The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice
The article deals with the discrete sine-Gordon equation that describes an infinite system of nonlinearly coupled nonlinear oscillators on a 2D-lattice with the external potential V (r) = K(1 - cos r). The main result concerns the existence of heteroclinic travelling waves solutions. Sufficient cond...
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irk-123456789-1458562019-02-02T01:23:10Z The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice Bak, S. The article deals with the discrete sine-Gordon equation that describes an infinite system of nonlinearly coupled nonlinear oscillators on a 2D-lattice with the external potential V (r) = K(1 - cos r). The main result concerns the existence of heteroclinic travelling waves solutions. Sufficient conditions for the existence of these solutions are obtained by using the critical points method and concentration-compactness principle. Статтю присвячено дискретному рiвнянню синус-Гордона, яке описує нескiнченну систему нелiнiйно зв'язаних нелiнiйних осциляторiв на двовимiрнiй гратцi iз зовнiшнiм потенцiалом V (r) = K(1 cos r). Основний результат стосується iснування розв язкiв у виглядi гетероклiнiчних рухомих хвиль. За допомогою методу критичних точок i принципу концентровано компактностi отримано достатнi умови iснування таких розв язкiв. 2018 Article The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice / S. Bak // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 1. — С. 16-26. — Бібліогр.: 19 назв. — англ. 1812-9471 DOI: https://doi.org/10.15407/mag14.01.016 Mathematics Subject Classification 2010: 34G20, 37K60, 58E50 http://dspace.nbuv.gov.ua/handle/123456789/145856 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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The article deals with the discrete sine-Gordon equation that describes an infinite system of nonlinearly coupled nonlinear oscillators on a 2D-lattice with the external potential V (r) = K(1 - cos r). The main result concerns the existence of heteroclinic travelling waves solutions. Sufficient conditions for the existence of these solutions are obtained by using the critical points method and concentration-compactness principle. |
| format |
Article |
| author |
Bak, S. |
| spellingShingle |
Bak, S. The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice Журнал математической физики, анализа, геометрии |
| author_facet |
Bak, S. |
| author_sort |
Bak, S. |
| title |
The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice |
| title_short |
The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice |
| title_full |
The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice |
| title_fullStr |
The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice |
| title_full_unstemmed |
The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice |
| title_sort |
existence of heteroclinic travelling waves in the discrete sine-gordon equation with nonlinear interaction on a 2d-lattice |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2018 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/145856 |
| citation_txt |
The Existence of Heteroclinic Travelling Waves in the Discrete Sine-Gordon Equation with Nonlinear Interaction on a 2D-Lattice / S. Bak // Журнал математической физики, анализа, геометрии. — 2018. — Т. 14, № 1. — С. 16-26. — Бібліогр.: 19 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
| work_keys_str_mv |
AT baks theexistenceofheteroclinictravellingwavesinthediscretesinegordonequationwithnonlinearinteractionona2dlattice AT baks existenceofheteroclinictravellingwavesinthediscretesinegordonequationwithnonlinearinteractionona2dlattice |
| first_indexed |
2025-07-10T22:41:20Z |
| last_indexed |
2025-07-10T22:41:20Z |
| _version_ |
1837301543485833216 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2018, Vol. 14, No. 1, pp. 16–26
doi: https://doi.org/10.15407/mag14.01.016
The Existence of Heteroclinic Travelling
Waves in the Discrete Sine-Gordon Equation
with Nonlinear Interaction on a 2D-Lattice
S. Bak
The article deals with the discrete sine-Gordon equation that describes
an infinite system of nonlinearly coupled nonlinear oscillators on a 2D-lattice
with the external potential V (r) = K(1 − cos r). The main result concerns
the existence of heteroclinic travelling waves solutions. Sufficient conditions
for the existence of these solutions are obtained by using the critical points
method and concentration-compactness principle.
Key words: discrete sine-Gordon equation, nonlinear oscillators, 2D-latt-
ice, heteroclinic travelling waves, critical points, concentration-compactness
principle.
Mathematical Subject Classification 2010: 34G20, 37K60, 58E50.
