stationary localized modes in the quintic nonlinear ... · g. l. alfimov 1, v. v. konotop2 and p....
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Stationary localized modes in thequintic nonlinear Schrodinger
equation with a periodic potential
G. L. Alfimov 1, V. V. Konotop2 and P. Pacciani2
1 Moscow Institute of Electronic Engineering, Zelenograd, Moscow, 124498,
Russia2 Centro de Fısica Teorica e Computacional and Departamento de Fısica,
Faculdade de Ciencias, Universidade de Lisboa, Lisbon
Portugal
arXiv.nlin.PS/0605035
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.1/29
Outline
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.2/29
Outline• The model and its physical applications
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.2/29
Outline• The model and its physical applications
• Analytical Results
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.2/29
Outline• The model and its physical applications
• Analytical Results
• Numerical Results
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.2/29
Outline• The model and its physical applications
• Analytical Results
- Sufficient conditions for collapse
- Lower bound for the number of particles
- Asymptotic behaviour of the solutions
• Numerical Results
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.2/29
Outline• The model and its physical applications
• Analytical Results
- Sufficient conditions for collapse
- Lower bound for the number of particles
- Asymptotic behaviour of the solutions
• Numerical Results
- Behaviour of the number of particles vs frequency
- Dynamics of solutions
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.2/29
Outline• The model and its physical applications
• Analytical Results
- Sufficient conditions for collapse
- Lower bound for the number of particles
- Asymptotic behaviour of the solutions
• Numerical Results
- Behaviour of the number of particles vs frequency
- Dynamics of solutions
• Conclusions
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.2/29
ModelThe Quintic Nonlinear Schrödinger Equation (QNLS)
iut + uxx − V (x)u− σ|u|4u = 0 ,
σ = +1 Repulsive case
σ = −1 Attractive case
V (x) external potential,|V (x)| ≤ V0, V (x) = V (x+ π),
V (x) = V (−x), with a local minimum placed atx = 0
It is a standard model of the nonlinear physics
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.3/29
Physical motivations iut+uxx−V (x)u−σ|u|4u = 0
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.4/29
Physical motivations iut+uxx−V (x)u−σ|u|4u = 0
• equation for a slow envelope of a small amplitude wave in a medium
with weak dispersion and intensity depending nonlinearityf(|u|2),whose first order term of the Taylor expansion is zero
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.4/29
Physical motivations iut+uxx−V (x)u−σ|u|4u = 0
• equation for a slow envelope of a small amplitude wave in a medium
with weak dispersion and intensity depending nonlinearityf(|u|2),whose first order term of the Taylor expansion is zero
• renewed interest was originated by suggestions of its use for description
of a one-dimensional gas of bosons
Kolomeisky, et. al., Phys. Rev. Lett.85, 1146 (2000)
M. D. Girardeau and E. M. Wright, Phys. Rev. Lett.84, 5239 (2000)
B. Damski, J. Phys. B37, L85 (2004)
J. Brand, J. Phys. B37, S287 (2004)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.4/29
Physical motivations iut+uxx−V (x)u−σ|u|4u = 0
• equation for a slow envelope of a small amplitude wave in a medium
with weak dispersion and intensity depending nonlinearityf(|u|2),whose first order term of the Taylor expansion is zero
