ∂Vk . Then 0 ∉ H ( λ, ∂Vk ) for any λ ∈ [ 0, 1 ].
Consequently, using the Brouwer topological degree theory [9], we derive
deg ( Pk ( LU – F ( U ) + h ), Vk , 0 ) = deg ( Ik , Vk , 0 ) = 1,
where Ik : Hk → Hk is the identity mapping. This gives a solution Uk ∈ Vk of
Pk ( LUk – F ( Uk ) + h ) = 0. (4.10)
Since H
2 is compactly embedded into H
0 [12], we can suppose that Uk → U0 ∈ H
2
in H
0. We note that
Pk u = c en m n m
n m k
, ,
,
2 1 2 1
2 1 2 1
− −
− − ≤
∑
when
u ( z, v ) = c en m n m
n m
, ,
,
2 1 2 1− −
∈
∑
Z
with cn m, = c n m− + − +1 1, . It is easy to check that Pk : H
r → Hk ⊂ Hr is an orthogonal
projection for any r ≥ 0. Then (4.10) gives ( LUk – f ( Uk ) + h, w ) 0 = 0 ∀w ∈ Hk1
and k ≥ k1
. But this means that (3.2) holds with U = Uk for any w ∈ Hk1
and k ≥
≥ k1
. Since N ( U ) = – F ( U )
is continuous from H
0 to itself, fixing k1 and passing
to the infinity with k → ∞, we see that (3.2) holds with U = U0 for any w ∈ Hk1
and
k1 ≥ 1. So U0 ∈ H
2 is a weak π-antiperiodic solution of (2.3) which is continuous.
Summarizing, we get the following result.
Theorem 4.1. Suppose f, h ∈ C2
( R, R ) satisfy (H1 ) and (H2 ). If
µ > 2 8L + χ
and one of the conditions I) and II) holds, and
h
c L
L2
2
2
2
2
216
< π( − )˜
, i.e., h 2 is sufficiently small,
where c̃ =
µ2
24k
– 8χ > 0 for I) and c̃ =
µ2
22 1( + )p
– 8χ > 0 for II), then for
any θ ∈ R, (2.3) has a unique π-antiperiodic solution U belonging to H
2, i.e.,
(2.3) is satisfied when generalized derivatives of U are considered.
Proof. Note that µ > 2 8L + χ and I), II) imply assumptions of Theorem 3.1.
So (2.3) has a unique weak π-antiperiodic solution U. Since also (4.5) is satisfied,
from the above consideration we also known that U ∈ H
2.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
136 M. FEČKAN
The theorem is proved.
The same arguments can be applied to show that if f, h ∈ C4
( R, R ) and h 2 is
sufficiently small in Theorem 4.1, then U ∈ H4 ⊂ C
2
( R, R ). So we get a unique
classical solution of (2.3). Indeed, we have for u ∈ Cπ
∞
f U f U f Uzzzz( ) ≤ ( ) + ( )4 0 0vvvv ≤
≤
L U U L U U U Uz z zz4
4
0
4
0 3
2
0
2
0
6+( ) + +( )v v vv +
+
4 32 0 0 2
2
0
2
0
L U U U U L U Uz zzz zz+( ) + +( )v vvv vv +
+
L U Uzzzz 0 0+( )vvvv , (4.11)
where Li : = max
x
if x
∈
( )( )
R
, i = 2, 3, 4. Using the Sobolev inequality (4.1) and the
Nirenberg ones like in [7, p. 273, 274; 12], we get from (4.11)
f U c L U L U L U L U( ) ≤ + +( ) +[ ]4 4 2
3
3 2
2
2 2 42 (4.12)
for a constant c > 0. Supposing either I) or II), from (4.4) and (4.12), we derive
( LU – f ( U ) + h, U ) 4 ≥
≥ c̃ L c L U L U L U U h U− − + +( )[ ] −2 4 2
3
3 2
2
2 2 4
2
4 4 . (4.13)
Since c̃ > 2L , the equation
c̃ L c L x L x L x− − ( + + )2 4
3
3
2
2 = 0
has a unique positive root x̃0 . Finally, we define a function G : [ 0, ∞ ) → [ 0, ∞ ) by
G ( x ) =
A
B
x
A
B
A Bx x x
A
B
2
24 2
0
2
for ,
for ,
≥
( − ) ≤ ≤
where constants A and B are given in (4.6) and c̃ =
µ2
24k
– 8χ > 0 for I) and c̃ =
=
µ2
22 1( + )p
– 8χ > 0 for II). Now we are ready to prove the following result.
