Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential
The nonlinear Schrodinger equation with attractive quintic nonlinearity in periodic potential in 1D, modelling a dilute gas Bose – Einstein condensate in a lattice potential, is considered and one family of exact stationary solutions is discussed. Some of these solutions have neither an analog in t...
Збережено в:
| Дата: | 2010 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2010
|
| Назва видання: | Нелінійні коливання |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/174969 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential / E. Kengne, R. Vaillancourt // Нелінійні коливання. — 2010. — Т. 13, № 4. — С. 533-545. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-174969 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1749692021-01-29T01:26:40Z Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential Kengne, E. Vaillancourt, R. The nonlinear Schrodinger equation with attractive quintic nonlinearity in periodic potential in 1D, modelling a dilute gas Bose – Einstein condensate in a lattice potential, is considered and one family of exact stationary solutions is discussed. Some of these solutions have neither an analog in the linear Schrodinger equation nor in the integrable nonlinear Schrodinger equation. Their stability is examined analytically and numerically. Розглянуто нелiнiйне рiвняння Шредiнгера з притягуючою нелiнiйнiстю п’ятого порядку в одновимiрному перiодичному потенцiалi, яке моделює розряджений газовий конденсат Бозе – Ейнштейна в решiтчатому потенцiалi, а також деяку сiм’ю точних стацiонарних розв’язкiв. Деякi з цих розв’язкiв не мають аналогiв серед розв’язкiв нi лiнiйного рiвняння Шредiнгера, нi iнтегровного нелiнiйного рiвняння Шредiнгера. Дослiджено стабiльнiсть таких розв’язкiв аналiтичними та чисельними методами. 2010 Article Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential / E. Kengne, R. Vaillancourt // Нелінійні коливання. — 2010. — Т. 13, № 4. — С. 533-545. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/174969 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The nonlinear Schrodinger equation with attractive quintic nonlinearity in periodic potential in 1D, modelling a dilute gas Bose – Einstein condensate in a lattice potential, is considered and one family of exact
stationary solutions is discussed. Some of these solutions have neither an analog in the linear Schrodinger
equation nor in the integrable nonlinear Schrodinger equation. Their stability is examined analytically and
numerically. |
| format |
Article |
| author |
Kengne, E. Vaillancourt, R. |
| spellingShingle |
Kengne, E. Vaillancourt, R. Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential Нелінійні коливання |
| author_facet |
Kengne, E. Vaillancourt, R. |
| author_sort |
Kengne, E. |
| title |
Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential |
| title_short |
Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential |
| title_full |
Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential |
| title_fullStr |
Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential |
| title_full_unstemmed |
Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential |
| title_sort |
stability of exact solutions of the cubic-quintic nonlinear schrödinger equation with periodic potential |
| publisher |
Інститут математики НАН України |
| publishDate |
2010 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/174969 |
| citation_txt |
Stability of exact solutions of the cubic-quintic nonlinear Schrödinger equation with periodic potential / E. Kengne, R. Vaillancourt // Нелінійні коливання. — 2010. — Т. 13, № 4. — С. 533-545. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT kengnee stabilityofexactsolutionsofthecubicquinticnonlinearschrodingerequationwithperiodicpotential AT vaillancourtr stabilityofexactsolutionsofthecubicquinticnonlinearschrodingerequationwithperiodicpotential |
| first_indexed |
2025-07-15T12:06:22Z |
| last_indexed |
2025-07-15T12:06:22Z |
| _version_ |
1837714566396510208 |
| fulltext |
UDC 517 . 9
STABILITY OF EXACT SOLUTIONS OF THE CUBIC-QUINTIC
NONLINEAR SCHRÖDINGER EQUATION WITH PERIODIC POTENTIAL
СТАБIЛЬНIСТЬ ТОЧНИХ РОЗВ’ЯЗКIВ РIВНЯННЯ ШРЕДIНГЕРА
З НЕЛIНIЙНIСТЮ ТРЕТЬОГО ТА П’ЯТОГО ПОРЯДКIВ
I ПЕРIОДИЧНИМ ПОТЕНЦIАЛОМ
E. Kengne
Univ. Ottawa
585 King Edward Ave., Ottawa, ON K1N 6N5, Canada
Univ. Dschang
P. O. Box 4509, Dauala, Cameroon
e-mail: ekengne6@yahoo.fr
R. Vaillancourt
Univ. Dschang
P.O. Box 4509, Douala, Republic of Cameroon
The nonlinear Schrödinger equation with attractive quintic nonlinearity in periodic potential in 1D, mode-
lling a dilute gas Bose – Einstein condensate in a lattice potential, is considered and one family of exact
stationary solutions is discussed. Some of these solutions have neither an analog in the linear Schrödinger
equation nor in the integrable nonlinear Schrödinger equation. Their stability is examined analytically and
numerically.