• renewed interest was originated by suggestions of its use for description
of a one-dimensional gas of bosons
Kolomeisky, et. al., Phys. Rev. Lett.85, 1146 (2000)
M. D. Girardeau and E. M. Wright, Phys. Rev. Lett.84, 5239 (2000)
B. Damski, J. Phys. B37, L85 (2004)
J. Brand, J. Phys. B37, S287 (2004)
• quasi-one-dimensional limit of the generalized Gross-Pitaevskii
equation when two-body interactions are negligibly small
Abdullaev, Salerno, Phys. Rev. A72, 033617 (2005)
Brazhnyi, Konotop, Pitaevskii, Phys. Rev. A73, 053601 (2006)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.4/29
Physical motivations
The stationary version of the QNLS describes the ground state of
the so-called Tonks-Girardeau gas
L. Tonks, Phys. Rev.50, 955 (1936)
M. Girardeau, J. Math. Phys.1, 516 (1960)
and it has been justified in
E.H. Lieb, R. Seiringer, J. Yngvason, Phys. Rev. Lett.91, 150401
(2003)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.5/29
Stationary Localized Solutions
iut + uxx − V (x)u− σ|u|4u = 0 ,
We seek for stationary localized solutions in the form
u(x, t) = exp(−iωt)φ(x)
ω is a constant referred below as frequency
φ(x) can be considered real and satisfying the equation
φxx − σφ5 + [ω − V (x)]φ = 0
with zero boundary conditions
lim|x|→∞
φ(x) = 0
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.6/29
Linear eigenvalue problemThe asymptotics of a localized solution are determined by the
linear eigenvalue problem
d2ϕαq
dx2+ [ωα(q) − V (x)]ϕαq = 0
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.7/29
Linear eigenvalue problemThe asymptotics of a localized solution are determined by the
linear eigenvalue problem
d2ϕαq
dx2+ [ωα(q) − V (x)]ϕαq = 0
which has a band spectrumωα(q) ∈[
ω(−)α , ω
(+)α
]
α = 1, 2, . . . referring to a band number
“+” and “–” standing respectively for the upper and lower
boundaries of theα-th band.
ϕαq are Bloch functions andq is the wave number in the reduced
Brillouin zone,q ∈ [−1, 1].
An interval(
ω(+)α , ω
(−)α+1
)
represents theα’s gap.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.7/29
Integrals of motionNumber of Particles
N =
∫
|u(x, t)|2dx
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.8/29
Integrals of motionNumber of Particles
N =
∫
|u(x, t)|2dx
Energy functional
E =1
2
∫(
|ux|2 −1
3|u|6 + V |u|2
)
dx
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.8/29
Integrals of motionNumber of Particles
N =
∫
|u(x, t)|2dx
Energy functional
E =1
2
∫(
|ux|2 −1
3|u|6 + V |u|2
)
dx
Remark: for σ = −1 andV (x) = 0 the numberN∗ =√
3π/2
separates the solutions which may collapse (N > N∗) and
solutions for which no collapse can occur (N < N∗)M. Weinstein, Comm. Math. Phys.87, 567 (1983)
G. Fibich and G. Papanicolaou, SIAM J. Appl. Math.60, 183 (1999)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.8/29
Stationary Localized SolutionsDefinitions
The stationary solutions withω < ω(−)1 (semi-infinite gap)
having the smallest number of particles at givenω will be
referred to asTownes Solitons(TSs):
the one-parametric family of solutions corresponding to the
lowest branch in the semi-infinite gap
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.9/29
Stationary Localized SolutionsDefinitions
The stationary solutions withω < ω(−)1 (semi-infinite gap)
having the smallest number of particles at givenω will be
referred to asTownes Solitons(TSs):
the one-parametric family of solutions corresponding to the
lowest branch in the semi-infinite gap
The stationary localized solutions withω ∈(
ω(+)α , ω
(−)α+1
)
will be
calledGap Solitons.
