aa r X i v : . [ phy s i c s . op ti c s ] S e p Solitonization of the Anderson Localization
Claudio Conti Department of Physics, Univiversity Sapienza, Piazzale Aldo Moro 5, 00185 Rome, Italy (Dated: November 9, 2018)We study the affinities between the shape of the bright soliton of the one-dimensional nonlinearSchroedinger equation and that of the disorder induced localization in the presence of a Gaussianrandom potential. With emphasis on the focusing nonlinearity, we consider the bound states of thenonlinear Schroedinger equation with a random potential; for the state exhibiting the highest degreeof localization, we derive explicit expressions for the nonlinear eigenvalue and for the localizationlength by using perturbation theory and a variational approach following the methods of statisticalmechanics of disordered systems. We numerically investigate the linear stability and “superlocal-izations”. The profile of the disorder averaged Anderson localization is found to obey a nonlocalnonlinear Schroedinger equation.
Introduction —
Solitons [1], and disorder induced An-derson states [2] are two apparently unrelated forms ofwave localization, the former being due to nonlinearity[3], the latter to linear disorder [4]. However, on closerinspection, they look similar for various reasons: theyare exponentially localized, they correspond to appropri-ately defined negative eigenvalues, they may be located inany position in space (which is homogeneous for solitarywaves, and populated by a random potential for Ander-son states). Furthermore, various recent investigationsdeal with the theoretical, numerical and experimentalanalysis of localized states in the presence of disorderand nonlinearity [5–19], as specifically in optics [20–22],in Bose-Eistein condensation (BEC) [23–26], and morerecently in for random lasers [27]. The nonlinear Ander-son localizations are expected to have a power (number ofatoms for BEC, pump fluence for random lasers or activecavities) dependent localization length, and eigenvalue,but also exist for a vanishing nonlinearity: in the lowfluence regime, they are Anderson localizations, but athigh fluence, it is expected that they are more related tosolitons. Such a situation resembles other forms of linearlocalization, as the multidimensional “localized waves”[28], which are “dressed” by the nonlinearity [29]; a keydifference with respect to localized waves is that Ander-son states are square-integrable, another feature in com-mon with bright solitons [30].Many authors investigated the effect of nonlinearity onAnderson localization, as, e.g., [31–35], here we report ona theoretical and numerical analysis that allows to deriveexplicit formulae describing the nonlinear dressing of thefundamental Anderson states, and the way they becomesolitons as the nonlinear effects are dominant. We showthat the disorder averaged profile of the nonlinear An-derson localization is given by the very same equationproviding the soliton shape, augmented by a power anddisorder dependent term. This equation defines a partic-ular highly nonlocal nonlinear response [36], and resultsin quantitative agreement with computations. In addi-tion, we numerically demonstrate that these states arestable with respect to small perturbations, and that this stability is driven by a novel kind of localization, whichwe address as “superlocalization”, resulting from the in-terplay of solitons and Anderson states.
Outline —
We review the nonlinear Schroedinger equa-tion with a Gaussian random potential; we describe theweak perturbation theory for small nonlinearity, whenlinear Anderson states are slightly perturbed; we considerthe strong perturbation theory, i.e., the regime wherethe disordered potential is negligible and the only formof localization is the bright soliton; we compare the twolimits with numerical simulations; we use a phase-spacevariational approach to derive results valid at any orderof nonlinearity and quantitatively in agreement with thetwo mentioned limits and with numerical analysis; thestability is finally numerically demonstrated.