Розглянуто нелiнiйне рiвняння Шредiнгера з притягуючою нелiнiйнiстю п’ятого порядку в од-
новимiрному перiодичному потенцiалi, яке моделює розряджений газовий конденсат Бозе – Ейн-
штейна в решiтчатому потенцiалi, а також деяку сiм’ю точних стацiонарних розв’язкiв. Де-
якi з цих розв’язкiв не мають аналогiв серед розв’язкiв нi лiнiйного рiвняння Шредiнгера, нi
iнтегровного нелiнiйного рiвняння Шредiнгера. Дослiджено стабiльнiсть таких розв’язкiв ана-
лiтичними та чисельними методами.
I. Introduction. It is well-known that a collapse phenomenon is observed in the Bose – Einstein
condensates (BECs) with attractive interaction if the number of atoms N exceeds a critical
value Nc, as in the case of atomic condensates with Li [1, 2]. In this case, experiments with
attractive two-body interaction have been performed [3, 4] with results consistent with the li-
mitation in the number of atoms and with the growth and collapse scenario. The nonlinear
Schrödinger (NLS) equation with cubic nonlinearity used to describe the BECs has stable
solutions in the one-dimensional (1D) case when the dispersion and nonlinearity effects can
effectively balance each other. In two and three dimensions, the focusing nonlinearity overcomes
the dispersion and a blow-up phenomenon occurs [5].
A few mechanisms, as the dispersion [6, 7] and nonlinearity management methods [5, 6, 8 –
12] have been suggested for the arrest of collapse. Based on the variational approach, method
of moments, and numerical simulations, the analysis showed that the nonlinearity management
method is effective in suppressing collapse in the 1D and 2D NLS equations with focusing cubic
nonlinearity. The arrest of collapse by using a strong cubic nonlinearity management scheme in
c© E. Kengne, R. Vaillancourt, 2010
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4 533
534 E. KENGNE, R. VAILLANCOURT
the 1D cubic-quintic nonlinear Schrödinger equations is of practical interest since it appears in
many branches of physics such as BEC. Here, it models the condensate with two- and three-
body interactions [13, 14]. In BEC, the variation of the atomic scattering length by the Feshbach
resonance technique leads to the oscillations of the mean-field cubic nonlinearity [15], but it also
induces variations of the quintic nonlinearity because the three-body interaction is dependent
on the scattering length [16].
In this paper we consider a quasi-one-dimensional BEC with two- and three-body interacti-
ons in a periodic potential. The governing equation is given by the NLS equation with a periodic
and with cubic and quintic terms
i
∂ψ
∂t
=
(
−1
2
∂2
∂x2
+ V (x) + λ2 |ψ|2 + λ3 |ψ|4
)
ψ, (1)
where ψ(x, t) represents the macroscopic wave function of the condensate and V (x) is an
external macroscopic potential. In this equation we assume dimensionless variables: the unit
of energy is ~ω/2, the unit of length is
√
~/(mω), and the unit of time is 1/ω. The parameters
λ2 and λ3 of the two- and three-body interactions in general can be complex quantities. In the
case of complex parameters, the imaginary parts of λ2 and λ3 describe the effects of inelastic
two- and three-body collisions on the dynamics of BEC’s, respectively. In this work, we do
not consider dissipative terms, and such cubic and quintic parameters are real. In general, the
parameter of the two-body interaction λ2 is proportional to the two-body scattering length as,
and is given by λ2 = 8πas [17].