They do not exist atV (x) ≡ 0
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.9/29
Sufficient condition for collapse
in the attractive case (σ = −1)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.10/29
Sufficient condition for collapse
in the attractive case (σ = −1)
y(t) =
∫
x2|u|2dx
z(t) = Im∫
xuuxdx ,
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.10/29
Sufficient condition for collapse
in the attractive case (σ = −1)
y(t) =
∫
x2|u|2dx
z(t) = Im∫
xuuxdx ,
Theorem– If the conditions
z(0) ≥ 0 and η = −4E0 − V1y1/2(0)N1/2 > 0 ,
whereE0 = E + V0N/2 and V1 = supx |Vx(x)|, are satis-
fied, theny(t) ≤ y(0) − 2η t2 and collapse occurs at finite time
T ≤ (y(0)/2η)1/2.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.10/29
Sketch of the proofAssuming that the solution exists for0 ≤ t < T , we obtain
d2y
dt2= −4
dz
dt= 16E − 4
∫
(2V + xVx)|u|2dx
y(t) is a decreasing positive function:0 < y(t) < y(0)
Therefore one obtains
y(t) − y(0) ≤ (8E0 + 2V1y1/2(0)N1/2)t2 − 4z(0)t
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.11/29
Sketch of the proofAssuming that the solution exists for0 ≤ t < T , we obtain
d2y
dt2= −4
dz
dt= 16E − 4
∫
(2V + xVx)|u|2dx
y(t) is a decreasing positive function:0 < y(t) < y(0)
Therefore one obtains
y(t) − y(0) ≤ (8E0 + 2V1y1/2(0)N1/2)t2 − 4z(0)t
Remark: Comparing with the conditions for collapse for homoge-
neous caseV1 = 0 we conclude that in the presence of a periodic
potential the collapse occurs at smaller values of energyE < 0
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.11/29
Existence of a minimum N
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.12/29
Existence of a minimum N∫
φ2xdx ≤ (ω + V0)N
[
1 +3
2
N2
N2∗
(σ − 1)
]−1
[φ `
φxx − σφ5 + [ω − V (x)] φ = 0´
, integrating and using the Gagliardo-Nirenberg
inequality:R
φ6dx ≤ (3N2/N2∗)
R
φ2xdx]
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.12/29
Existence of a minimum N∫
φ2xdx ≤ (ω + V0)N
[
1 +3
2
N2
N2∗
(σ − 1)
]−1
[φ `
φxx − σφ5 + [ω − V (x)] φ = 0´
, integrating and using the Gagliardo-Nirenberg
inequality:R
φ6dx ≤ (3N2/N2∗)
R
φ2xdx]
• The stationary solutions atω < −V0 haveN > N∗/√
3
• Combining with the inequality(supx φ)4 ≤ 4N∫
φ2xdx
limit of small number of particles,N → 0, if available,
implies smallness of the amplitude of the function
supx |φ| → 0.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.12/29
Existence of a minimum N∫
φ2xdx ≤ (ω + V0)N
[
1 +3
2
N2
N2∗
(σ − 1)
]−1
[φ `
φxx − σφ5 + [ω − V (x)] φ = 0´
, integrating and using the Gagliardo-Nirenberg
inequality:R
φ6dx ≤ (3N2/N2∗)
R
φ2xdx]
• The stationary solutions atω < −V0 haveN > N∗/√
3
• Combining with the inequality(supx φ)4 ≤ 4N∫
φ2xdx
limit of small number of particles,N → 0, if available,
implies smallness of the amplitude of the function
supx |φ| → 0.