The model —
We consider the one-dimensionalSchroedinger equation with random potential V ( x ): iψ t = − ψ xx + V ( x ) ψ − χ | ψ | ψ ≡ N [ ψ ], (1)where χ = 1 ( χ = −
1) corresponds to the focusing (de-focusing) case. V ( x ) is Gaussianly distributed such that h V ( x ) V ( x ′ ) i = V δ ( x − x ′ ).The linear states ( χ = 0) ψ = ϕ exp( − iEt ) are given by − ϕ xx + V ( x ) ϕ = L ϕ = Eϕ , (2)and we consider the lowest energy localized states withnegative E ; these are ϕ n ( x ) with ( ϕ n , ϕ m ) = δ nm and δ nm the Kronecker delta. We denote as E the lowestnegative energy of the linear ( χ = 0) fundamental state ϕ . We stress that ϕ is the fundamental linear statewith unitary norm, and in the following we will use statessuch that P = R | ψ | dx as the solutions of the nonlinearequation (1); P measures the strength of the nonlinearityas χ = ± l = | R ϕ dx | R ϕ dx = P R ϕ dx . (3)For example, for an exponentially localized state ϕ ( e ) = √ P ϕ e with ϕ e = exp ( − | x | / ¯ l ) / p ¯ l/
2, we have l = ¯ l . l is the linear localization length of the fundamental state √ P ϕ . We recall that for the linear problem, the numberof states per unit length is known and the mean valuefor the energy can be approximated by E L ∼ = − V / / E L is the mean linear negative value of theenergy levels of a Gaussian random potential, and willbe used below as the appropriate limit for P →
0. Wealso recall that in the linear case, the energy scales likethe inverse squared localization length [4], as also foundbelow when P → The Lyapunov functional —
The nonlinear Andersonstates for a specific disorder realization V ( x ) are the so-lutions of N ( ϕ ) = − ϕ xx + V ( x ) ϕ − χϕ = Eϕ , (4)which are obtained numerically, as detailed below. Thenonlinear states correspond to the extrema of the Lya-punov functional F = Z {| ψ x | + [ V ( x ) − E ] | ψ | − χ | ψ | } dx (5)with the Hamiltonian H = F + EP .As E = ( ϕ, N [ ϕ ]) / ( ϕ, ϕ ) = ( ϕ, N [ ϕ ]) /P , one has, forthe solutions of (4), F = χ ( ϕ , ϕ ) /
2, that is F = χ Z ϕ dx = χ P l , or 1 l = 2 FχP , (6)which show that a connection between the Lyapunovfunctional F and the localization length exists. Weak perturbation theory —
For small P , standard per-turbation theory [15] on √ P ϕ ( x ) gives E = E − χ Pl + O ( P ), (7)where l is the linear localization length in Eq.(3). For χ = 1, E < χ = − ϕ (1) the standard first order cor-rection to the linear state ϕ , we find at order O ( P ) forthe localization length: l = ( ϕ,ϕ ) ( ϕ ,ϕ ) ∼ = ( ϕ ,ϕ ) +4 P ( ϕ ,ϕ (1) ) = l (cid:20) − P ( ϕ ,ϕ (1) )( ϕ ,ϕ ) (cid:21) == l − χP l P n> ( ϕ n ,ϕ ) E n − E = l (cid:16) − χ PP (cid:17) (8)Eq.(8) predicts that l increases (decreases) with P in the defocusing (focusing) case; P = (cid:2) l P n> ( ϕ n , ϕ ) / ( E n − E )) (cid:3) − gives the powerlevel such that, when χ = 1, l vanishes, and this isdefined as the critical power for the transition to a soli-tonic regime, where the weak expansion is expected notto be valid. P depends on the linear eigenstates of thepotential and comes from the lowest order perturbationexpansion of the localization length. Summarizing, the weak expansion allows to affirm thattwo critical powers can be defined: in the defocusingcase, there is a power P = | E | l at which the eigen-value changes sign, corresponding to a nonlinearity thatdestroys the Anderson states; in the focusing case there isa power P = P at which the localization length vanishes,this is the nonlinearity level needed to the bound statefor resembling a bright soliton (i.e., for the “solitoniza-tion” of the Anderson state). In the weak expansion thesecritical powers are dependent on the specific disorder re-alization and have a statistical distribution. In a latersection, we report a variational approach that allows toderive closed expressions for the peak of these distribu-tions P C , which depends only on V ; we will limit to thefocusing case, as the defocusing one requires a separatedtreatment. Strong perturbation theory —
