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PHYSICAL REVIEW A 86, 061801(R) (2012)

Solitonization of the Anderson localization

Claudio ContiDepartment of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy

(Received 16 March 2012; published 12 December 2012)

We study the affinities between the shape of the bright soliton of the one-dimensional nonlinear Schroedingerequation and the shape of the disorder-induced localization with a Gaussian random potential. For the mostlocalized state, we derive expressions for the nonlinear eigenvalue and for the localization length. We investigatethe stability and superlocalizations.

DOI: 10.1103/PhysRevA.86.061801 PACS number(s): 42.65.Tg, 03.75.Lm, 05.45.Yv, 71.23.An

Introduction. Solitons [1] and disorder-induced Andersonstates [2] are two apparently unrelated forms of wave lo-calization, the former being due to nonlinearity [3] and thelatter to linear disorder [4]. However, on closer inspection,they have some similarities: They are exponentially localized,they correspond to appropriately defined negative eigenvalues,and they may be located in any position in space. Variousrecent investigations reported on the theoretical, numerical,and experimental analyses of localized states in the presenceof disorder and nonlinearity [5–17], specifically in optics[18–20], in Bose-Eistein condensation (BEC) [21–24], andmore recently for random lasers [25]. In the presence ofnonlinearity, disorder-induced localizations are expected tohave eigenvalue and localization length dependent on power(or number of atoms for BEC and pump energy for lasers).These states, however, also exist for a negligible nonlinearity:Hence, in the low fluence regime, they are linear Andersonlocalizations, but at high fluence, it is expected that theybecome related to solitons.

This resembles other forms of linear localization, as themultidimensional “localized waves” [26], which are “dressed”by nonlinearity [27]. At variance with these localized waves,the Anderson states are square integrable, another feature theyhave in common with bright solitons.

Many authors investigated the effect of nonlinearity onAnderson localization, including, for example, Refs. [28–32];here we report on a theoretical and numerical analyses thatallow us to derive explicit formulas describing the nonlineardressing of the ground Anderson state and the way it progres-sively becomes similar to a soliton as the strength of nonlineareffects increases. We show that the disorder-averaged profileof the nonlinear Anderson localization is given by the sameequation providing the soliton shape augmented by a power-and disorder-dependent term. This equation includes a highlynonlocal nonlinear response [33] and results in quantitativeagreement with computations. In addition, we numericallydemonstrate that these states are stable with respect to smallperturbations and that this stability is driven by a novel kind oflocalization (“superlocalization”), resulting from the interplayof solitons and Anderson states.

The model. We consider the one-dimensional Schroedingerequation with random potential V (x):

iψt = −ψxx + V (x)ψ − χ |ψ |2ψ ≡ N [ψ], (1)

where χ = 1 (χ = −1) corresponds to the focusing (de-focusing) case. V (x) has a Gaussian distribution with〈V (x)V (x ′)〉 = V 2

0 δ(x − x ′).

For a given realization of the disorder V (x), the linear states(χ = 0) ψ = ϕn exp(−iEnt) are given by

−ϕn,xx + V (x)ϕn = Lϕn = Enϕ, (2)

with (ϕn,ϕm) = δnm, where δnm is the Kronecker δ, andeigenvalues En. En < 0 for localized states, and E0 is thelowest negative energy.

The localization length is calculated by the inverse partici-pation ratio:

l = | ∫ ϕ2dx|2∫ϕ4dx

= P 2∫ϕ4dx

. (3)

For example, for an exponentially localized state ϕ(e) = √Pϕe

with ϕe = exp (−2|x|/l)/√

l/2, we have l = l. l0 is the linearlocalization length of

√Pϕ0. For the linear problem, the

number of states per unit length is known and the meannegative value of E can be approximated by EL

∼= −V4/3

0 /3;furthermore, the energy scales like the inverse squared local-ization length [4].

The Lyapunov functional. The nonlinear Anderson statesfor a specific disorder realization V (x) are the solutions of

N (ϕ) = −ϕxx + V (x)ϕ − χϕ3 = Eϕ, (4)

which are obtained numerically below; these correspond to theextrema of the Lyapunov functional

F =∫ {

|ψx |2 + [V (x) − E]|ψ |2 − χ

2|ψ |4

}dx (5)

with Hamiltonian H = F + EP .As E = (ϕ,N [ϕ]) /(ϕ,ϕ) = (ϕ,N [ϕ]) /P , for Eq. (4), F =