Small amplitude limit of the QNLS
[V. A. Brazhnyi and V. V. Konotop, Mod. Phys. Lett. B18627 (2004)
and references therein]Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.12/29
Minimal number of particlesWe look for a solution of the QNLS in a form
ψ = ǫ1/2ψ1 + ǫ3/2ψ2 + · · ·
ǫ≪ 1 andψj are functions of slow variablesxp = ǫpx, tp = ǫpt
ψ1 = A(x1, t2)ϕα(x0) exp(−iωαt0)
ωα stands either forω(+)α (if σ = 1) or for ω(−)
α+1 (if σ = −1) and
ϕα stands for the respective Bloch function.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.13/29
Minimal number of particlesThe slow envelopeA(x1, t2) solves the QNLS equation
i∂A
∂t2+
1
2Mα
∂2A
∂x21
− σχα|A|4A = 0
χα =∫ π
0|ϕα(x)|6dx is theeffective nonlinearity
M−1α = d2ωα(q0)/dq
2 is theeffective mass(q0 = 0 andq0 = 1
for the center and the boundary of the Brillouin zone)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.14/29
Minimal number of particlesThe slow envelopeA(x1, t2) solves the QNLS equation
i∂A
∂t2+
1
2Mα
∂2A
∂x21
− σχα|A|4A = 0
The soliton solution is known:
A =31/4ǫ1/2
χ1/4α
exp(−iσǫ2t)√
cosh(
ǫx√
8|Mα|)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.14/29
Minimal number of particlesThe slow envelopeA(x1, t2) solves the QNLS equation
i∂A
∂t2+
1
2Mα
∂2A
∂x21
− σχα|A|4A = 0
The soliton solution is known:
A =31/4ǫ1/2
χ1/4α
exp(−iσǫ2t)√
cosh(
ǫx√
8|Mα|)
Thusǫ2 can be interpreted as a frequency detuning to the gap,
ǫ2 = |ω − ωα|
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.14/29
Minimal number of particlesThe slow envelopeA(x1, t2) solves the QNLS equation
i∂A
∂t2+
1
2Mα
∂2A
∂x21
− σχα|A|4A = 0
The soliton solution is known:
A =31/4ǫ1/2
χ1/4α
exp(−iσǫ2t)√
cosh(
ǫx√
8|Mα|)
The number of particles
N ≈ N = ǫ
∫
|A|2dx =π√
3
2√
2|χαMα|
is determinedonlyby the lattice parametersStationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.14/29
Minimal number of particlesThe slow envelopeA(x1, t2) solves the QNLS equation
i∂A
∂t2+
1
2Mα
∂2A
∂x21
− σχα|A|4A = 0
The soliton solution is known:
A =31/4ǫ1/2
χ1/4α
exp(−iσǫ2t)√
cosh(
ǫx√
8|Mα|)
Note the difference with the case of nonlinear Schrödinger
equation with cubic nonlinear term: the number of particlesN
tends to zero together with the detuning:N ∝√
|ωα − ω|
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.14/29
Minimal number of particlesThe slow envelopeA(x1, t2) solves the QNLS equation
i∂A
∂t2+
1
2Mα
∂2A
∂x21
− σχα|A|4A = 0
The soliton solution is known:
A =31/4ǫ1/2
χ1/4α
exp(−iσǫ2t)√
cosh(
ǫx√
8|Mα|)
The number of particles
N ≈ N = ǫ
∫
|A|2dx =π√
3
2√
2|χαMα|
is determinedonlyby the lattice parametersStationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.14/29
Numerical results - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
xφ
-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
ω(−)1 ≈ −0.9368
(A) ω = −1.2, (B) ω = −1.1, (C) ω = −0.9374, (D) ω = −0.95. ω∗ ≈ 1.0005
x
φ
-2 0 20
1
2 (a)ω
N
-12 -10 -82.2
2.4
2.6
2.8N*
ω
N
-12 -10 -84.4
4.8
5.2
5.62N*
x
φ
-4 -2 0 2 40
1
2 (b)ω
N
-12 -10 -86.8
7.6
8.43N*
x
φ
-4 -2 0 2 40
1
2 (c)
V0 = 1 solid line,V0 = 3 dashed line,V0 = 5 dotted line, corresponding to 1st, 2nd, 3rd
branches. In (a), (b), (c) the shapes of the solutions forV0 = 3 andω = −12.175
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.15/29
Numerical results - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
x
φ-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.16/29
Numerical results - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
x
φ-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
• The number of particles corresponding to TSs (the lowest
branch) achieves itsnon-zero minimum, showing
nonmonotonic behaviour
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.16/29
Numerical results - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
x
φ-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
• The number of particles corresponding to TSs (the lowest
branch) achieves itsnon-zero minimum, showing
nonmonotonic behaviour
• Increase (decrease) of the number of particles in a TS does
not necessarily lead either to collapse or to spreading out of
the solution
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.16/29
Numerical results - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
x
φ-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
• Existence of the local minimum on the lowest curveN(ω),
at a frequency shifted from the edge toward the gap, does
not contradict the small amplitude approximation. The
small parameterǫ1/2 ∼ 0.1 corresponds to the frequency
detuning|ω − ωα| ∼ 10−4, therefore the region of validity
of asymptotic expansions is very narrow: the reshaping of a
soliton from the envelope soliton in C to an intrinsic
localized mode in B occurs already at quite small frequency
detuning Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.16/29
Numerical results - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
x
φ-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
• The whole lower branch lies below the critical valueN∗.