1. Introduction
In the paper, we study the discrete sine-Gordon equation that describes the
dynamics of an infinite system of nonlinearly coupled nonlinear oscillators on a
two-dimensional lattice. Let qn,m be a generalized coordinate of the (n,m)-th
oscillator at the time t. It is assumed that each oscillator interacts nonlinearly
with its four nearest neighbors. The equation of motion of the system considered
is of the form
q̈n,m = V ′(qn+1,m − qn,m)− V ′(qn,m − qn−1,m) + V ′(qn,m+1 − qn,m)
− V ′(qn,m − qn,m−1)−K sin(qn,m), (n,m) ∈ Z2, (1)
where K > 0. Equations (1) form an infinite system of ordinary differential
equations.
System (1) can be considered as a 2D version of the Frenkel–Kontorova model
(see, e.g., [11]). Notice that this system represents a wide class of systems called
lattice dynamical systems extensively studied in last decades. In this area of re-
search, a great attention is paid to an important specific class of solutions called
travelling waves solutions. A comprehensive presentation of the results on trav-
elling waves for 1D Fermi–Pasta–Ulam lattices is given in [19]. The existence
c© S. Bak, 2018
https://doi.org/10.15407/mag14.01.016
The Existence of Heteroclinic Travelling Waves in the Discrete sine-Gordon . . . 17
of periodic travelling waves in the Fermi–Pasta–Ulam system on a 2D-lattice is
studied in [4]. On the other hand, some results on the chains of oscillators are also
known in the literature. In particular, in [14] they are obtained by means of bifur-
cation theory, while in [1] and [2] the existence of periodic and solitary travelling
waves is studied by means of the critical point theory. In papers [3,10,12,13], trav-
elling waves for infinite systems of linearly coupled oscillators on a 2D-lattice are
studied. Paper [18] is devoted to periodic and homoclinic travelling waves for the
infinite one-dimensional chain of nonlinearly coupled nonlinear particles. In [6],
a result on the existence of subsonic periodic travelling waves for the system of
nonlinearly coupled nonlinear oscillators on a 2D-lattice is obtained, and in [7],
supersonic periodic travelling waves for these systems are studied. Paper [15]
contains a result on the existence of heteroclinic travelling waves for the dis-
crete sine-Gordon equation with linear interaction. In [16], periodic, homoclinic
and heteroclinic travelling waves for such systems with nonlinear interaction are
studied. In paper [5], a result on the existence of periodic travelling waves for
the discrete sine-Gordon equation with nonlinear interaction on a 2D-lattice is
obtained. [8] is devoted to the existence of heteroclinic travelling waves for the
discrete sine-Gordon equation with linear interaction on a 2D-lattice.
2. The problem statement
A travelling wave solution of equation (1) is a function of the form
qn,m(t) = u(n cosϕ+m sinϕ− ct) ,
where the profile function u(s) of the wave, or simply profile, satisfies the equation
c2u′′(s) = V ′(u(s+ cosϕ)− u(s))− V ′(u(s)− u(s− cosϕ))
+ V ′(u(s+ sinϕ)− u(s))− V ′(u(s)− u(s− sinϕ))−K sin(u(s)). (2)
The constant c 6= 0 is called the speed of the wave. If c > 0, then the wave moves
to the right, otherwise to the left.
An important role is played by the quantity c1 defined by the equation
c21 := 2 sup
|r|<6π
∣∣∣∣V (r)
r2
∣∣∣∣ .
We consider the case of heteroclinic travelling waves. The profile function of
this wave satisfies the conditions:
lim
s→−∞
u(s) = −π and lim
s→+∞
u(s) = π. (3)
In what follows, a solution of equation (2) is understood as a function u(s)
from the space C2(R) satisfying equation (2) for all s ∈ R.
18 S. Bak
3. Variational setting
To equation (2), we associate the functional
J(u) :=
∫ +∞
−∞
[
c2
2
(u′(s))2 − V (u(s+ cosϕ)− u(s))
−V (u(s+ sinϕ)− u(s)) +K(1 + cos(u(s)))
]
ds, (4)
defined on the Hilbert space
E := {u ∈ H1
loc(R) : u′ ∈ L2(R)}
with the scalar product
(u, v)E = u(0)v(0) +
∫ +∞
−∞
u′(s)v′(s) ds.
It is not so difficult to verify that the critical points of the functional J are the
solutions of equation (2).
Now we introduce the following notation:
M−π,π = {u ∈ E : u(−∞) = −π, u(+∞) = π},
Au(s) := u(s+ cosϕ)− u(s),
Bu(s) := u(s+ sinϕ)− u(s).