In the absence of the three-body interaction parameter (λ3 = 0), exact solutions of Eq. (1)
have been constructed with the experimentally generated potential V (x) = V0 sin2(x) and their
stability was investigated in [18, 21]. In fact, a potential more general than sinusoidal potential
was considered: V (x) = −V0 sn2(x, k), where sn(x, k) denotes the Jacobian elliptic sine functi-
on with elliptic modulus 0 ≤ k ≤ 1 [19]. For a potential of this form, some exact solutions were
also found and their stability has been analyzed in [20], taking λ2 = −1. Currently, no experi-
ments are being performed where a BEC with two- and three-body interaction is trapped in 1D
periodic potential. Although motivated by the developments in BECs, in this paper we consider
Eq. (1) with attractive three-body interaction (λ3 < 0). Thus we consider
i
∂ψ
∂t
=
(
−1
2
∂2
∂x2
+ V (x) + λ2 |ψ|2 − |ψ|4
)
ψ. (2)
As in BECs with two-body interaction [21 – 23], the proper choices for the potential allow for
the construction of a large class of exact solutions. The external potential considered in this
work is a generalization of the sinusoidal, standing light wave potential [24]:
V (x) = A0 +B0 sn2(lx, k). (3)
For B0 = 0, Eq. (2) becomes a cubic-quintic nonlinear Schrödinger equation and hence is
integrable [25, 26]. In the limit as k → 1−0, V (x) becomes an array of well-separated hyperbolic
secant potential barriers or wells, while in the limit as k → +0 it becomes purely sinusoidal.
Because sn(lx, k) is a periodic function with period
4K(k)l =
4
l
∫ π/2
0 dz√
1− k2 sin2 z
,
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
STABILITY OF EXACT SOLUTIONS OF THE CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION . . . 535
the external potential V (x) is also a periodic function with period 2K(k). This period approaches
infinity as k → 1 − 0. Thus, as k → 1 − 0, the potential (3) is a periodic lattice of separated
peaks or troughs. Hence, by changing the parameter k, various interesting regimes of BECs are
considered. This is the reason for considering potential of the form (3). The parameters A0, B0
and l are introduced to facilitate the construction of exact solutions and, for simplicity, we take
l > 0. Although this exact expression for the potential is necessary to allow the construction
of exact solutions, it is its qualitative features, that is, its periodicity and amplitude, that are the
most important.
The paper is outlined as follows. In the next section we derive and consider various properti-
es and limit of explicit solutions of Eq. (2) with potential (3). In Section III we develop the
analytic framework for linear stability properties of solutions of Section II. In section IV, the
results of the numerical simulations are discussed. A brief summary of the results concludes the
paper in Section V.
II. Stationary solutions. Equation (2) with V (x) = const is an integrable cubic-quintic
nonlinear Schrödinger equation of which many explicit solutions are known [25 – 27]. If V (x) 6=
6= const the cubic-quintic NLS equation is not integrable, and only small classes of explicit
solutions can be obtained. A judicious choice of the potential V (x) allows for the cancellation
of the nonlinear terms in Eq. (2) so that exact solutions can be constructed. Of course, one
can always find a suitable potential V (x) by solving Eq. (2) for V (x), given a certain ψ(x, t).
This results in a time-dependent potential and hence is not of interest. In this section, we give a
dictionary of the families of exact solutions we were able to construct. These families are built
as the families of exact solutions found for the two-body interaction case, discussed in [18, 20].
For the exact solutions, the density of the condensate |ψ(x, t)|2 is a linear function of either
dn(x, k) or sn(lx, k), or cn(lx, k), where sn(lx, k) and cn(lx, k) are the Jacobian elliptic sine
and cosine functions, respectively. These solutions are given by ψ(x, t) = r(x) exp (iθ(x)− ωt) ,
where r(x) and θ(x) are two real functions to be determined and ω is the chemical potential of
the condensate. Inserting this ansatz into Eq. (2) yields
r3r′′ + 2ωr4 − 2V r4 − 2λ2r
6 + 2r8 − C2 = 0. (4)
The parameter C is defined via the relation θ′(x) = C/r2(x), which expresses conservation of
angular momentum. Null angular momentum solutions, which constitute an important special
case, satisfy C = 0 and we choose θ(x) = 0. First we discuss the solution with dn(lx, k).