This means that, unlike the homogeneous QNLS equation,
small enough perturbations of stationary solutions in
presence of a periodic lattice do not result in collapsing
behaviour
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.16/29
Numerical results - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
x
φ-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
• In the cubic nonlinear Schrödinger equation the number of
particles is a monotonic function of the frequency detuning:
Aα ∝ |ω − ωα|1/2 and the small parameter isε instead of
ε1/2 obtained above
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.16/29
Numerical results - Semi-infinity gap,σ = −1
x
φ
-2 0 20
1
2 (a)ω
N
-12 -10 -82.2
2.4
2.6
2.8N*
ω
N
-12 -10 -84.4
4.8
5.2
5.62N*
x
φ
-4 -2 0 2 40
1
2 (b)ω
N
-12 -10 -86.8
7.6
8.43N*
x
φ
-4 -2 0 2 40
1
2 (c)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.17/29
Numerical results - Semi-infinity gap,σ = −1
x
φ
-2 0 20
1
2 (a)ω
N
-12 -10 -82.2
2.4
2.6
2.8N*
ω
N
-12 -10 -84.4
4.8
5.2
5.62N*
x
φ
-4 -2 0 2 40
1
2 (b)ω
N
-12 -10 -86.8
7.6
8.43N*
x
φ
-4 -2 0 2 40
1
2 (c)
• Different branches of the solutions at large negative
frequencies, independently on the potential depth, tend toa
chain of equidistant Townes solitons, the number of atoms
of each one approachingN∗
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.17/29
Numerical results - Semi-infinity gap,σ = −1
x
φ
-2 0 20
1
2 (a)ω
N
-12 -10 -82.2
2.4
2.6
2.8N*
ω
N
-12 -10 -84.4
4.8
5.2
5.62N*
x
φ
-4 -2 0 2 40
1
2 (b)ω
N
-12 -10 -86.8
7.6
8.43N*
x
φ
-4 -2 0 2 40
1
2 (c)
• The lowest branch solution has its counterpart for the
homogeneous QNLS equation, the coupled solitonic states
of the higher branches do not exist atV (x) ≡ 0. This
means that a lattice of arbitrary depth can be treated as a
singular perturbation allowing soliton binding
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.17/29
Numerical results - Semi-infinity gap,σ = −1
x
φ
-2 0 20
1
2 (a)ω
N
-12 -10 -82.2
2.4
2.6
2.8N*
ω
N
-12 -10 -84.4
4.8
5.2
5.62N*
x
φ
-4 -2 0 2 40
1
2 (b)ω
N
-12 -10 -86.8
7.6
8.43N*
x
φ
-4 -2 0 2 40
1
2 (c)
• Conjecture: "quantization" of the number of particles.
