Nonlinear Schrodinger equation with chaotic, random, and nonperiodic nonlinearity
W. B. Cardoso, S. A. Leao, A. T. Avelar, D. Bazeia, M. S. Hussein
aa r X i v : . [ qu a n t - ph ] D ec Nonlinear Schr¨odinger equation with chaotic, random, and nonperiodic nonlinearity
W. B. Cardoso, S. A. Le˜ao, A. T. Avelar, D. Bazeia, and M. S. Hussein Instituto de F´ısica, Universidade Federal de Goi´as, 74001-970, Goiˆania, GO, Brazil. Departamento de F´ısica, Universidade Federal da Para´ıba, 58051-970, Jo˜ao Pessoa, PB, Brazil. Departamento de F´ısica Matem´atica, Instituto de F´ısica,Universidade de S˜ao Paulo, 05314-970, S˜ao Paulo, SP, Brazil
In this paper we deal with a nonlinear Schr¨odinger equation with chaotic, random, and nonperiodic cubicnonlinearity. Our goal is to study the soliton evolution, with the strength of the nonlinearity perturbed in thespace and time coordinates and to check its robustness under these conditions. Comparing with a real system,the perturbation can be related to, e.g., impurities in crystalline structures, or coupling to a thermal reservoirwhich, on the average, enhances the nonlinearity. We also discuss the relevance of such random perturbationsto the dynamics of Bose-Eisntein Condensates and their collective excitations and transport.
PACS numbers: 42.65.Tg; 42.25.Dd; 05.45.Pq
The nonlinear Schr¨odinger equation (NLSE) is the math-ematical vehicle that describes the evolution of solitonic so-lutions for different nonlinear systems, such as, fiber optics[1], bulk medium and photonic crystals [2], Langmuir wavesin plasmas [3], wave function of Bose-Einstein condensates(BECs) [4], and others.A special case involving the NLSE consists in variable co-efficients modulated in the spatial and/or temporal coordi-nates. The control of these coefficients allow us to obtain newdistinct solutions. In this context, [5, 6] have recently pro-posed a treatment of BECs using similarity transformations toconstruct explicit nontrivial solutions of the cubic and cubic-quintic NLSE with potentials and nonlinearities dependingboth on time and on the spatial coordinates. Also, thermaleffects on nonlinearities can change its form, presenting newsolutions [7]. In BECs, the s -wave scattering length of inter-atomic collisions determines the strength of the nonlinearitycoefficient [4], and it can be controlled using the Feshbachresonance (FR) [8] via external magnetic [9, 10] or optical[11] fields. The FR mechanism allows for a practical meansto manipulate the nonlinearity [12].Although the control of the nonlinearity has been very ef-fective, noises can appear in the system management or addedto it. In this way, the perturbations can change the nonlinear-ity, and thus influencing possible changes in the noise-free so-lutions. The inclusion of spatial random potential in the studyof BEC dynamics has been proposed in Refs. [13, 14, 15, 16]and recently was tested experimentally [17]. When random-ness is introduced in the BEC dynamics, through an opticalspeckle, one may be able to study Anderson localization inthe context of BEC and superfluidity [13]. It was clearlydemonstrated in these papers that in the presence of disorderthe condensate’s expansion in 1D waveguides is inhibited, andthe collective dipole and quadrupole oscillations are stronglydamped. In a way, this is similar to the damping of collectivestates in nuclei and metal clusters [18], where the randomnessis internal.A natural question arises as to what would be the conse-quence of having the disorder present directly in the nonlin-earity term? For BEC, this implies a point two-body inter- action (t-matrix) with a random component. Would this addor remove some of the effects of the speckle potential? Inthis connection, [19] have recently proposed an NLSE in thepresence of random nonlinearity. Specifically, these authorsconsider the effects of random time modulation of the non-linearity coefficient on the dynamics of solitary waves in theNLSE. On the other hand, to our knowledge, chaotic perturba-tions in the nonlinearity term in the NLSE have not been fullyconsidered yet, though it is a common knowledge that severalphysical systems do exhibit chaotic behavior.Classically chaotic systems appear to behave as randomsystems. Tiny differences in the initial state of the system canlead to enormous differences in the final state even over fairlysmall time scales [20]. This happens even though these sys-tems are deterministic, meaning that their future dynamics arefully determined by their initial conditions with no randomelements involved. This behavior is known as deterministicchaos.The dynamical systems theory (DST) is an area whose in-terest lies mainly in nonlinear phenomena, the source of clas-sical chaos. DST groups use several concepts to the study ofchaos, such as Lyapunov exponents, fractal dimension, bifur-cation, and symbolic dynamics among other elements [20].Recently, other approaches have been considered, such as in-formation dynamics and entropic chaos degree [21]. For ex-ample, given C , C n can be the n -th iterate of the quadraticfunctions: C n ( µ ) = C n − + µ ; sine functions: C n ( µ ) = µ sin( C n − ; logistic functions C n ( µ ) = µC n − (1 − C n − ) ;exponential functions: C n ( µ ) = µ exp( C n − ) ; doublingfunction defined on the interval [0 , : C n = 2 C n − mod ,and so on, µ being a parameter. It is worth recalling that all thefunctions in the above list are familiar to researchers in DST.For example, for some values of µ , it is known that some ofthese functions can behave in quite a chaotic manner [20]. Inwhat follows we make a distinction between chaos and ran-domness, though both concepts indicate disorder.The major thrust of our paper is to verify the influence of thedifferent types of perturbations in the nonlinearity of a systemgoverned by NLSE. In this sense, we investigate the chaotic,random, and nonperiodic nonlinearity perturbation. We knowthat these perturbations are different, however, to what extentthe overall effect is universal (independent on the details ofthe random perturbations), and how can it modify a solitonicsolution? We purports to supply some answers to the above.Differently from Ref. [19], which considers a random timemodulation on a certain point (generating a Gaussian distri-bution), here we consider a constant background nonlinearityperturbed by a random function that interferes in both spatialand temporal coordinates. Surprisingly, some of the solutionsfound here can move depending of the amount of perturbedpoints in the nonlinearity. This fact is similar to those studiedby the thermal effects on the nonlinearity [7].Firstly we consider the NLSE given by iψ t = − ψ xx + g ( x, t ) | ψ | ψ, (1)where ψ = ψ ( x, t ) , x , and t dimensionless, and g ( x, t ) is thefunction that describe the nonlinearity of the system. Here weconsider g ( x, t ) = G (1 + σ ( x, t )) , (2)where G and Gσ ( x, t ) are the nonlinear parameter and the co-efficient generated by a chaotic, random, or nonperiodic gen-erator, respectively. Eq. (1) describes, e.g., the density of par-ticles in a Bose-Einstein condensates when it is free of exter-nal potentials, in a configuration type cigar-shaped; the spatialpulse propagation in bulk crystals that present Kerr-effect inthe nonlinearity or temporal pulse propagation in nonlinearoptical fiber; etc.We investigate the evolution of the solution for the NLSEvia numerical simulations, based on the split-step finite dif-ferences (SSFD) method with a time-step and space-step sizeof ∆ t = 0 . and ∆ x = 0 . , respectively; we use N torepresent the number of points in space. The core of SSFD isbased on the Crank-Nicolson algorithm [22]. To control thesenumerical simulations we looked for the conserved quantitygiven by (power) P = N X x =1 | ψ ( x, t ) | . (3)To calculate the error we use the comparative form E r = 1 N N X x =1 (cid:16) | ψ ( x, | − | ψ ( x, t f ) | (cid:17) , (4)where t f is the final time of the evolution. The equation aboverepresents a mean distance between the input and output state.When we take G = − and σ ( x, t ) = 0 in (2), we can write ψ = e iµt sech( x ) (5)as solution of (1), with µ = 1 . This solution will be taken asinitial condition for our simulations of the Eq. (1).For our numerical simulations we consider the chaotic per-turbation in the nonlinearity given by the logistic function, the FIG. 1: Plots of the perturbed nonlinearity for the (a) chaotic, (b)random, (c) and nonperiodic functions. random perturbation is generated by random algorithm simu-lator, and the nonperiodic perturbations are generated by thefunction α (cos(5 x ) / √ x ) / , where α assumes thevalues of the perturbation. In Figs. 1a, 1b, and 1c we display aform of the chaotic, random, nonperiodic nonlinearity as func-tions of space for a generic time, respectively. Experimentally,this perturbation in the nonlinearity can be constructed, e.g.,in a crystal with impurities which are altered chaotically, ran-domly, or non-periodically. It remains to be seen what thepresence of Gσ ( x, t ) implies, though one would guess that itamounts to taking into account, within the mean field, Gross-Pitaevskii, description, the effects of the many-body correla-tions.We use (5) as input state in (1) to verify its evolution in