According to Lemma 3.1 from [10],
‖Au(s)‖L2(R) ≤ | cosϕ| · ‖u′(s)‖L2(R), u ∈ E,
‖Bu(s)‖L2(R) ≤ | sinϕ| · ‖u′(s)‖L2(R), u ∈ E.
Then the functional J can be expressed in the form
J(u) :=
∫ +∞
−∞
[
c2
2
(u′(s))2 − V (Au(s))− V (Bu(s)) +K(1 + cos(u(s)))
]
ds. (5)
Throughout the paper we will assume that the interaction potential V (r)
satisfies the following conditions:
(i) V (r) ∈ C1(R), V (0) = 0 and V (r) ≥ 0 for all r ∈ R;
(ii) limr→±∞ V (r) = +∞;
(iii) there exists finite limr→0
∣∣∣V (r)
r2
∣∣∣ ;
(iv) the wave speed c satisfies c2 > c21.
The following lemma can be obtained by a straightforward calculation (see [15]
for details).
The Existence of Heteroclinic Travelling Waves in the Discrete sine-Gordon . . . 19
Lemma 3.1. Let v0 : R → [−π, π] be a monotone function in C∞(R) such
that v0(s) = −π for s < −1 and v0(s) = π for s > 1. Define the functional Ψ :
H1(R)→ R by
Ψ(v) := J(v0 + v)
and suppose that assumptions (i)–(iv) are satisfied. Then the following holds:
(i1) Ψ(v) < +∞ for all v ∈ H1(R) (equivalently, J(u) < +∞ for all u of the
form u = v0 + v for some v ∈ H1(R));
(ii1) J(u) = +∞ for all u ∈ M−π,π which are not of the form u = v0 + v for
some v ∈ H1(R). In particular, a minimizer u of J on M−π,π can be
expressed as u = v0 + v for some v ∈ H1(R);
(iii1) Ψ ∈ C1 on H1(R);
(iv1) let v ∈ H1(R) be a critical point of Ψ and set u := v0 + v. Then u, v ∈
C2(R), and u is a solution of (2) with boundary conditions (3).
Let F be a non-negative function in C∞(R) such that
F (r) = 0, if |r| ≤ 5π
2 ,
F (r) ≥ 4
∣∣∣∫ 2r
0 |V
′(x)|dx
∣∣∣ and F (r) ≥ 2K, if |r| ≥ 3π,
1
2 ≤ 1 + cos r + 1
2KF (r), if |r| ∈
(
5
2π, 3π
)
.
(6)
Now we define the modified functional J̃ : E → R ∪ {∞} by
J̃(u) :=
∫ +∞
−∞
[
c2
2
(u′(s))2 − V (Au(s))− V (Bu(s))
+K(1 + cos(u(s))) + F (u(s))
]
ds. (7)
Remark 3.2. Obviously, J̃(u) = J(u) for all u ∈ E with norm
‖u‖L∞(R) ≤
5
2
π.
Now we denote the modified potential of interaction by
Ṽ (r) =
∣∣∣∣∫ r
0
|V ′(x)|dx
∣∣∣∣ .
Then from (6) for all |r| ≥ 3π, we have
V (2r) ≤ Ṽ (2r) ≤ 1
4
F (r). (8)
Hence, by (ii), F (r)→ +∞ for r → ±∞.
The lemma below can be found in [16, Lemma 2.5].
20 S. Bak
Lemma 3.3. Let W ∈ C1(R) be such that W (±π) = 0 and W (ξ) > 0 for
|ξ| < π, and let
I(u) :=
∫ +∞
−∞
[(u′(s))2 +W (u(s))]ds.
Then the minimum of I on M−π,π is attained and
min
u∈M−π,π
I(u) = 2
∫ π
−π
√
W (ξ) dξ =: ϑ.
Moreover, with the same ϑ,
inf
T>0
inf
u∈H1(−T,T )
{∫ T
−T
[(u′(s))2 +W (u(s))] ds : u(−T ) = −π, u(T ) = π
}
= ϑ.