Theorem 4.2. Suppose (H1 ) and (H2 ) for f, h ∈ C
4
( R, R ). If
µ > 2 8L + χ
and one of the conditions I) and II) holds, and
h G x2 0< ( )˜ , i.e., h 2 is sufficiently small,
then for any θ ∈ R, (2.3) has a unique classical π-antiperiodic solution U.
Proof. From h 2 < G x( )˜0 we infer the existence of 0 < κ < min , ˜A
B
x
2 0
and R >> 1 such that (4.7) holds for any U ∈ Cπ
∞ with U 2 = κ, and for any U ∈
∈ Cπ
∞ with U 2 ≤ κ and U 4 = R, the following inequality holds:
( LU – f ( U ) + h, U ) 4 > 0. (4.14)
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PERIODIC MOVING WAVES ON 2D LATTICES … 137
Next, like in [13], we take a convex set
Wk : = U H U U Rk∈ ≤ ≤{ }2 4κ, .
We consider the homotopy H ( λ, U ) defined in (4.8). We recall that Pk : H
r → Hk ⊂
⊂ H
r is an orthogonal projection for any r ≥ 0. On the boundary ∂Wk of Wk
, either
U 2 = κ which, by (4.7), implies (4.9) for any λ ∈ [ 0, 1 ], or U 4 = R which, by
(4.14), implies ( H ( λ, U ), U ) 4 > 0 for any λ ∈ [ 0, 1 ]. Summarizing, we see that 0
∉ ∉ H ( λ, ∂Wk ) for any λ ∈ [ 0, 1 ]. Consequently, using the Brouwer topological
degree theory [9], we derive
deg ( Pk ( LU – F ( U ) + h ), Wk , 0 ) = deg ( Ik , Wk , 0 ) = 1.
This gives a solution Uk ∈ Wk of Pk ( LUk – F ( Uk ) + h ) = 0 for any k ∈ N. So we
can suppose that weakly Uk Æ U ∈ H
4, and so strongly Uk → U in H
2. The rest of
the proof is the same as for Theorem 4.1, consequently, U ∈ H
4 is a unique weak π-
antiperiodic solution of (2.3). Since U ∈ H
4 ⊂ C2
( R
2, R ), we get a unique classical
solution of (2.3).
The theorem is proved.
Remark 4.1. i) We note that in Theorem 4.2 we need only to control the norm
h 2 in spite of the fact that h ∈ C
4
( R
2, R ). For instance, for
h z
p z
p
q
q
p q, , , :
sin sin
ε ε ε( ) = ( + )
( + )
+ ( + )
( + )
v
v2 1
2 1
2 1
2 12 2 , p, q ∈ Z, ε ≠ 0,
we have hp q, ,ε 2
= ε π, while hp q, ,ε 4
→ ∞ as | p | + | q | → ∞ . So for ε ≠ 0
sufficiently small, Theorem 4.2 can be applied with h = hp q, ,ε for any p, q ∈ Z.
ii) Next, it seems to be awkward to find the constant c in (4.12) and subsequently
the root x̃0 , for this reason we present Theorem 4.1 with concrete and explicit values
of involved constants.
