Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media
In the framework of nonlinear Schrödinger equation, we analytically studied the nonlinear localized states in the system with two potential holes in the cases of linear and nonlinear media in the holes as well as their linear and nonlinear environment. All the possible solutions for the system are f...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1405522018-07-11T01:23:09Z Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media Gerasimchuk, V.S. Gerasimchuk, I.V. Dranik, N.I. In the framework of nonlinear Schrödinger equation, we analytically studied the nonlinear localized states in the system with two potential holes in the cases of linear and nonlinear media in the holes as well as their linear and nonlinear environment. All the possible solutions for the system are found and studied. The frequency dependences of the field amplitudes for all types of possible stationary localized states are obtained. 2016 Article Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media / V.S. Gerasimchuk, I.V. Gerasimchuk, N.I. Dranik // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 2. — С. 168-176. — Бібліогр.: 11 назв. — англ. 1812-9471 Mathematics Subject Classification 2000: 35Q55; 37K10. http://dspace.nbuv.gov.ua/handle/123456789/140552 DOI: doi.org/10.15407/mag12.02.168 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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In the framework of nonlinear Schrödinger equation, we analytically studied the nonlinear localized states in the system with two potential holes in the cases of linear and nonlinear media in the holes as well as their linear and nonlinear environment. All the possible solutions for the system are found and studied. The frequency dependences of the field amplitudes for all types of possible stationary localized states are obtained. |
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Article |
| author |
Gerasimchuk, V.S. Gerasimchuk, I.V. Dranik, N.I. |
| spellingShingle |
Gerasimchuk, V.S. Gerasimchuk, I.V. Dranik, N.I. Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media Журнал математической физики, анализа, геометрии |
| author_facet |
Gerasimchuk, V.S. Gerasimchuk, I.V. Dranik, N.I. |
| author_sort |
Gerasimchuk, V.S. |
| title |
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media |
| title_short |
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media |
| title_full |
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media |
| title_fullStr |
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media |
| title_full_unstemmed |
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media |
| title_sort |
solutions of nonlinear schrödinger equation with two potential wells in linear/nonlinear media |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2016 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/140552 |
| citation_txt |
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells in Linear/Nonlinear Media / V.S. Gerasimchuk, I.V. Gerasimchuk, N.I. Dranik // Журнал математической физики, анализа, геометрии. — 2016. — Т. 12, № 2. — С. 168-176. — Бібліогр.: 11 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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AT gerasimchukvs solutionsofnonlinearschrodingerequationwithtwopotentialwellsinlinearnonlinearmedia AT gerasimchukiv solutionsofnonlinearschrodingerequationwithtwopotentialwellsinlinearnonlinearmedia AT dranikni solutionsofnonlinearschrodingerequationwithtwopotentialwellsinlinearnonlinearmedia |
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2025-07-10T10:42:57Z |
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2025-07-10T10:42:57Z |
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1837256337203920896 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2016, vol. 12, No. 2, pp. 168–176
Solutions of Nonlinear Schrödinger Equation with Two
Potential Wells in Linear/Nonlinear Media
V.S. Gerasimchuk1, I.V. Gerasimchuk1,2, and N.I. Dranik1
1National Technical University of Ukraine “Kyiv Polytechnic Institute”
37 Peremohy Ave., Kyiv 03056, Ukraine
E-mail: viktor.gera@gmail.com
2Institute of Magnetism, National Academy of Sciences of Ukraine and Ministry of Education
and Science of Ukraine, 36-b Vernadsky Blvd., Kyiv 03142, Ukraine
E-mail: igor.gera@gmail.com
Received February 7, 2015
In the framework of nonlinear Schrödinger equation, we analytically stud-
ied the nonlinear localized states in the system with two potential holes in
the cases of linear and nonlinear media in the holes as well as their linear
and nonlinear environment. All the possible solutions for the system are
found and studied. The frequency dependences of the field amplitudes for
all types of possible stationary localized states are obtained.
Key words: nonlinear Schrödinger equation, nonlinear localized states,
potential well.
Mathematics Subject Classification 2010: 35Q55; 37K10.
1. Introduction
In the paper, we consider the actual problem of analytical research of the
character of localization of nonlinear stationary waves propagating in an anhar-
monic medium along thin plane-parallel layers with different physical properties.
As is well known [1, 2], the nonlinearity of the medium can give rise to new phys-
ical effects such as dependence of the transparency of a medium on the power of
the transmitted wave, spatial localization of nonlinear waves in periodic arrays
of optical waveguides, etc. From the point of view of technological applications,
of special interest are layered and modulated structures of various types.