Lemma 3.4. Assume conditions (i)–(iv) hold. Then for all u ∈ E,
J̃(u) ≥
∫ +∞
−∞
[
c2 − c21
2
(u′(s))2 +K(1 + cos(u(s)) +
1
2
F (u(s))
]
ds, (9)
and the functional J̃ is bounded from below on M−π,π. Moreover,
8
√
(c2 − c21)K < inf
u∈M−π,π
J̃(u) < 8c
√
K. (10)
Proof. Since
|Au(s)| ≤ |u(s+ cosϕ)|+ |u(s)| ≤ 2 max{|u(s+ cosϕ)|, |u(s)|},
|Bu(s)| ≤ |u(s+ sinϕ)|+ |u(s)| ≤ 2 max{|u(s+ sinϕ)|, |u(s)|},
then for every k > 0,
{s ∈ R : |Au(s)| > k} ⊆
{
s ∈ R : max{|u(s+ cosϕ)|, |u(s)|} > k
2
}
⊆
{
s ∈ R : |u(s+ cosϕ)| > k
2
}
∪
{
s ∈ R : |u(s)| > k
2
}
,
{s ∈ R : |Bu(s)| > k} ⊆
{
s ∈ R : max{|u(s+ sinϕ)|, |u(s)|} > k
2
}
⊆
{
s ∈ R : |u(s+ sinϕ)| > k
2
}
∪
{
s ∈ R : |u(s)| > k
2
}
.
Making use of (8) and the monotonicity of the potential Ṽ on (−∞, 0) and on
(0,+∞), we have∫
{s∈R:|Au(s)|>6π}
V (Au(s))ds ≤
∫
{s∈R:|Au(s)|>6π}
Ṽ (Au(s)) ds
≤
∫
{s∈R:|Au(s)|>6π}
Ṽ (2 max{|u(s+ cosϕ)|, |u(s)|}) ds
The Existence of Heteroclinic Travelling Waves in the Discrete sine-Gordon . . . 21
≤
∫
{s∈R:max{|u(s+cosϕ)|,|u(s)|}>3π}
1
4
F (max{|u(s+ cosϕ)|, |u(s)|}) ds
≤ 2
∫
{s∈R:|u(s)|>3π}
1
4
F (u(s))ds ≤ 1
2
∫ +∞
−∞
F (u(s)) ds. (11)
Similarly, ∫
{s∈R:|Bu(s)|>6π}
V (Bu(s))ds ≤ 1
2
∫ +∞
−∞
F (u(s))ds. (12)
By the definition of c1, we obtain∫
{s∈R:|Au(s)|≤6π}
V (Au(s)) ds ≤
∫
{s∈R:|Au(s)|≤6π}
c21
2
(Au(s))2 ds
≤
∫ +∞
−∞
c21
2
(Au(s))2 ds,∫
{s∈R:|Bu(s)|≤6π}
V (Bu(s)) ds ≤
∫
{s∈R:|Bu(s)|≤6π}
c21
2
(Bu(s))2 ds
≤
∫ +∞
−∞
c21
2
(Bu(s))2 ds.
Then it follows from (11) and (12) that
J̃(u) ≥
∫ +∞
−∞
[
c2
2
(u′(s))2 − c21
2
(Au(s))2 − c21
2
(Bu(s))2
+K(1 + cos(u(s))) + F (u(s))
]
ds
−
∫
{s∈R:|Au(s)|>6π}
V (Au(s))ds−
∫
{s∈R:|Bu(s)|>6π}
V (Bu(s)) ds
≥
∫ +∞
−∞
[
c2 − c21
2
(u′(s))2 +K(1 + cos(u(s))) +
1
2
F (u(s))
]
ds
for all u ∈ E, and (9) holds true.
Applying Lemma 3.3 to the functional
I1(u) =
c2 − c21
2
∫ +∞
−∞
[(u′(s))2 +W1(u(s))] ds,
where
W1(x) :=
2K
c2 − c21
[1 + cosx+
1
2K
F (x)],
and making use of (9), we obtain
inf
u∈M−π,π
J̃(u) ≥
(
c2 − c21
) ∣∣∣∣∫ π
−π
√
W1(x) dx
∣∣∣∣
=
√
2(c2 − c21)K
∣∣∣∣∫ π
−π
√
1 + cosx+ 0 dx
∣∣∣∣ = 8
√
(c2 − c21)K.
22 S. Bak
Furthermore, since V ≥ 0, we have
J̃(u) ≤ c2
2
∫ +∞
−∞
[
(u′(s))2 +
2
c2
(
K(1 + cos(u(s))) +
3
2
F (u(s))
)]
ds.