5. Damped and periodically forced systems. In this section, we consider the
infinite system of ODEs
˙̇ ˙, , , ,u u u f u h tn m n m n m n m= − + ( ) − ( ) + ( )δ χ µ∆ , ( n, m ) ∈ Z2, (5.1)
on the two-dimensional integer lattice Z2 for f ∈ C1
( R, R ), h ∈ C ( R, R ), δ > 0,
χ > 0, µ > 0 under conditions (H1 ) and (H2 ). Inserting (2.2) into (5.1), we get
ν µν µ δ µ ν2 22U z U z U z U z U zzz z z( ) − ( ) + ( ) + ( ) − ( )( ), , , , ,v v v v vv vv v =
= ( ( ( + ) + ( − )χ θ θU z U zcos , cos ,v v +
+ U z U z U z f U z h( + ) + ( − ) − ( ) − ( ) + ( )) ( )sin , sin , , ,θ θv v v v v4 . (5.2)
Now we write (5.2) as follows:
˜ ˜LU N U h+ ( ) + = 0 (5.3)
with
̃LU : = − ( ) + ( ) − ( )ν µν µ2 22U z U z U zzz z, , ,v v vv vv + δ ν µ( )( ) − ( )U z U zz , ,v vv
(5.4)
and
Ñ U( ) : = χ θ θ( ( + ) + ( − )U z U zcos , cos ,v v +
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
138 M. FEČKAN
+ U z U z U z f U z( + ) + ( − ) − ( ) − ( )) ( )sin , sin , , ,θ θv v v v4 .
We have
˜
,Len m = ˜
, ,λn m n me with ˜
,λn m : = ( − ) + ( − )n m i n mν µ δ ν µ2 . Clearly, N :
H
0 → H
0 is Lipschitz continuous with constant L + 8χ. A weak π-antiperiodic
solution U ∈ H
0 of (5.2) is formulated like in (3.2), so we omit that formula.
Theorem 5.1. Suppose (H1 ) and (H2 ) hold. If one of the following conditions
holds:
a) ν = µ 2 1
2
p
k
+
for some p ∈ Z and k ∈ N such that µ δ µ4 2 2 24+ k >
> 16 84 2k L( + )χ ,
b) ν = µ 2
2 1
k
p +
for some k ∈ Z and p ∈ Z+ such that µ δ µ4 2 2 22 1+ ( + )p >
> ( + ) ( + )2 1 84 2p L χ ,
then for any θ ∈ R, (5.2) has a unique weak π-antiperiodic solution.
Proof. We expand u ∈ H
0 in the Fourier series
u ( z, v ) = c en m n m
n m
, ,
,
2 1 2 1− −
∈
∑
Z
, c cn m n m, ,= − + − +1 1.
Then u 0
2 = cn mn m ,,
2
∈∑ Z
and ̃Lu =
c en m n m n mn m , , ,,
λ̃2 1 2 1 2 1 2 1− − − −∈∑ Z
. If a)
holds, then we have
˜
,λ µ δ µ
2 1 2 1
4
4
2
2
216 4n m
k k
− − ≥ + .
If b) holds, then we have
˜
,λ µ δ µ
2 1 2 1
4
4
2
2
22 1 2 1n m
p p
− − ≥
( + )
+
( + )
.
Consequently, L̃−1
: H
0 → H
0 satisfies
L̃−1 ≤
4
4
2
4 2 2 2
k
kµ δ µ+
under condition a),
L̃−1 ≤
( + )
+ ( + )
2 1
2 1
2
4 2 2 2
p
pµ δ µ
under condition b).
In both cases we get L̃ ( + )L 8χ < 1, so rewriting (5.3) as a fixed point problem
U = − ( ) −− −˜ ˜ ˜L L1 1N U h, (5.5)
and applying the Banach fixed point theorem to (5.5), we get the desired unique weak
π-antiperiodic solution of (5.2).
The theorem is proved.
Of course, other results of Sections 3 and 4 can be extended for (5.2), but since it is
straightforward, we omit details.