In the limitω → −∞ there exists an infinite number of
different stationary solutions of the QNLS equation with a
periodic potential, each of them havingnN∗, number of
particles (wheren is an integer). Thus there exist no upper
bound on the number of particles which can be loaded in
lattice Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.17/29
Stability - Semi-infinity gap,σ = −1
x
φ
-2 0 20
1
2 (a)ω
N
-12 -10 -82.2
2.4
2.6
2.8N*
ω
N
-12 -10 -84.4
4.8
5.2
5.62N*
x
φ
-4 -2 0 2 40
1
2 (b)ω
N
-12 -10 -86.8
7.6
8.43N*
x
φ
-4 -2 0 2 40
1
2 (c)
Intoducing a perturbation of the initial form(1 + δ)φ(x), whereδ
was±0.01 andφ(x) was given by one of the distributions shown
in the figure we did not observe any significant change in the be-
haviour of the modes
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.18/29
Stability - Semi-infinity gap,σ = −1
ω
N
-1.3 -1.2 -1.1 -10.5
1
1.5
2
2.5
3
A
BC
D
ω∗
N*
x
φ-10 0 100
0.4
0.8 A
x
φ
-10 0 100
0.4
0.8B
x
φ
-100 0 100
0.1
0.2 C
x
φ
-20 0 200
0.25
0.5 D
In the case (D), a decrease of the amplitude (δ = −0.01) resulted
in spreading out of the solution, confirming the results reported
earlier in F. K. Abdullaev and M. Salerno, Phys. Rev. A72,
033617 (2005). Increase of the amplitude resulted in a oscilla-
tion of the shape of the mode, which was neither collapsing nor
reaching any stationary state
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.18/29
Gap Solitons -first gap ω(+)1 ≈ −0.733, ω
(−)2 ≈ 2.1659
ω
N
0 1 23
4.5
6
7.5
9
A
B
CD
ω∗
xφ
-10 0 10-1
0
1 A
x
φ
-50 0 50
-1
0
1 Bx
φ
-50 0 50-0.5
0
0.5 C
x
φ
-10 0 10-1
0
1 D
σ = −1, even solutions(A) ω = 1.26, (B) ω = −0.73, (C) ω = 2.15, (D) ω = 1.33. ω∗ ≈ 1.256
ω
N
0 1 20
3
6
9
AB
C
D
N*
x
φ
-30 0 30-0.6
0
0.6 A
x
φ
-60 0 60-0.3
0
0.3 Bx
φ
-90 0 90
0
1C
x
φ
-15 0 15
0
0.5
D
σ = +1, odd solutions(A) ω = −0.7105, (B) ω = −0.73, (C) ω = 2.16, (D) ω = −0.5
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.19/29
Gap Solitons -first gap
ω
N
0 1 23
4.5
6
7.5
9
A
B
CD
ω∗
xφ
-10 0 10-1
0
1 A
x
φ
-50 0 50
-1
0
1 Bx
φ
-50 0 50-0.5
0
0.5 C
x
φ
-10 0 10-1
0
1 D
ω
N
0 1 20
3
6
9
AB
C
D
N*
x
φ
-30 0 30-0.6
0
0.6 A
x
φ
-60 0 60-0.3
0
0.3 Bx
φ
-90 0 90
0
1C
x
φ
-15 0 15
0
0.5
D
• One observes nonzero minima for the number of particles,
which are necessary for creation of gap solitons and
achieved at nonzero detuning toward the gap
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.20/29
Gap Solitons -first gap
ω
N
0 1 23
4.5
6
7.5
9
A
B
CD
ω∗
xφ
-10 0 10-1
0
1 A
x
φ
-50 0 50
-1
0
1 Bx
φ
-50 0 50-0.5
0
0.5 C
x
φ
-10 0 10-1
0
1 D
ω
N
0 1 20
3
6
9
AB
C
D
N*
x
φ
-30 0 30-0.6
0
0.6 A
x
φ
-60 0 60-0.3
0
0.3 Bx
φ
-90 0 90
0
1C
x
φ
-15 0 15
0
0.5
D
• One observes nonzero minima for the number of particles,
which are necessary for creation of gap solitons and
achieved at nonzero detuning toward the gap
• The number of particles for the localized modesgrows
infinitelywhen frequency approaches one of the gap edges.