Now, we apply Lemma 3.3 to the functional
I2(u) =
c2 − c21
2
∫ +∞
−∞
[(u′(s))2 +W2(u(s))] ds,
where
W2(x) :=
2K
c2
[1 + cosx+
3
2K
F (x)].
As a consequence, we obtain
inf
u∈M−π,π
J̃(u) ≤ c2
∣∣∣∣∫ π
−π
√
W2(x)dx
∣∣∣∣ < 8c
√
K,
from which inequalities (10) follow.
The following lemma can be proved in the same way as Lemma 2.7 from [16].
Lemma 3.5. Assume conditions (i)–(iv) hold. Let ũ ∈M−π,π be a minimizer
of J̃ on M−π,π, then
‖ũ‖L∞(R) ≤
3
2
π + δ,
where
δ :=
4c21
c2 − c21 + c
√
c2 − c21
. (13)
In particular, if the speed c is large enough to ensure δ < π, then ‖ũ‖L∞(R) ≤ 5
2π.
4. Main result
In order to prove the main result, we need the following version of the
concentration-compactness principle obtained in [15, Lemma 4.1] (see [16,17,19]
for other versions of this principle).
Given T > 1 and η ∈ R, we define a truncated version of J̃ by
J̃T (u, η) :=
∫ 1
0
∫ η+T−1+τ
η−T+τ
c2
2
(u′(s))2 ds dτ −
∫ η+T−1
η−T
V (Au(s)) ds
−
∫ η+T−1
η−T
V (Bu(s)) ds+
∫ η+T− 1
2
η−T+ 1
2
[
K
(
1 + cos(u(s))
)
+
3
2
F (u(s))
]
ds.
Lemma 4.1 (Concentration-compactness). Assume conditions (i)–(iv) hold.
Let (un) ⊂ M−π,π be a minimizing sequence for J̃ on M−π,π, and let c be large
enough to ensure δ < π for δ defined in (13). Then there exists a subsequence,
still denoted by (un), such that one of the following holds:
The Existence of Heteroclinic Travelling Waves in the Discrete sine-Gordon . . . 23
(i2) (concentration) there is a sequence (ηn) ⊂ R such that for all small enough
ε > 0 there exists T > 0 such that
|J̃(un)− J̃T (un, ηn)| < ε
for every n ∈ N;
(ii2) (vanishing) for all T > 0,
lim
n→∞
sup
η∈R
J̃T (un, η) = 0;
(iii2) (dichotomy) there exists ε1 > 0 such that for every 0 < ε < ε1 there are
(fn), (gn) ⊂ E such that
|un − (fn + gn − π)| ≤ ε, |J̃(un)− (J̃(fn) + J̃(gn)| ≤ ε,
lim
n→∞
dist(supp(f ′n), supp(g′n)) = +∞, lim
n→∞
J̃(fn) = α, lim
n→∞
J̃(gn) = β,
for some 0 < α, β < infu∈M−π,π J̃(u) (π is needed in the first inequality to
ensure J(fn) < +∞ and J(gn) < +∞).
Lemma 4.2. Under the assumptions of Lemma 4.1, the functional J̃ has a
minimizer on M−π,π.
Proof. By Lemma 3.4, the functional J̃ is bounded from below on M−π,π.
Let (un) ⊂ M−π,π be a minimizing sequence. Then, by Lemma 4.1, the subse-
quence exists, still denoted by (un), which satisfies either of the following criteria:
concentration, vanishing or dichotomy.
Vanishing is impossible (see the proof of Lemma 5.1 in [15]).
We will show that dichotomy is also impossible. Indeed, as fn, gn ∈ E and
J̃(fn), J̃(gn) < +∞, the analogous statement of Lemma 3.1 (with J replaced by
J̃) shows that fn(±∞), gn(±∞) ∈ {±π}. Since fn + gn − π ∈ M−π,π, then only
fn(−∞) = fn(+∞) or only gn(−∞) = gn(+∞). In the first case, we set ũn :=
gn and in the second case, ũn := fn. Then (ũn) ⊂M−π,π and, by (iii2), possibly
after passing to a subsequence, we have
lim
n→∞
J̃(ũn) < inf
u∈M−π,π
J̃(u) = lim
n→∞
J̃(un).
We obtained a contradiction to the assumption that (un) ⊂M−π,π is a minimiz-
ing sequence of J̃ .