For large P , we write thesolution by a multiple scale expansion as ϕ = P η ( P x )and Eq.(4), at the highest order in P ( x P ≡ P x , E =( ϕ, N [ ϕ ]) /P ≡ P E P ), reduces to − d ηdx P − χη = E P η , (9)where E P is the eigenvalue scaled by P . For large P , thenonlinear Anderson states are asymptotically describedby the solitary-wave solutions in a manner substantiallyindependent of V ( x ). For χ = 1, Eq.(9) is satisfied bythe fundamental bright soliton [3] and, correspondingly,we have ϕ = √− E/ cosh( √− Ex ).We stress that, in this expansion, E < E = E S = − P /
16 and l = l S = 12 /P , the subscript referring to the soliton trends. E S is the “nonlinear eigenvalue” for the soliton,which is determined by P , while l S is the correspondinglocalization length in this strong perturbation expansion,i.e., when neglecting the linear potential V ( x ). Note alsothat this trend at high power is also expected for thehigher order states ϕ n . Nonlinear dressing —
We detail the transition from thelinear Anderson states to the solitary wave. We limitto the focusing case χ = 1 hereafter (the case χ = − x → x/x , V → V x and ψ → ψ/x , such that we canlimit the size of system to x ∈ [ − π, π ], over which period-ical boundary conditions are enforced. The prolongationof the linear states to the nonlinear case is not trivial.We start from a linear localized state ( χ = 0) and weprolong to χ > χ , we rescale ϕ , by using the scaling propertiesof Eq.(1), so that it corresponds to χ = 1, and we cal-culate P , H , l and E . In figure 1a we show the shapeof the fundamental solution (lowest negative eigenvalue)for increasing power P . The inset shows the projectionof the numerically retrieved nonlinear localization withthe fundamental soliton sech profile: as P increases the −2 0 20.20.40.60.81 position x non li nea r A nde r s on s t a t e
10 20 30−80−60−40−200 non li nea r e i gen v a l ue E power P0 10 201 power Psoliton fraction P=0.01P=27.4 weakstrongannealednumerical(a) (b)
FIG. 1: (Color online) (a) Plot of the nonlinear Andersonstates | ϕ | /max ( | ϕ | ) for different P ( V = 4), two values ofthe powers are indicated corresponding to the blue and dashedlines; the inset shows the projection on the soliton profile; (b)nonlinear eigenvalue E versus P ( V = 4) for several disorderrealizations (cyan thin lines), compared with the strong (redthick line) and weak (green thick line, only shown for a singlerealization) expansions, and with the result from the annealedphase-space variational approach (black thick line). l o c a l i z a t i o n l e n g t h l power P0 10 20 30012345 0 10 20 30102030 P c o u n t s (a) (b)annealednumerical bright soliton(strong)linear (weak) P P C FIG. 2: (Color online) (a) Localization length versus P ( V = 1): the results after various disorder realizations areshown (cyan thin lines) and compared with the weak (greenline, only shown for a single realization with the correspond-ing P indicated), with the strong expansion (red thick line),and with annealed phase-space approach (black thick line);(b) distribution of the critical power P (200 disorder realiza-tions), the red vertical line is the analytical result for P C , asgiven in the text ( V = 2). shape of the disorder induced localization is progressivelysimilar to the soliton. The trend of the eigenvalue E isshown in Fig.1b, compared with the weak (low P ) andstrong expansions (high P ), and with Eq.(17) below: forlow P we have a linear trend, Eq.(7), while the trendfollows the solitonic one E S = − P /
16 for high P .In Fig.2a we show the calculated localization length com-pared with the strong perturbation theory and, for a sin-gle realization, with the weak perturbation theory, with P given by the intercept with the horizontal axis. When P increases, the localization length deviates from the lin-ear trend, and follows the bright soliton l S = 12 /P athigh P for all the considered realizations. The phase-space/variational approach —