In nonlinear optics, where layered and modulated structures can be applied
in optical communication systems, optical fibers, photonic crystals, optical delay
c© V.S. Gerasimchuk, I.V. Gerasimchuk, and N.I. Dranik, 2016
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells
lines [1], the studying of localized states in the system with two linear/nonlinear
defect layers, for instance, in optical switches [4, 5], and in periodic modulated
structures [1, 3] are very actual. Also of importance is the study of the adsorp-
tion of polymer chains in the systems with interfaces modeled by δ-functions or
potential wells (in some cases, by nonlinear or/and asymmetric ones) [6–8]. The
study of the structures with potential wells of this type is also important in the
theory of Bose–Einstein condensation [9].
In the present paper, in the framework of nonlinear Schrödinger equation we
study analytically the character of localization of nonlinear stationary waves in a
model system, which is a medium with linear or nonlinear properties at presence
of the potential in the form of two rectangular wells also with linear and nonlinear
properties. All possible solutions of the nonlinear Schrödinger equation for this
system are found and studied under the conditions of continuity of the wave
function and its first derivative at the boundaries of the potential wells and the
environment.
The exact solutions are found and the character of localization of nonlinear
stationary waves is studied for all possible combinations: (1) continuous linear
medium in the system; (2) nonlinear medium in the potential wells and linear
medium in the surrounding regions; (3) medium with linear properties in the
potential wells and nonlinear medium in the surrounding regions; (4) continu-
ous nonlinear medium in the system. The frequency dependences of the field
amplitudes for all types of possible stationary localized states are obtained.
2. Formulation of the Problem
We describe a model system consisting of two equal potentials U0 in the form
of symmetrical rectangular wells with the width d and placed at the distance
2a from each other. Let us consistently consider all the possible cases when the
medium inside the wells (a < |x| < a + d), as well as the environment, has linear
and nonlinear properties.
To solve this problem, we divide the system of potentials under consideration
along the axis of coordinates into five regions as shown in Fig. 1. We will seek
solutions separately for each region (the potential in each region is assumed to be
constant) and then make a “cross-linking” of these regions, taking into account
the equality of wave functions and their derivatives at the boundaries. Thus, the
problem reduces to finding the solutions Ψi(i = 1, 2, . . . , 5) of the one-dimensional
stationary nonlinear Schrödinger equation in the form
λΨ = −d2Ψ
dx2
+ 2Ψ3 − U0 Ψ, (1)
satisfying the following boundary conditions on the boundaries of the regions of
a partition:
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 169
V.S. Gerasimchuk, I.V. Gerasimchuk, and N.I. Dranik
Ψ1|−a−d−0 = Ψ4|−a−d+0, Ψ′
1|−a−d−0 = Ψ′
4|−a−d+0,
Ψ4|−a−0 = Ψ3|−a+0, Ψ′
4|−a−0 = Ψ′
3|−a+0,
Ψ3|a−0 = Ψ5|a+0, Ψ′
3|a−0 = Ψ′
5|a+0,
Ψ5|a+d−0 = Ψ2|a+d+0, Ψ′
5|a+d−0 = Ψ′
2|a+d+0.
(2)
The eigenfunction Ψ is corresponded to the eigenvalue λ.
I IV III V II
U
-a-d -a a a+d x
� λ–
-U
0
Fig. 1. Investigated system with potential wells.
Notice that in the regions (I–III) the values of λ can be only non-positive,
λ ≤ 0, and in the regions (IV–V) the values of λ are in the range of −U0 ≤ λ ≤ 0.
Let us consider each of the cases separately.
3. Linear Medium in the Potential Wells and Linear
Environment
For a given system, Eq. (1) in the respective regions is reduced to the equa-
tions
λΨ1,2,3 = −d2Ψ1,2,3
dx2
, (3)
λΨ4,5 = −d2Ψ4,5
dx2
− U0 Ψ4,5. (4)
170 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells
The general solution of Eqs. (3) has the form
Ψ1,2,3 = A1,2,3 e−
√
|λ|x + B1,2,3 e
√
|λ|x. (5)
Due to the finiteness of the wave function, the solutions Ψ1 and Ψ2 must vanish
at x →∞, thus the coefficients of these solutions A1 and B2 should be set to be
equal to zero. Accounting for symmetry of the wave functions Ψ2(x) = Ψ1(−x)
and Ψ3(x) = Ψ3(−x) allows us to simplify the form of the solutions of (3):
Ψ1 = A e
√
|λ|x, Ψ2 = Ae−
√
|λ|x, (6)
Ψ3 = A3
(
e−
√
|λ|x + e
√
|λ|x
)
= 2A3ch
(√
|λ|x
)
, (7)
where A ≡ A2 = B1 and A3 = B3.
The solution of Eq. (4) is conveniently written in the form
Ψ4,5 = C4,5 cos εx + D4,5 sin εx, (8)
where ε2 = U0 − |λ|.