The quantities associated with this solution will be denoted with the subscript 1. The quantities
associated with the sn(lx, k) and cn(lx, k) solutions will be subscripted by 2 and 3, respectively.
In order to find some restrictions on the domain of the parameters of the solutions, we will use
the fact that both sn(lx, k) and cn(lx, k) have zero average as functions of x and lie in [−1, 1],
while dn(lx, k) has nonzero average, and its range is
[√
1− k2, 1
]
.
To find the dn(lx, k) solutions, we set
r21(x) = A1 +B1 dn(lx, k). (5)
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
536 E. KENGNE, R. VAILLANCOURT
Inserting Eq. (5) into Eq. (4) yields
A1 =
3k2l2 − 8B0
8 (k2l2 − 2B0)
λ2, B2
1 =
3k2l2 − 8B0
8k2
,
ω1 =
32B2
0 + 9k4l4 − 36k2l2B0
32 (k2l2 − 2B0)
2 λ2
2 +
B0 + k2A0
k2
+
l2
(
k2 − 2
)
8
, (6)
C2
1 =
(
λ2
2 −
24
(
k2l2 − 2B0
)2 (
l2k2 − 4B0
)
k2 (3k2l2 − 8B0)
2
)
×
×
(
λ2
2 −
8
(
k2l2 − 2B0
)2 (1− k2
)
k2 (3k2l2 − 8B0)
)
k2l2
(
3k2l2 − 8B0
)2
512 (k2l2 − 2B0)
3 .
The freedom in choosing the potential gives five free parameters, λ2, A0, B0, k, and l. The
requirements that r21(x),B
2
1 , and C2
1 be nonnegative imposes conditions on the domain of these
parameters. The condition on the sign of r21(x) and B2
1 gives
3k2l2
8
> B0 and
λ2 ≥ −
4
(
k2l2 − 2B0
)√
1− k2
k
√
2 (3k2l2 − 8B0)
for B1 > 0, and
3k2l2
8
> B0 and
λ2 ≥
4
(
k2l2 − 2B0
)
k
√
2 (3k2l2 − 8B0)
for B1 < 0. Solving the inequality C2
1 ≥ 0, one obtains the supplementary conditions on the
domain of parameters λ2, A0, B0, k, and l for the validity of solutions (5), (6).
The sn(lx, k) solutions are found by substituting
r22(x) = A2 +B2 sn(lx, k) (7)
in Eq. (4). Equating different powers of sn(lx, k) imposes the following constraints on the
parameters:
A2 =
3k2l2 − 8B0
8 (k2l2 − 2B0)
λ2, B2
2 =
8B0 − 3k2l2
8
,
ω2 = A0 +
32B2
0 − 36k2l2B0 + 9k4l4
32 (k2l2 − 2B0)
2 λ2
2 +
l2
(
1 + k2
)
8
, (8)
C2
2 =
l2
(
8B0 − 3k2l2
)3
2048 (k2l2 − 2B0)
4
(
8
(
k2l2 − 2B0
)3
8B0 − 3k2l2
− λ2
2
)(
k2λ2
2 −
8
(
k2l2 − 2B0
)3
8B0 − 3k2l2
)
.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
STABILITY OF EXACT SOLUTIONS OF THE CUBIC-QUINTIC NONLINEAR SCHRÖDINGER EQUATION . . . 537
The freedom in choosing the potential gives five free parameters, λ2, A0, B0, k, and l. The
requirements that r22(x),B
2
2 , and C2
2 be nonnegative imposes conditions on the domain of these
parameters. For the positivity of both r22(x) and B2
2 , we obtain that either B0 >
k2l2
2
and
λ2 ≥
4
(
2B0 − k2l2
)√
2 (8B0 − 3k2l2)
> 0 or
3k2l2
8
< B0 <
k2l2
2
and λ2 ≤
4
(
2B0 − k2l2
)√
2 (8B0 − 3k2l2)
< 0. The
complementary conditions on the values of the parameters λ2, A0, B0, k, and l for the validity
of solutions (7), (8) are obtained by solving the inequality C2
2 ≥ 0.