In the weak ex-pansion Eq.(8), valid as P →
0, the power P for the transition to the soliton depends on the realization ofthe disorder and has a statistical distribution shown inFig.2b. The histogram of P in Fig.2b is found not tosubstantially change for a number of realizations greaterthan 100.On the other hand, the strong expansion Eq.(9), validas P → ∞ , completely neglects the random potential,and the result is independent of the strength of disorder.Here we introduce an approach based on the statisticalmechanics of disordered systems [38] valid at any orderin P . The first step is to define an appropriate measurebased on the fact that the nonlinear Anderson state (4)maximizes a weight in the space of all the possible ψ .Following the fact the nonlinear bound states extremizethe Lyapunov functional F , and that, for these states, F scales like 1 /l after Eq.(6), we consider a Boltzmann likeweight: T l = exp( − L/l ) with L determined in the follow-ing. Note that exp( − L/l ) is the transmission of a slab ofdisordered material with length L and localization length l [4]. For a specific disorder realization, we introduce themeasure ρ [ ψ ] = 1 Z exp (cid:18) − Ll (cid:19) (10)with Z the “partition function” Z = Z exp (cid:18) − Ll (cid:19) d [ ψ ], (11)such that R ρ [ ψ ] d [ ψ ] = 1. The inverse localization lengthis calculated as an average over the whole functionalspace of ψ [after Eq.(6)]: h l i ≡ Z l ρ [ ψ ] d [ ψ ] = 1 Z Z l exp (cid:18) − LFP (cid:19) d [ ψ ]. (12)In (12), the solution of (4) is that providing the highestcontribution to the weighted average among all the ψ .Our aim is to find an equation for such a state after av-eraging over the disorder V ( x ); this averaging is denotedby an over-line: h l i = − ∂ L log ( Z ) ∼ = − ∂ L logZ . (13)In (13) we used the so-called annealed average log( Z ) ∼ =log Z ,[39] whose validity is to be confirmed a posteriori .We find Z = R exp (cid:0) − LF eff /P (cid:1) d [ ψ ], being F eff ≡ Z (cid:20) | ψ x | −
12 (1 + 2 LV P ) | ψ | − E | ψ | (cid:21) dx . (14)This effective Lyapunov functional F eff is extremized by − ψ xx − (cid:18) LV P (cid:19) | ψ | ψ = Eψ (15)with the constraint P = R | ψ | dx , which gives E as afunction P . Eq.(15) generalizes the strong perturbationlimit Eq.(9), retrieved for V = 0 or P → ∞ , to a finitepotential V and P .Eq.(15) shows that the role of the disorder is to alterthe nonlinear response, namely to increase the strengthof the nonlinear coefficient, such that solitary waves areobtained at smaller power than in the ordered case. Con-versely, the linear localizations can be seen as the nonlin-ear Anderson states in the limit of vanishing power, thatis a form of solitons only due to disorder. In Eq.(15), P explicitly appears; this is due to the fact that the averageover disorder introduces nonlocality [36] in the model. Inthe defocusing case a result similar to (15) is found, witha nonlinear coefficient changing sign at high power, de-noting the absence of localization for large P , as we willreport in future work.In the focusing case, by using the fundamental sech soli-ton of Eq(15), we find the corresponding eigenvalue, de-noted as E C : E C = − P (cid:18) LV P (cid:19) (16)with the localization length l C = 3 / √− E C . E C is the P and V dependent eigenvalue for a generic L , and l C isthe corresponding localization length; note that accord-ing to this analysis a measurement of l C directly provides E C .In the next step we determine L by imposing the cor-rect asymptotic value in the linear limit: as P → E C → E L ∼ = − V / /
3, which furnishes L = 2 V − / P/ √
3; conversely, in the large P limit onerecovers the expected expression E C → E S = − P / E C = − P (cid:18) P C P (cid:19) , (17)with the only parameter P C = 4 V / / √