We note that for sufficiently large δ > 0, equation (5.2) has a weak π-antiperiodic
solution. Indeed, if
δµ χ> ( + )2 8L , (5.6)
then condition a) is satisfied with k = 1 and any p ∈ Z, and condition b) holds as well
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
PERIODIC MOVING WAVES ON 2D LATTICES … 139
with p = 0 and any k ∈ Z. So we get an expected result that if the damping δ > 0 is
sufficiently large, for instance, if (5.6) holds, then the system (5.1) has a (weak)
periodic moving wave solution. Moreover, they are infinitely many in any direction
( cos θ, sin θ ). Indeed, for different values of ν in (5.2), the above derived weak
solutions are different. To show this, suppose that (5.2) has a weak π-antiperiodic
solution U for two parameters ν1 ≠ ν2 with the same µ, χ , δ and θ satisfying
assumptions of Theorem 5.1. Then according to (5.5) we get
U =
− ( ) − = − ( ) −− −( ) ( )˜ ˜ ˜ ˜L L1
1
2
1N U h N U h , (5.7)
where
L1 2
1
,
− are linear maps of (5.4) for parameters µ, δ , ν1 , 2 with eigenvalues
˜
, , ,λ2 1 2 11 2n m− − , respectively. Then we derive
Im ˜
, ,λ δ ν µ2 1 2 11 12 1 2 1n m n m− − = ( − ) − ( − )( ) ≠
≠ δ ν µ λ( )( − ) − ( − ) = − −2 1 2 12 2 1 2 1 2n m n mIm ˜
, , .
So for any n, m ∈ Z we see that ˜
, ,λ2 1 2 11n m− − ≠ ˜
, ,λ2 1 2 1 2n m− − . But then
˜ ˜L1
1− h ≠
˜ ˜L2
1− h
for any 0 ≠ h̃ ∈ H
0. Clearly Ñ U( ) – h ≠ 0 in (5.7), since otherwise U = 0 and then
h = 0, which is excluded in (H2 ). But then
˜ ˜L1
1− ( )( ) −N U h ≠
˜ ˜L2
1− ( )( ) −N U h , which
contradicts to (5.7). So solutions in Theorem 5.1 are different for different values of ν,
i.e., we have infinitely many of them.
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Received 15.10. 07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
|
| id | umjimathkievua-article-3143 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:01Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/b4/60bc6a08fc82be19e998cea7fa7521b4.pdf |
| spelling | umjimathkievua-article-31432020-03-18T19:46:36Z Periodic moving waves on 2D lattices with nearest-neighbor interactions Періодичні рухомі хвилі нa двовимірних ґратках із взаємодіями найближчих сусідів Fečkan, M. Фечкан, М. We study the existence of periodic moving waves on two-dimensional periodically forced lattices with linear coupling between nearest particles and with periodic nonlinear substrate potentials. Such discrete systems can model molecules adsorbed on a substrate crystal surface. Вивчено питання існування періодичних рухомих хвиль на двовимірних періодично збурених ґратках із лінійним зчепленням між найближчими частинками та з періодичними нелінійними потенціалами підкладинки. Такі дискретні системи можуть моделювати молекули, що адсорбуються на кристалічну поверхню підкладинки. Institute of Mathematics, NAS of Ukraine 2008-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3143 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 1 (2008); 127–139 Український математичний журнал; Том 60 № 1 (2008); 127–139 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3143/3034 https://umj.imath.kiev.ua/index.php/umj/article/view/3143/3035 Copyright (c) 2008 Fečkan M. |
| spellingShingle | Fečkan, M. Фечкан, М. Periodic moving waves on 2D lattices with nearest-neighbor interactions |
| title | Periodic moving waves on 2D lattices with nearest-neighbor interactions |
| title_alt | Періодичні рухомі хвилі нa двовимірних ґратках із взаємодіями найближчих сусідів |
| title_full | Periodic moving waves on 2D lattices with nearest-neighbor interactions |
| title_fullStr | Periodic moving waves on 2D lattices with nearest-neighbor interactions |
| title_full_unstemmed | Periodic moving waves on 2D lattices with nearest-neighbor interactions |
| title_short | Periodic moving waves on 2D lattices with nearest-neighbor interactions |
| title_sort | periodic moving waves on 2d lattices with nearest-neighbor interactions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3143 |
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