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Localized modes in the nonlinearSchr odinger equation with periodicnonlinearity and periodic potential
Vladimir Konotop
University of Lisbon, PORTUGAL
Yu. V. Bludov , VVK, Phys. Rev. A 74, 043616 (2006)
Yu. V. Bludov, V.A. Brazhnyi , VVK, Phys Rev A 76 (2007) (in press;
cond-mat:0706.0079)
F. Kh. Abdullaev , Yu. V. Bludov, S. V. Dmitriev , P. G. Kevrekidis , VVK,
(submitted; cond-mat: 0707.2512)Nonlinear Schrodinger equation with periodic coefficients – p. 1/17
Outline
Physical applications
Modulational instability and localized modes
Delocalizing transition
Lattices: "Tight-binding" approximation
Nonlinear Schrodinger equation with periodic coefficients – p. 2/17
EM waves in stratified media
∂2E
∂x2+∂2E
∂z2+ω2
c2n2E = 0,
where n = n0 + n1(x, z) + n2(x, z)|E|2 + n4(x, z)|E|4.In the parabolic approximation E(x, z) = eikzA(x, z), k = ω
cn0,
Az ≪ kA
2ik∂A
∂z+∂2A
∂x2+k2
(
2n1
n0
+n2
1
n20
+2n2
n0
|A|2 +
(
n22
n20
+2n4
n0
)
|A|4)
A = 0
Nonlinear Schrodinger equation with periodic coefficients – p. 3/17
EM waves in stratified media
∂2E
∂x2+∂2E
∂z2+ω2
c2n2E = 0,
where n = n0 + n1(x, z) + n2(x, z)|E|2 + n4(x, z)|E|4.In the parabolic approximation E(x, z) = eikzA(x, z), k = ω
cn0,
Az ≪ kA
2ik∂A
∂z+∂2A
∂x2+k2
(
2n1
n0
+n2
1
n20
+2n2
n0
|A|2 +
(
n22
n20
+2n4
n0
)
|A|4)
A = 0
Nonlinear Schrodinger equation with periodic coefficients – p. 3/17
EM waves in stratified media
∂2E
∂x2+∂2E
∂z2+ω2
c2n2E = 0,
where n = n0 + n1(x, z) + n2(x, z)|E|2 + n4(x, z)|E|4.In the parabolic approximation E(x, z) = eikzA(x, z), k = ω
cn0,
Az ≪ kA
2ik∂A
∂z+∂2A
∂x2+k2
(
2n1
n0
+n2
1
n20
+2n2
n0
|A|2 +
(
n22
n20
+2n4
n0
)
|A|4)
A = 0
Let n1(x, z) ≡ n1(x), n2(x, z) ≡ n2(x), and n4 ≡ 0, then
2ik∂A
∂z+∂2E
∂x2+ 2k2
(
n1(x)
n0
+n2(x)
n0
|A|2)
A = 0
or after renormalization iψt = −ψxx + V (x)ψ +G(x)|ψ|2ψNonlinear Schrodinger equation with periodic coefficients – p. 3/17
BEC in an optical lattice
Heisenberg equation
i~Ψt = − ~2
2m∆Ψ + Vext(r)Ψ + g(r)Ψ†
ΨΨ
Assumption: Ψ = Ψ + ψ with Ψ ≈ 〈N |Ψ|N + 1〉Mean-field approximation (Gross-Pitaevskii equation)
i~Ψt = − ~2
2m∆Ψ + Vext(r)Ψ + g(r)|Ψ|2Ψ
One-dimensional limit
iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψ[Fedichev, Kagan, Shlyapnikov, PRL, 77, 2913 (1996); Abdullaev and Garnier, PRA 72,061605 (2005)]
Nonlinear Schrodinger equation with periodic coefficients – p. 4/17
Boson-fermion mixture in OL
Heisenberg equations
i~Ψt = − ~2
2mb
∆Ψ + VbΨ + g1Ψ†ΨΨ + g2Φ
†ΦΨ ,
i~Φt = − ~2
2mf
∆Φ + Vf Φ + g2Ψ†ΨΦ
Mean-field approximation: Ψ = 〈Ψ〉, 〈Φ†Φ〉 = ρ0(r) + ρ1(r, t)
[Tsurumi, Wadati, J. Phys. Soc. Jap. 69 97 (2000); Bludov, Konotop, PRA 74, 043616 (2006)]
i~∂Ψ
∂t= − ~
2
2mb
∆Ψ + Vb(r)Ψ + gbb|Ψ|2Ψ + gbfρΨ
∂2ρ1
∂t2= ∇
[
ρ0(r)∇(
(6π2)2/3~
2
3m2fρ
1/30 (r)
ρ1 +gbfmf
|Ψ|2)]
.