Thus (i2) holds. Hence, given ε > 0, there exists a sequence (ηn) ⊂ R and
T0 > 0 such that
|J̃(un)− J̃T0(un, ηn)| < ε.
Let wn(s) = un(ηn + s). The sequence (wn) is bounded in E. Indeed, by (9),
‖w′n‖L2(R) = ‖u′n‖L2(R) ≤
2
c2 − c21
J(un),
24 S. Bak
and by Lemma 3.5,
|wn(0)| ≤ 3
2
π + δ.
Hence, (wn) contains a subsequence, still denoted by (wn), that converges weakly
to some limit u ∈ E. The convergence is uniform on [−T0, T0], and
‖u′‖L2(−T0,T0) ≤ lim
n→∞
inf ‖w′n‖L2(−T0,T0) .
Since the functions V (u), 1 + cosu and F (u) belong to C1(R) and therefore are
Lipschitz continuous for |u| ≤ 3
2π + δ, there exists n0 ∈ N such that for all n >
n0, ∣∣∣∣(J̃(u)− c2
2
‖u′‖L2(R)
)
−
(
J̃T0(wn)− c2
2
‖u′‖L2(−T0,T0)
)∣∣∣∣ ≤ ε.
In fact, this inequality holds for all T > T0 instead of T0. By Lemma 3.1, u ∈
M−π,π. Furthermore, as T 7→ J̃T (wn, 0) is non-decreasing for every n ∈ N, we
obtain that J̃T (wn, 0) ≤ J̃(wn). Then,
J̃(u) = lim
T→∞
J̃T (u, 0) ≤ lim
T→∞
lim
n→∞
inf J̃T (wn, 0)
≤ lim
T→∞
lim
n→∞
J̃(wn) = lim
n→∞
J̃(wn) = lim
n→∞
J̃(un),
and thus u is a minimizer of the functional J̃ on M−π,π.
The following theorem is the main result of the paper.
Theorem 4.3. Assume conditions (i)–(iv) hold. Suppose that c is large
enough to ensure δ < π for δ defined by (13). Then equation (2) has a solu-
tion u that satisfies boundary conditions (3).
Proof. By Lemma 3.1, the modified functional J̃ has a minimizer u∗ ∈M−π,π.
We have to show that u∗ is a solution of equation (2) with boundary conditions
(3). We define the functional Ψ̃ similarly to Ψ but in terms of J̃ . Then the
function υ∗ = u∗ − υ0 minimizes Ψ̃ on H1(R). Since the embedding H1(R) ⊂
L∞(R) is continuous, we have that
‖υ0 + υ‖L∞(R) <
5
2
π
for all υ in the neighborhood ∆ ⊂ H1(R) of υ∗. Then, by Remark 3.2, for all
υ ∈ ∆,
Ψ(υ) = J(υ0 + υ) = J̃(υ0 + υ) = Ψ̃(υ),
and υ∗ minimizes Ψ as well as Ψ̃ in ∆. In particular, υ∗ is a local minimizer of the
functional Ψ on H1(R), i.e., υ∗ is a critical point of Ψ. Hence, by Lemma 3.1 (iv1),
u∗ = υ0 +υ∗ is the solution of equation (2) that satisfies boundary conditions (3).
The Existence of Heteroclinic Travelling Waves in the Discrete sine-Gordon . . . 25
References
[1] S.M. Bak, Traveling waves in chains of oscillators, Mat. Stud. 26 (2006), 140–153
(Ukrainian).
[2] S.M. Bak, Periodic traveling waves in chains of oscillators, Commun. Math. Anal.
3 (2007), 19–26.
[3] S.M. Bak, Existence of periodic traveling waves in a system of nonlinear oscillators
on a two-dimensional lattice, Mat. Stud. 35 (2011), 60–65 (Ukrainian).
[4] S.M. Bak, Existence of periodic traveling waves in the Fermi–Pasta–Ulam system
on a two-dimensional lattice, Mat. Stud. 37 (2012), 76–88 (Ukrainian).
[5] S.M. Bak, Periodic traveling waves in the discrete sine-Gordon equation on 2D-
lattice, Mat. Komp. Model. Ser.: Fiz.-Mat. Nauky 9 (2013), 5–10 (Ukrainian).