By satisfying the boundary conditions (2) and taking into account the sym-
metry of the functions Ψ4(−x) = Ψ5(x), the coefficients C and D in (8) can
be expressed in terms of the coefficients A and A3 which are connected by the
relation
A = 2A3
[
ch
(√
|λ| a
)
cos εd + µ sh
(√
|λ| a
)
sin εd
]
, (9)
where µ =
√
|λ|
ε .
Finally, using the normalization condition for a linear medium, N =
+∞∫
−∞
|Ψ|2 dx
= 1, we find the solutions of the Schrödinger equation in each of the five regions:
Ψ1,2(x) =
ch2
(√
|λ| a
)
+ µ2 sh2
(√
|λ| a
)
− µ2 sh2
(√
|λ| a
)
√
Ω
√
(1 + µ2)
√
ch2
(√
|λ| a
)
+ µ2 sh2
(√
|λ| a
) e
√
|λ| (a+d±x), (10)
Ψ3 =
ch
(√
|λ|x
)
√
Ω
, (11)
Ψ4,5 =
ch
√
|λ| a cos εa− µsh
√
|λ| a sin εa√
Ω
cos εx
∓ ch
√
|λ| a sin εa + µ sh
√
|λ| a cos εa√
Ω
sin εx, (12)
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 171
V.S. Gerasimchuk, I.V. Gerasimchuk, and N.I. Dranik
where it is assumed that
Ω = a + d
(
ch2
√
|λ| a + µ2 sh2
√
|λ| a
)
+
1√
λ
(
sh
(
2
√
|λ| a
)
+ ch2
√
|λ| a
)
+2
µ
ε
sh
√
|λ| a
(
ch
√
|λ| a− sh
√
|λ| a
)
ch2
√
|λ| a + µ2 sh2
√
|λ| a . (13)
4. Nonlinear Medium in Potential Wells and Linear
Environment
It is obvious that the solutions in the regions with a linear medium do not
differ from the corresponding solutions (6), (7). The equations that describe the
media in potential wells become nonlinear,
λΨ4,5 = −d2Ψ4,5
dx2
+ 2Ψ3
4,5 − U0 Ψ4,5, (14)
and can be reduced to the equation for the Duffing oscillator with soft nonlin-
earity. Similarly to the Duffing equation, in the main anharmonic approximation
they describe small-amplitude oscillations in the symmetric potential and have
periodic solutions in the energy interval 0 < E < E0 = ε4
8 [10].
By multiplying (14) for Ψ by 2Ψ′ and then integrating, we reduce it to the
form
(Ψ′)2 = Ψ4 − ε2 Ψ2.
Separating the variables in the resulting equation and using an integral of motion
(energy), we can get its solution in an implicit form
Ψ∫
0
dΨ√
2E + Ψ4 − ε2 Ψ2
= x− x0. (15)
Expressing the energy E by the dimensionless parameter q =
√
ε4
8E−
√
ε4
8E − 1
and changing the variable according to Ψ = ε q√
1+q2
, an implicit solution (15) can
be transformed to the elliptic integral of the first kind:
F
(
arcsin
Ψ
Ψ∗ , q
)
=
Ψ
Ψ∗∫
0
dz√
(1− z2) (1− q2z2)
=
ε (x− x0)√
1 + q2
, (16)
where Ψ∗ = ε q√
1+q2
corresponds to the amplitude of the oscillations of the Duffing
pendulum, and q is the elliptic integral modulus.
172 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells
Finally, by expressing the sine of the amplitude through elliptic sine sn(u, q) ≡
sin am(u, q) and taking into account the condition of periodicity sn(u + K, q) =
cn(u,q)
dn(u,q) [11], the solution (16) can be written in an explicit form:
Ψ = γq
cn(γx, q)
dn(γx, q)
, (17)
where γ = ε√
1+q2
.
Under the requirement that the solutions in linear (6), (7) and nonlinear
(17) media be “cross-linked” on the borders of the regions, from the boundary
conditions (2), we obtain
Ψ1,2 = qγ
cn (γ(a + d), q)
dn (γ(a + d), q)
e
√
|λ|(a+d±x), (18)
Ψ3 =
√
|λ| q2 γ2 cn2(γa, q) dn2(γa, q)− q2 q′4 γ4 sn2(γa, q)√
|λ| dn2(γa, q)
ch
(√
|λ|x
)
, (19)
Ψ4,5(x) = γ q
cn(γx, q)
dn(γx, q)
, (20)
where γ ≡ γ4 = γ5, q ≡ q4 = q5, and q′ =
√
1− q2. The parameter q as a
function of the parameters of the system a, d and λ is determined by the equality
√
|λ| cn (γ (a + d), q) = q′2 γ
sn (γ(a + d), q)
dn (γ(a + d), q)
. (21)
Solutions (18)–(21) allow us to determine the full number of elementary ex-
citations due to the nonlinearity of the medium in potential wells:
N = 2 γ2 d− 2 γ [E (am(γ(a + d), q), q)− E (am(γa, q), q)]
+
q2 γ2
√
|λ|
cn2 (γ(a + d), q)
dn2 (γ(a + d), q)
+
(
2a + sh2
√
|λ| a
)
×|λ| q
2 γ2 cn2(γa, q) dn2(γa, q)− q2 q′4 γ4 sn2(γa, q)
|λ|3/2 dn4(γa, q)
+2 γ q2 [Λ1 (γ(a + d), q)− Λ1(γa, q)] . (22)
Here Λ1(γa, q) = sn(γa,q) cn(γa,q)
dn(γa,q) , E(ϕ, q) is the elliptic integral of the second kind,
and am(ϕ, q) = arcsin [sn(ϕ, q)] is the elliptic amplitude [11].
Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2 173
V.S. Gerasimchuk, I.V. Gerasimchuk, and N.I. Dranik
5. Linear Medium in the Potential Wells and Nonlinear
Environment
In this case, the environment outside the potential wells is described by the
nonlinear equations in the form
λΨ1,2,3 = −d2Ψ1,2,3
dx2
+ 2 Ψ3
1,2,3, (23)
the solutions of which in the symmetric regions I and II are given by
Ψ1,2 = ∓ ε0
sh (ε0(x∓ x0))
, (24)
where ε2
0 = |λ| and x0 > −(a + d).
We will seek the solutions of Eq. (23) in region III in the form
Ψ =
A
cn(Bx)
.
Because of the correlation between the constants, A = q′B, where B = γ3 =
ε0√
2 q2−1
and 1√
2
< q = q3 ≤ 1, we obtain
Ψ3 =
q′ γ3
cn(γ3 x, q3)
. (25)
The linear medium in potential wells is described by equations (4). The so-
lutions of these Eqs. (8), which satisfy the boundary conditions (2), for nonlinear
environment take the form
Ψ4,5 =
q′ γ3
cn (γ3 a, q)
[γ3 Λ2 (γ3 a, q) sin ε(a± x) + cos ε(a± x)] , (26)
where Λ2 (γ3 a, q) = sn (γ3 a,q) dn (γ3 a,q)
cn (γ3 a,q) .
The total number of elementary excitations is given by the expression
N = 2
√
ε2
0 +
q′23 γ2
3 α2
cn2 (γ3 a, q3)
− ε0 + γ3 [q′2 (γ3, q) + Λ2 (γ3 a, q)
−E (am(γ3 a, q), q)] +
q′2 γ2
3
2 cn2 (γ3 a, q)
[
d
(
1 + γ2
3 Λ2
2 (γ3 a, q)
)
+
1
2 ε
[4 γ3 Λ2 sin2 ε d + (1− γ2
3 Λ2
2) sin 2 ε d]
]}
, (27)
174 Journal of Mathematical Physics, Analysis, Geometry, 2016, vol. 12, No. 2
Solutions of Nonlinear Schrödinger Equation with Two Potential Wells
where the parameter α = cos εd−γ3 Λ2(γ3a, q3) sin εd depends on the parameters
of both potential wells and is conditioned by the nonlinear medium surrounding
the potential wells.
6. Nonlinear Medium in the Potential Wells and Nonlinear
Environment
Let the studied model system be a continuous nonlinear medium both in the
potential wells and around them. The Schrödinger equation in the corresponding
regions is reduced to (23) and (14). Their solutions have already been found in
Secs. 4 and 5:
Ψ1,2 = ∓ ε0
sh (ε0(x∓ x0))
, Ψ3 =
q′3 γ3
cn(γ3x, q3)
, Ψ4,5 = q γ
cn (γx, q)
dn (γx, q)
. (28)
The total number of elementary excitations is determined by the formula
N = 2
{
−ε0 + γ3
[
γ2d + q′2(γ3, q) + Λ2(γ3a, q)− E (am(γ3a, q), q)
]
+ γ [E (am (γ(a + d), q) , q)− E (am(γa, q), q)]
− γq2 [Λ1 (γ(a + d), q)− Λ1(γa, q)] +
√
ε2
0 + q2γ2
cn2 (γ(a + d), q)
dn2 (g(a + d), q)
}
. (29)
7. Conclusions
In the paper, we studied the possible localized states in the system containing
two potential wells for all possible combinations of linear and nonlinear media in
potential wells and regions outside them. The obtained results can be useful for
the study of localized states in systems with defects/interfaces in various fields of
physics, physical chemistry, biophysics, etc. In particular, they can be used for
the description of adsorption of polymer chains at the interfaces, in fiber optics,
where layered and modulated media are used in fiber communication systems,
optical switches, delay lines, in the theory of Bose–Einstein condensation, etc.
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