ρ0(r) is an unperturbed (Thomas-Fermi) distribution of fermions
Nonlinear Schrodinger equation with periodic coefficients – p. 5/17
BF mixture in OL
1D limit [Bludov, Konotop, PRA 74, 043616 (2006)]
iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψU(x) ≡ U0(x) + U1(x), G(x) = G0 − G1
1/3(x),
Here U0(x) = Vb(x)/(~2κ2/2mb), U1 = 4πκabfmb/m,
G0 = 4abbNb/κa2, and G1 = 2 (6/π)1/3 (abf/a)
2(mfmb/m2)Nb
(x) = ρ0(x)/κ
Λ = κabb
Nonlinear Schrodinger equation with periodic coefficients – p. 6/17
NLS equation with periodic nonlinearity
iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψ,
U(x) = U(x+ π), G(x) = G(x+ π)
The linear eigenvalue problem
−d2ϕ
(σ)n
dx2+ U(x)ϕ(σ)
n = E (σ)n ϕ(σ)
n ,(
E (−)α , E (+)
α
)
is α’s gap
Multiple scale expansion: ψ ≈ ǫA(τ, ξ)ϕ(σ)n (x) where ξ = ǫx
and τ = ǫ2t are the slow variables, ǫ ∼ |µ− E (σ)n | ≪ 1
NLS equation: iAτ = −(2M(σ)n )−1Aξξ + χ
(σ)n |A|2A
M(σ)n = [d2E (σ)
n /dk2]−1 is the effective massχ
(σ)n =
∫ π
0G(x)|ϕ(σ)
n (x)|4dx is the effective nonlinearity
For the modulational instability: M (σ)α χ
(σ)α < 0
Nonlinear Schrodinger equation with periodic coefficients – p. 7/17
NLS equation with periodic nonlinearity
If γ(−)2 < Λ < γ
(+)1 : gap
solitons do not exist
Since Gm = minG(X) < 0,in the limit E → −∞, thereexists a soliton:
φS(X) ≈ e−iET
p
2|E|/|Gm|
cosh(Xp
|E|)
If χ(−)1 > 0 in the semi-infinite
band there exists a minimalnumber of bosons necessaryfor creation of a localizedmode [Sakaguchi, Malomed,Phys. Rev. A 72, 046610(2005) ]
Nonlinear Schrodinger equation with periodic coefficients – p. 8/17
NLS equation with periodic nonlinearity
All bright localized modes are real [Alfimov, VVK, Salerno, Europhys. Lett. 58, 7
(2002), review Brazhnyi, VVK, Mod. Phys. Lett. B 14 627 (2004)]
Nonlinear Schrodinger equation with periodic coefficients – p. 9/17
NLS equation with periodic nonlinearity
The first lowest gap
Nonlinear Schrodinger equation with periodic coefficients – p. 10/17
Delocalizing transition
Decrease of the linear lattice potential in the NLS equation results in delocalizingtransition in 2D and 3D, but no transition occurs in 1D. [Kalosakas, Rasmussen,Bishop, PRL 89, 030402 (2002); Baizakov, Salerno, PRA 69, 013602 (2004).]
iψt = −ψxx + U(x)ψ + |ψ|2ψ
V (x) = −A cos(2x)
k
E
0 0.2 0.4 0.6 0.8 10
2
4
6
8
10
12
14
A= 3
α=0
α=1
Nonlinear Schrodinger equation with periodic coefficients – p. 11/17
Delocalizing transition
Consider iψt = −ψxx + U(x)ψ + G(x)|ψ|2ψ
Recall M (σ)n = [d2E (σ)
n /dk2]−1, and χ(σ)n =
∫ π
0G(x)|ϕ(σ)
n (x)|4dxIf M (+)
n χ(+)n > 0 and M (−)
n χ(−)n > 0, small amplitude solitons
cannot exist at the both gap edges.
Let
U(x) = −V cos(2x)
G(x) = G− cos(2x)
A stationary solution:
ψ(x, t) → ψ(x)e−iµtG
(σ)n = G
(σ)n (V ) is the value of G at which
χ(σ)n becomes zero.
Nonlinear Schrodinger equation with periodic coefficients – p. 12/17
Delocalizing transition
Let V is changing and G is fixed
The chemical potential is µ = (E + Enl)/N where
E =
∫(
|ψx|2 + U(x)|ψ|2 +1
2
∫
G(x)|ψ|4)
dx
Enl =1
2
∫
G(x)|ψ|4dx
Nonlinear Schrodinger equation with periodic coefficients – p. 13/17
Delocalizing transition
Let G is changing and V is fixed
Nonlinear Schrodinger equation with periodic coefficients – p. 14/17
Tight-binding approximation
Wannier functions
wnα(x) =1√2
∫ 1
−1
ϕαq(x)e−iπnq dq
The expansion: ψ(x, t) =∑
n,α cnα(t)wnα(x) leads to
icnα − cnαω0α − (cn−1,α + cn+1,α)ω1α−∑
n1,n2,n3
cn1αcn2αcn3αWnn1n2n3αααα = 0
where Eαq =∑
n ωnαeiπnq, ωnα = 1
2
∫ 1
−1Eαqe−iπnqdq
W nn1n2n3αα1α2α3
=
∫ ∞
−∞
G(x)wnα(x)wn1α1(x)wn2α2(x)wn3α3(x)dx
Nonlinear Schrodinger equation with periodic coefficients – p. 15/17
Tight-binding approximation
icn = ω0cn + ω1(cn−1 + cn+1) +W0|cn|2cn+W1
(
|cn−1|2cn−1 + σcn−1c2n + 2σ|cn|2cn−1
+2|cn|2cn+1 + cn+1c2n + σ|cn+1|2cn+1
)
+W2
(
2|cn−1|2cn + cnc2n−1 + cnc
2n+1 + 2|cn+1|2cn
)
,
with
Wj =
∫ ∞
−∞
G(x)wj1α(x)w4−j0α (x)dx j = 1, 2
Nonlinear Schrodinger equation with periodic coefficients – p. 16/17
Tight-binding approximation
t
x
0 20 40 60 80 100
−10
−5
0
5
10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
t
n
0 20 40 60 80 100
−10
−5
0
5
10 0
0.2
0.4
0.6
0.8
cn(t) = A (−i)n exp(−iω0t)Jn(2ω1t)
Nonlinear Schrodinger equation with periodic coefficients – p. 17/17