This phenomenon is explained by thealgebraicasymptotic
of the gap soliton, which corresponds to the border of the
band:φ ∝ 1/x1/2 asx→ ∞
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.20/29
Gap Solitons - AsymptoticsLet us denote byωα the respective gap boundary, such that
ωα = ω(−)α+1 if σ = 1 andωα = ω
(+)α if σ = −1
Let ϕ0 be the periodic Bloch function corresponding toωα,
ϕ0(x) = ϕ0(x+ 2π), normalized by the relation∫ 2π
0ϕ2
0dx = 1.
We seek a solution in the form of the formal asymptotic series
φ = γ( ϕ0
x1/2+
ϕ1
x3/2+
ϕ2
x5/2+ · · ·
)
whereγ is a constant to be found.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.21/29
Gap Solitons - AsymptoticsRecurrent formula forϕn (n = 1, 2...), the first step of which
leads tod2ϕ1
dx2+[ ϕ1
x3/2− V (x)
]
ϕ1 =dϕ0
dx
A solution of this equation can be represented in the form
ϕ1 = ϕ1(x) + c1ϕ0(x)
c1 is a constant
ϕ1 satisfies the orthogonality condition∫ 2π
0
ϕ1ϕ0 = 0
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.22/29
Gap Solitons - Asymptotics
Considering the terms proportional tox−5/2, we obtain
d2ϕ2
dx2+ [ωα − V (x)]ϕ2 = σγ4ϕ5
0 + 3dϕ1
dx− 3
4ϕ0
for which the orthogonality condition yields
σγ4
∫ 2π
0
ϕ60(x)dx+
3
4
∫ π
0
ϕ20dx− 3
∫ 2π
0
ϕ0dϕ1
dxdx = 0
Taking into account the representation ofϕ(x) as well as
normalization condition forϕ0 we compute
γ =
(
3σ1 − 4
∫ 2π
0ϕ1
ϕ0
dxdx
4∫ 2π
0ϕ6
0dx
)1/4
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.23/29
Gap Solitons - AsymptoticsNext, representingϕ2 = ϕ2 + c2ϕ0, considering the terms
corresponding tox−7/2 we obtain
d2ϕ3
dx2+ [ωα − V (x)]ϕ3 = 5
dϕ2
dx− 5γ4ϕ4
0ϕ1 −15
4ϕ1
From the orthogonality condition we obtain the coefficientc1
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.24/29
Gap Solitons - AsymptoticsContinuing the described procedure yields the terms of the
expansion forφ up to any order and all of them are
unambiguously defined.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.24/29
Gap Solitons - AsymptoticsContinuing the described procedure yields the terms of the
expansion forφ up to any order and all of them are
unambiguously defined.
The logarithmicdivergence of the number of particlesN takes
place asω → ωα,
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.24/29
Gap Solitons - AsymptoticsContinuing the described procedure yields the terms of the
expansion forφ up to any order and all of them are
unambiguously defined.
The logarithmicdivergence of the number of particlesN takes
place asω → ωα,
Remark: a similar expansion for the cubic NLS equation gives the
decay∝ x−1, which implies finiteness of the number of particles
for the solution on the gap boundary.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.24/29
Stability - first gap, σ = −1
ω
N
0 1 23
4.5
6
7.5
9
A
B
CD
ω∗
x
φ-10 0 10
-1
0
1 A
x
φ
-50 0 50
-1
0
1 Bx
φ
-50 0 50-0.5
0
0.5 C
x
φ
-10 0 10-1
0
1 D
σ = −1, even solutions
All branches of the spectrum are located above the critical value
N∗, which suggests possibility of observing instability and col-
lapse of the respective solutions.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.25/29
Stability - first gap
Dynamics of gap solitons in attrac-
tive case, starting with perturbed ini-
tial conditions given by the stationary
solution withω = 2.15 multiplied by
a factor(1 + δ) with δ = 0.01
Dynamics of gap solitons in repul-
sive case, starting with perturbed ini-
tial conditions given by the stationary
solution withω = 2.15 multiplied by
a factor(1 + δ) with δ = 0.01
Negativeδ in the both cases resulted in spreading out solutions
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.26/29
Stability - first gap
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.27/29
Stability - first gap
The dynamics shows three different stages of evolution, the first two
being similar for both attractive and repulsive cases.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.27/29
Stability - first gap
Growth of the amplitudes and shrinking of the width of the solutions.