3. Correspond-ingly, the localization length is l C = 12 /P (1 + P C /P ) , (18)which also gives l S = 12 /P for large P , and the weaklimit Eq.(8) with l = 12 /P C and P = P C . P C is the critical power for the transition from the Anderson local-izations to the solitons and is determined by the strengthof disorder V . In the linear limit P →
0, Eqs.(18) and(17) reproduces the known power-independent link be-tween the localization length and energy E = E C = − /l [4]. We stress that l c is the localization lengthof the state that mostly contribute to Z .In Figures 1 and 2, we compare this theoretical approachwith the numerical simulations at any value of P ; resultsfor various V are used to show that quantitative agree-ment is found in all of the considered cases. position x e v o l u t i o n t −2 0 20123 −2 0 2 position x−2 0 2−2 0 2−2 0 2013 ϕ −2 0 2 −2 0 20510 −2 0 2 0 10 2020406080 power P Ω (a) (b) (c)(d) (e) FIG. 3: (Color online) Stability of the nonlinear Andersonstates: (a) dashed, nonlinear state at power P = 7; full lines,two different superlocalizations arbitrarily vertically shifted(blu Ω = 11,green Ω = 14); (b) as in (a) for P = 23 (blu Ω =59, green Ω = 61); (c) eigenvalues versus power P ( V = 10);(d) evolution of the nonlinear states with P = 3 .
54 with 10%amplitude noise; (e) as in (d) with P = 13 .
2; the white linecorresponds to the arbitrarily scaled initial profile ϕ ( V = 2). Stability and superlocalization —
We consider the stabil-ity of the nonlinear localization: we calculate the eigen-values of the linearized problem following the Vakhitovand Kolokolov formulation[30, 40, 41]. We write ψ =( ϕ + δψ ) exp( − iEt ), with δψ = [ u ( x ) + iv ( x )] exp(Ω t ),where ϕ is a solution of Eq.(4) and u and v are real val-ued. Eq.(4) is linearized as − Ω u = L L u , (19)with the operators L = − ∂ x − E + V ( x ) − ϕ ( x ) and L = − ∂ x − E + V ( x ) − ϕ ( x ) . The stable (unstable)eigenvalues correspond to Ω > < V ( x ). We numerically solve Eq.(19) andfind that no unstable states are present for the consid-ered disorder realizations and values of V , demonstrat-ing that the nonlinear Anderson states are indeed stablewith respect to linear perturbations. The interesting is-sue is that in regions far from the nonlinear bound states(where ϕ ∼ = 0), Eq.(19) still admits non trivial solutions,corresponding to L L ∼ = L = [ − ∂ x + V ( x ) − E ] , suchthat the linear Anderson states correspond to Ω ∼ = 0. As P increases, the location of these states drifts towards thecenter of the nonlinear localization and this coupling re-sults into a power dependent Ω (examples are given inFig.3a,b,c). These can be taken as “superlocalizations”due to interplay between disorder and solitons. The sta-bility of the nonlinear Anderson states is also verified bytheir numerically calculated t -evolutions in the presenceof a perturbation, as shown in Fig.3d,e. Conclusions —
We reported on a theoretical approachon nonlinear Anderson localization demonstrating thestrong connection between solitons and disorder inducedlocalization. By a variational formulation we derivedclosed formulae for the fundamental state providing thetrend of the nonlinear eigenvalue and the localizationlength at any power level in quantitative agreementwith numerical simulations. Disorder averaged nonlin-ear Anderson localization is found to obey a nonlocalSchroedinger equation with a disorder dependent nonlin-earity. Such an equation, in the linear limit, reproducesthe linear Anderson states. The reported approach canbe extended to the multidimensional case and to othernonlinearities.The research leading to these results has received fundingfrom the European Research Council under the EuropeanCommunity’s Seventh Framework Program (FP7/2007-2013)/ERC grant agreement n.201766, project
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