χ (ϕ2,ϕ2)/2, that is

F = χ

2

∫ϕ4dx = χ

2

P 2

l, or

1

l= 2F

χP 2, (6)

revealing that a connection between F and l exists.Weak perturbation theory. For small P , standard perturba-

tion theory [13] on√

Pϕ0(x) gives

E = E0 − χP

l0+ O(P 2), (7)

where l0 is the linear localization length in Eq. (3). Forχ = 1, E < 0 decreases as P increases; for χ = −1 E < 0increases as P increases and eventually changes sign. With

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CLAUDIO CONTI PHYSICAL REVIEW A 86, 061801(R) (2012)

ϕ = √P [ϕ0 + Pϕ(1) + O(P 2)], we find, at order O(P ),

l = (ϕ,ϕ)2

(ϕ2,ϕ2)∼= 1(

ϕ20 ,ϕ

20

) + 4P(ϕ3

0 ,ϕ(1)

)= l0

[1 − 4P

(ϕ3

0 ,ϕ(1)

)(ϕ2

0 ,ϕ20

)]

= l0 − 4χP l20

∑n>0

(ϕn,ϕ

30

)2

En − E0= l0

(1 − χ

P

P0

). (8)

Equation (8) predicts that l increases (decreases) with P in thedefocusing (focusing) case; P0 = [4l0

∑n>0(ϕn,ϕ

30)2/(En −

E0)]−1 gives the power level such that, when χ = 1, l vanishes.P0 is the critical power for the transition to a solitonic regime,where the weak expansion is expected not to be valid. P0

depends on the linear eigenstates of a specific realization ofV (x).

Two critical powers can be defined: (i) in the defocusingcase, a power P = |E0|l0 at which E changes sign, corre-sponding to a nonlinearity destroying Anderson states; and(ii) in the focusing case, a power P = P0 at which l vanishes,when the bound state resembles a bright soliton (i.e., the powerfor the “solitonization” of the Anderson state). These criticalpowers are dependent on the disorder realization and have astatistical distribution. We report below a variational approachfor the peak of these distributions PC , only depending on V0;we limit this discussion to the focusing case, as the defocusingcase requires a separated treatment.

Strong perturbation theory. For large P , we write ϕ =Pη(Px), and Eq. (4), at the highest order in P (xP ≡ Px,E = (ϕ,N [ϕ]) /P ≡ P 2EP ), reduces to

− d2η

dx2P

− χη3 = EP η, (9)

where EP is the eigenvalue scaled by P 2. For large P , thenonlinear Anderson states are described by the solitary-wavesolutions independently of V (x). For χ = 1, Eq. (9) is satisfiedby the bright soliton ϕ = √−2E/ cosh(

√−Ex), with E < 0as for the linear Anderson states, with E = ES = −P 2/16 andl = lS = 12/P . The quadratic scaling of E with respect to P ,at high power, is also expected for ϕn>0.

Nonlinear dressing. We limit this discussion to the focusingcase χ = 1 hereafter (the case χ = −1 will be reportedelsewhere). We numerically solve Eq. (4), which is invariantwith respect to the scaling x → x/x0, V → V x2

0 , and ψ →ψ/x2

0 , such that we can limit the size to x ∈ [−π,π ] withperiodical boundary conditions.

The prolongation of the linear states to the nonlinear caseis not trivial. We start from a linear localized state (χ = 0) andwe prolong to χ > 0 by a pseudospectral Newton-Raphsonalgorithm using χ as prolongation parameter; this furnishessolutions with χ = ±1 at variance with the scaling in Eq. (4).For each χ , we rescale ϕ, using the scaling properties of Eq. (1),to map the solution to the case χ = ±1, and we calculate theresulting P , H , l, and E. In Fig. 1(a) we show the shapeof the ground-state solution (lowest negative eigenvalue) forincreasing P . The inset shows the projection of the numericallyretrieved nonlinear localization with the fundamental solitonsech profile: As P increases the disorder-induced localization

−2 0 2

0.2

0.4

0.6

0.8

1

position x

nonl

inea

r A

nder

son

stat

e

10 20 30−80

−60

−40

−20

0

nonl

inea

r ei

genv

alue

E

power P

0 10 20

1

power P

soliton fraction

P=0.01

P=27.4

weak

strong

annealednumerical

(a) (b)

FIG. 1. (Color online) (a) For a realization of disorder, thenonlinear Anderson states |ϕ|/max(|ϕ|) for different P (V0 = 4),two values of P are indicated (blue [gray] and black dashed lines);the inset shows the projection on the soliton profile. (b) E vs P

(V0 = 4) for several realizations (cyan [light gray] thin lines), withthe strong (red [gray] thick line) and weak (green [gray] thick line, fora single realization) expansions, and with the annealed phase-spacevariational approach (black thick line).

is progressively more similar to the soliton. The projectionis calculated as

∫ϕ(x)/ cosh(

√Px/4)dx/

√8P , so that if ϕ

is equal to the soliton when V = 0, the projection is unitary.E(P ) is shown in Fig. 1(b), compared with the weak andstrong expansions, and with Eq. (18) below: At low P , E(P )is a linear function, while E(P ) ∼= ES at high P .