For the cn(lx, k) solutions, we substitute
r23(x) = A3 +B3 cn(lx, k) (9)
into Eq. (4) and obtain
A3 =
3k2l2 − 8B0
8 (k2l2 − 2B0)
λ2, B2
3 =
3k2l2 − 8B0
8
,
ω3 =
32B2
0 − 36k2l2B0 + 9k4l4
32 (k2l2 − 2B0)
2 λ2
2 +
l2
(
1− 2k2
)
8
+A0 +B0, (10)
C2
3 =
k2l2
(
3k2l2 − 8B0
)3
2048 (k2l2 − 2B0)
4
(
−λ2
2 +
k4l4
(
3k2l2 − 8B0
)3
256 (k2l2 − 2B0)
4
)
×
×
(
k2l4
(
1− k2
) (
8B0 − 3k2l2
)3
4096 (k2l2 − 2B0)
4 − λ2
2
)
.
The freedom in choosing the potential gives five free parameters, λ2, A0, B0, k, and l. The
requirements that r23(x),B
2
3 , and C2
3 be nonnegative imposes conditions on the domain of these
parameters. For the positivity of both r23(x) and B2
3 , we must have
3k2l2
8
> B0 and λ2 ≥
≥
4
(
k2l2 − 2B0
)√
2 (3k2l2 − 8B0)
> 0. This means that the cn(lx, k) solutions do not exist for the BECs
with attractive two-body-interaction. Other restrictions on the region of validity of solution (9),
(10) are obtained by imposing the nonnegativity of C2
3 .
The null angular momentum solutions case. Null angular momentum corresponds toCj = 0.
Since for each of solutions rj(x), j = 1, 2, 3, C2
j has three factors, one of which is different from
zero (because Bj 6= 0 for non plane wave solutions), and each factor that can be zero is a
quadratic function of λ2 (see the third equation of Eqs. (6), (8), and (10), there are at most four
possible choices of λ2 for which this occurs.
(I) For the solutions r1(x), we have
λ2 ∈
± 2
(
k2l2 − 2B0
)
k (3k2l2 − 8B0)
√
6 (l2k2 − 4B0),±
2
(
k2l2 − 2B0
)
k
√
2 (1− k2)
3k2l2 − 8B0
if l2k2/4 > B0,
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
538 E. KENGNE, R. VAILLANCOURT
and
λ2 ∈
±2
(
k2l2 − 2B0
)
k
√
2 (1− k2)
3k2l2 − 8B0
if l2k2/4 ≤ B0 < 3k2l2/8.
(II) For the solutions r2(x) with condition r22(x) ≥ 0, we obtain
λ2 = 2
(
2B0 − k2l2
)√2
(
k2l2 − 2B0
)
8B0 − 3k2l2
if
k2l2 − 1
2
≥ B0 >
3
8
k2l2 and k2l2 − 4 > 0, and
λ2 =
2
(
2B0 − k2l2
)
k
√
2
(
k2l2 − 2B0
)
8B0 − 3k2l2
if
k2
(
l2 − 1
)
2
≥ B0 >
3
8
k2l2 and l2 − 4 > 0.
We note that Eqs. (9) and (10) do not define any null angular momentum cn(lx, k) solutions
with r23(x) ≥ 0 for all x.
The trigonometric limit. In the limit as k → +0, the Jacobian elliptic functions reduce to
trigonometric functions and V (x) = A0 + B0 sin2(lx). In the limit as k → +0, Eq. (8) gives
a negative C2
2 , and this means that Eqs. (7) and (8) do not give any trigonometric solution.
Passing in the limit as k → +0 in Eqs. (9) and (10), we find the trigonometric solutions
r23±(x) =
λ2
2
±
√
−B0 cos(lx), ω3 =
λ2
2
4
+
l2
8
+A0 +B0, B0 < 0,
which is a null angular momentum solution. For the positivity of r23±(x), we must have λ2 ≥
≥ 2
√
−B0. Some trigonometric solutions are illustrated in Fig. 1.
ISSN 1562-3076. Нелiнiйнi коливання, 2010, т . 13, N◦ 4
|