[6] S.M. Bak, Existence of the subsonic periodic traveling waves in the system of nonlin-
early coupled nonlinear oscillators on 2D-lattice, Mat. Komp. Model. Ser.: Fiz.-Mat.
Nauky 10 (2014), 17–23 (Ukrainian).
[7] S.M. Bak, Existence of the supersonic periodic traveling waves in the system of
nonlinearly coupled nonlinear oscillators on 2D-lattice, Mat. Komp. Model. Ser.:
Fiz.-Mat. Nauky 12 (2015), 5–12 (Ukrainian).
[8] S.M. Bak, Existence of heteroclinic traveling waves in a system of oscillators
on a two-dimensional lattice, Mat. Metodi Fiz.-Mekh. Polya 57 (2014), 45–52
(Ukrainian); Engl. transl.: J. Math. Sci. (N.Y.) 217 (2016), 187–197.
[9] S.N. Bak, Existence of solitary traveling waves for a system of nonlinear cou-
pled oscillators on a two-dimensional lattice, Ukräın. Mat. Zh. 69 (2017), 435–444
(Ukrainian); Engl. transl.: Ukrainian Math. J. 69 (2017), 509–520.
[10] S.N. Bak and A.A. Pankov, Traveling waves in systems of oscillators on two-
dimensional lattices, Ukr. Mat. Visn. 7 (2010), 154–175 (Ukrainian); Engl. transl.:
J. Math. Sci. (N.Y.) 174 (2011), 437–452.
[11] O.M. Braun and Y.S. Kivshar, The Frenkel–Kontorova Model. Concepts, Methods,
and Applications. Texts and Monographs in Physics, Springer–Verlag, Berlin, 2004.
[12] M. Fečkan and V. Rothos, Travelling waves in Hamiltonian systems on 2D lattices
with nearest neighbour interactions, Nonlinearity 20 (2007), 319–341.
[13] G. Friesecke and K. Matthies, Geometric solitary waves in a 2D math-spring lattice,
Discrete Contin. Dyn. Syst. Ser. B 3 (2003), 105–114.
[14] G. Ioos and K. Kirchgässner, Travelling waves in a chain of coupled nonlinear oscil-
lators, Comm. Math. Phys. 211 (2000), 439–464.
[15] C.-F. Kreiner and J. Zimmer, Heteroclinic travelling waves for the lattice sine-
Gordon equation with linear pair interaction, Discrete Contin. Dyn. Syst. 25 (2009),
915–931.
[16] C.-F. Kreiner and J. Zimmer, Travelling wave solutions for the discrete sine-Gordon
equation with nonlinear pair interaction, Nonlinear Anal. 70 (2009), 3146–3158.
[17] P.-L. Lions, The concentration–compactness principle in the calculus of variations.
The locally compact case, I, II, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984),
223–283.
26 S. Bak
[18] P.D. Makita, Periodic and homoclinic travelling waves in infinite lattices, Nonlinear
Anal. 74 (2011), 2071–2086.
[19] A. Pankov, Travelling Waves and Periodic Oscillations in Fermi–Pasta–Ulam Lat-
tices. Imperial College Press, London, 2005.
Received June 22, 2017.
S. Bak,
Vinnytsia Mykhailo Kotsiubynskyi State Pedagogical University, 32 Ostrozkogo St., Vin-
nytsia, 21001, Ukraine,
E-mail: sergiy.bak@gmail.com
Iснування гетероклiнiчних рухомих хвиль в
дискретному рiвняннi синус-Ґордона на двовимiрнiй
ґратцi
С. Бак
Статтю присвячено дискретному рiвнянню синус-Ґордона, яке опи-
сує нескiнченну систему нелiнiйно зв’язаних нелiнiйних осциляторiв на
двовимiрнiй ґратцi iз зовнiшнiм потенцiалом V (r) = K(1−cos r). Основ-
ний результат стосується iснування розв’язкiв у виглядi гетероклiнi-
чних рухомих хвиль. За допомогою методу критичних точок i принципу
концентрованої компактностi отримано достатнi умови iснування таких
розв’язкiв.
Ключовi слова: дискретне рiвняння синус-Ґордона, нелiнiйнi осциля-
тори, двовимiрна ґратка, гетероклiнiчнi рухомi хвилi, критичнi точки,
принцип концентрованої компактностi.
mailto:sergiy.bak@gmail.com
Introduction
The problem statement
Variational setting
Main result
|