In Fig. 2(a) we show the calculated localization lengthcompared with the strong perturbation theory and, for a singlerealization, with the weak perturbation theory, with P0 givenby the intercept with the horizontal axis. When P increases,the localization length deviates from a straight line and followslS = 12/P at high P for all the considered realizations.

The phase-space variational approach. Here we introducean approach based on the statistical mechanics of disorderedsystems [34] valid at any order in P . We first define a phase-space measure based on the fact that the nonlinear bound stateis an extremum F ; as F scales like 1/l after Eq. (6), weconsider a Boltzmann-like weight: Tl = exp(−L/l) with L

determined in the following. This is equivalent to introducinga weight like exp(−βF ) with β as the scaling parameter to

loca

lizat

ion

leng

th l

power P0 10 20 30

0

1

2

3

4

5

0 10 20 30

10

20

30

P0

coun

ts

(a) (b)annealed

numerical

bright soliton(strong)

linear (weak)

P0PC

FIG. 2. (Color online) (a) l vs P (V0 = 1): Various disorderrealizations are shown (cyan [gray] thin lines) and compared with theweak (green [gray] line, for a single realization with P0 indicated),with the strong expansion (red [gray] thick line), and with thevariational approach (black thick line). (b) Distribution of P0 (200realizations); the red (light gray) vertical line is the analytical PC .See text (V0 = 2).

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SOLITONIZATION OF THE ANDERSON LOCALIZATION PHYSICAL REVIEW A 86, 061801(R) (2012)

be identified. Note that exp(−L/l) is the transmission of adisordered slab with length L and localization length l [4].

For a specific disorder realization, we define

ρ[ψ] = 1

Zexp

(−L

l

)(10)

with Z being the “partition function”

Z =∫

exp

(−L

l

)d[ψ], (11)

such that∫

ρ[ψ]d[ψ] = 1. The inverse localization length iscalculated as an average over the whole functional space of ψ

[after Eq. (6)]:⟨1

l

⟩≡

∫1

lρ[ψ]d[ψ] = 1

Z

∫1

lexp

(−2LF

P 2

)d[ψ].

(12)

In relation (12), the solution of Eq. (4) is that providing thehighest contribution to the weighted average among all the ψ .Our aim is to find an equation for such a state after averagingover the disorder V (x):⟨

1

l

⟩= −∂Lln(Z) ∼= −∂LlnZ. (13)

In Eq. (13) we used the so-called annealed average ln(Z) ∼=ln Z [35], whose validity is to be confirmed a posteriori.Letting D[V ] = d[V ] exp[− ∫

dx V (x)2

2V 20

],

Z =∫

D[V ]d[ψ]e− 2LF

P 2 =∫

d[ψ]e− 2LFeff

P 2 , (14)

with

Feff ≡∫ [

|ψx |2 − 1

2

(1 + 2LV 2

0

P 2

)|ψ |4 − E|ψ |2

]dx.

(15)

When making the variation of Feff , we note that E appears asthe Langrange multiplier of

∫ |ψ |2dx and E is determined bythe condition

∫ |ψ |2dx = P . This gives

−ψxx −(

1 + 2LV 20

P 2

)|ψ |2ψ = Eψ (16)

with the constraint P = ∫ |ψ |2dx, giving E as a function P .Equation (16) generalizes the strong perturbation limit Eq. (9),retrieved for V0 = 0 or P → ∞.

Equation (16) shows that the role of the disorder is toincrease the strength of the nonlinear coefficient, such thatsolitary waves are obtained at smaller P than in the orderedcase (V0 = 0). The linear localizations can be seen as thenonlinear Anderson states in the limit of vanishing power, thatis, a form of solitons only due to disorder, corresponding to the

case 2LV 20

P 2 � 1. In Eq. (16), P explicitly appears as the averageover the disorder-introduced nonlocality [33] in the model. Inthe defocusing case a result similar to that of Eq. (16) is found,with a nonlinear coefficient changing sign at high P , denotingthe absence of localization for large P , as we will report infuture work.