This "quasi-collapse" behavior is explained by the fact that in both
cases the dynamics is approximately described by the equation
i ∂A∂t2
+ 12Mα
∂2A∂x2
1− σχα|A|4A = 0, which is also an effective QNLS
equation, and thus corresponds to collapsing solutions.
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.27/29
Stability - first gap
It is not authentic collapse, because after the particles are gathered
mainly in a few (actually two) potential wells, that equation is not
applied any more and the systems is similar to a dimer model. A dimer
can be proven to the dynamics of the number of particles insensitive to
the sign of the nonlinearity
A. Sacchetti, SIAM J. Math. Anal.35, 1160 (2004)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.27/29
Stability - first gap
After several oscillations in attractive case we observed collapse. In the
repulsive case oscillatory behaviour was observed for longer times, also
showing three typical stages of the evolution, the first and second ones
being the same as in the attractive case and the last one corresponding
to decay of the pick amplitude and spreading out of the pulse
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.27/29
ConclusionsThe presence of the periodic potential results in qualitative
changes in behavior of the QNLS:
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.28/29
ConclusionsThe presence of the periodic potential results in qualitative
changes in behavior of the QNLS:• it modifies the sufficient condition for the collapse,
requiring negative energies with larger absolute values
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.28/29
ConclusionsThe presence of the periodic potential results in qualitative
changes in behavior of the QNLS:• it modifies the sufficient condition for the collapse,
requiring negative energies with larger absolute values• it allows the existence of infinite families of the localized
solutions
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.28/29
ConclusionsThe presence of the periodic potential results in qualitative
changes in behavior of the QNLS:• it modifies the sufficient condition for the collapse,
requiring negative energies with larger absolute values• it allows the existence of infinite families of the localized
solutions• it can bind Townes solitons (the latter expressed by the
conjecture about quantization of the number of particles in
the limit of large negative frequencies)
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.28/29
ConclusionsThe presence of the periodic potential results in qualitative
changes in behavior of the QNLS:• it modifies the sufficient condition for the collapse,
requiring negative energies with larger absolute values• it allows the existence of infinite families of the localized
solutions• it can bind Townes solitons (the latter expressed by the
conjecture about quantization of the number of particles in
the limit of large negative frequencies)• it allows storage of an infinite number of atoms, either by
coupling Townes soliton or by allowing existence of gap
solitons
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.28/29
ConclusionsThe presence of the periodic potential results in qualitative
changes in behavior of the QNLS:• it modifies the sufficient condition for the collapse,
requiring negative energies with larger absolute values• it allows the existence of infinite families of the localized
solutions• it can bind Townes solitons (the latter expressed by the
conjecture about quantization of the number of particles in
the limit of large negative frequencies)• it allows storage of an infinite number of atoms, either by
coupling Townes soliton or by allowing existence of gap
solitons• it enforces the stability of Townes solitons allowing them to
have under-critical number of particlesStationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.28/29
ConclusionsThe localized modes were shown to be very different from their
counterpart in the cubic nonlinear Schrödinger equation, where
the number of particles can go to zero and is bounded from
above.
The localized modes in the quintic nonlinear Schrödinger
equation
• require a minimal nonzero number of particles
• may have arbitrarily large number of atoms
Stationary localized modes in the quintic nonlinear Schrodinger equation with a periodic potential – p.29/29
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