In the focusing case, by using the fundamental sech solitonof Eq. (16), we find the eigenvalue, denoted as EC :

EC = −P 2

16

(1 + 2LV 2

0

P 2

)2

(17)

with the localization length lC = 3/√−EC . Measuring lC

directly provides EC .We determine L by imposing the correct asymptotic value in

the linear limit: As P → 0, it must be EC → EL∼= −V

4/30 /3,

which furnishes L = 2V−4/3

0 P/√

3; conversely, in the largeP limit one recovers the expected expression EC → ES =−P 2/16.

Summarizing, we find for the nonlinear eigenvalue

EC = −P 2

16

(1 + PC

P

)2

, (18)

with the only parameter PC = 4V2/3

0 /√

3. Correspondingly,the localization length is

lC = 12/P

(1 + PC/P ), (19)

which also gives lS = 12/P for large P , and the weak limitEq. (8) with l0 = 12/PC and P0 = PC . PC is the criticalpower for the transition from the Anderson localizations tothe solitons and is determined by the strength of disorder V0.For P → 0, Eqs. (19) and (18) gives the known link betweenl0 and energy E = −9/l2

0 [4]In Figs. 1 and 2, we compare EC and lC with the numerical

simulations; quantitative agreement is found in all of theconsidered cases at any P .

Stability and superlocalization. We consider thestability of the nonlinear localization: We calculatethe eigenvalues of the linearized problem followingRefs. [36–38]. We write ψ = (ϕ + δψ) exp(−iEt), with δψ =

position x

evol

utio

n t

−2 0 2

0

1

2

3−2 0 2

position x−2 0 2−2 0 2

−2 0 20

1

3

ϕ

−2 0 2 −2 0 20

5

10

−2 0 2 0 10 20

20406080

power P

Ω

(a) (b) (c)

(d) (e)

FIG. 3. (Color online) Stability of the nonlinear Anderson states:(a) dashed, nonlinear state at power P = 7; full lines, two differentsuperlocalizations arbitrarily vertically shifted (blue [gray] � = 11,green [gray] � = 14); (b) as in panel (a) for P = 23 (blue [gray]� = 59, green [gray] � = 61); (c) eigenvalues ves power P (V0 =10); (d) evolution of the nonlinear states with P = 3.54 with 10%amplitude noise; and (e) as in panel (d) with P = 13.2; the white linecorresponds to the arbitrarily scaled initial profile ϕ (V0 = 2).

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CLAUDIO CONTI PHYSICAL REVIEW A 86, 061801(R) (2012)

[u(x) + iv(x)] exp(�t), where ϕ is a solution of Eq. (4) and u

and v are real valued. Equation (4) is linearized as

−�2u = �2u = L1L0u, (20)

with the operators L0 = −∂2x − E + V (x) − ϕ(x)2 and L1 =

−∂2x − E + V (x) − 3ϕ(x)2. The positive (negative) eigenval-

ues of L1L0 correspond to stable (unstable) modes of thenonlinear Anderson states with �2 > 0 and � imaginary(�2 < 0 and � real). As it happens for the standard solitons,the bound-state profile ϕ is also an eigenvalue of Eq. (20),with � = 0 due to the gauge invariance; conversely, otherneutral modes due to translational invariance are lost becauseof the symmetry-breaking potential V (x). We numericallysolve Eq. (20) and find that no unstable states are present for theconsidered disorder realizations and values of V0; the nonlinearAnderson states are stable with respect to linear perturbations.In regions far from the nonlinear bound states (where ϕ ∼=0), Eq. (20) still admits nontrivial solutions, correspondingto L1L0

∼= L2 = [−∂2x + V (x) − E]2; linear Anderson states

correspond to �2 ∼= 0. As P increases, the location of thesestates drifts towards the center of the nonlinear localization

with a-power dependent �2 [examples are given in Figs. 3(a),3(b), and 3(c)]. These can be taken as “superlocalizations”due to interplay between disorder and solitons. The stability isalso verified by the t evolutions with perturbation, as shown inFigs. 3(d) and 3(e).

Conclusions. We reported on a theoretical approach onnonlinear Anderson localization, demonstrating the strongconnection between solitons and disorder-induced localiza-tion. By a variational formulation, we derived closed formulasfor the nonlinear eigenvalue and the localization length for theground state, at any power level, in agreement with numericalsimulations. Disorder-averaged nonlinear Anderson localiza-tion is found to obey a nonlocal Schroedinger equation. Thereported approach can be extended to the multidimensionalcase and to other nonlinearities.

Acknowledgments. The research leading to these results hasreceived funding from the European Research Council underthe European Community’s Seventh Framework Program(FP7/2007-2013)/ERC Grant Agreement No. 201766, ProjectLight and Complexity (COMPLEXLIGHT). We acknowledgesupport from the Humboldt Foundation.

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