thepresence of the above mentioned perturbations. In Fig. 2a weconsider the chaotic perturbation via σ ( x, t ) between ± . and ± . ( ). The chaotic perturbation is obtained consid-ering points affected within the interval − ≤ x ≤ in space, that changes the spatial profile of the nonlinearity.These perturbed points are changed by a new function after atime t = 20 , and so we will have temporal points affectedin the interval ≤ t ≤ . Fig. 2b displays the soliton am-plitude (height) of the solution ( | ψ | ) at the position x = 0 .Note that it becomes vanishingly small after t = 60 owing toits motion. However, the soliton is stable in this range of per-turbation. We calculated the error in the power of . × − and the comparative error of E r = 9 . .On the other hand, when we consider a chaotic perturba-tion of in the value of the amplitude, we found that thesoliton practically disappears. This case is shown in Fig. 3.In the Figs. 3a and 3b we display the | ψ | and the height inthe position x = 0 , respectively. The error in the power was . × − with a mean distance E r = 9 . . In this case themean distance is due to the moving and the vanishing pattern,i.e., the output state can not be at the same position as that ofthe input state.Now, when we consider a random perturbation of the non-linearity, obtained here by an algorithm of random numbergeneration, we note a different behavior of the solutions. Herethe soliton remains stable even for of perturbation, differ-ently from the case of the chaotic perturbation. In Fig. 4a weplot the soliton solution considering of random pertur-bation in the nonlinearity using the same arguments presentedfor the chaotic case, i.e., affected points in space versus temporal points into the range shown. Fig. 4b displaysthe amplitude of the soliton at x = 0 . This perturbation is FIG. 2: Plots of the soliton evolution | ψ ( x, t ) | with of chaoticperturbation in the nonlinearity. In (a) is displayed the solution andits profile (top panel) and (b) its height at position x = 0 .FIG. 3: Plots of the soliton evolution | ψ ( x, t ) | with of chaoticperturbation in the nonlinearity (see text for details). In (a) is dis-played the solution and its profile (top panel) and (b) its height atposition x = 0 . responsible for moving the soliton. The error in the power isof . × − with the mean distance between the input andoutput states of − . For of random perturbation the os-cillation of the soliton is more evidenced when compared withthe case of of random perturbation. Figs. 5a and 5b showthe | ψ | and the amplitude for x = 0 , respectively. The ro-bustness is guaranteed, differently form the chaotic case. Theerrors in power and the mean distance are . × − and − , respectively.To conclude our study we investigate the effects of the non-periodic perturbation in the nonlinearity. From Fig. 1c oneobserves that this perturbation seems the most well-behavedcompared to the other two types. This fact is reflected in thebehavior of the solution which remains practically with thesame form as that of the input state. This occurs even whenit suffers of nonperiodic perturbation, as can be seen inFig. 6. The error in the power is of . × − and the meandistance is of . .With the results presented here, we verify that, under simi-lar conditions, chaotic, random, and nonperiodic perturbationsin the nonlinearity can present distinct features, and some-times results in vanishing solitons, as verified when the systemsuffers chaotic perturbations.In summary, in the present work we have studied the ef-fects of chaotic, random, and nonperiodic perturbations in thenonlinearity on the soliton evolution via NLSE. We consid-ered cubic nonlinearity with strength perturbed chaotically,randomly, or nonperiodically. In the chaotic case we foundthat moving solitons can be destroyed when they are perturbedwith in the intensity of the nonlinearity. On the otherhand, when the system engenders random perturbation this FIG. 4: Plots of the soliton evolution | ψ ( x, t ) | with of ran-dom perturbation in the nonlinearity (see text for details). In (a) isdisplayed the solution and its profile (top panel) and (b) its height atposition x = 0 .FIG. 5: Plots of the soliton evolution | ψ ( x, t ) | with of ran-dom perturbation in the nonlinearity (see text for details). In (a) isdisplayed the solution and its profile (top panel) and (b) its height atposition x = 0 . does not occur. The soliton solution remains stable, how-ever now it can move. Finally, when we look for the non-periodic perturbation we found that it displays robust solu-tions with no apparent influence on the solitons. In a way,we can say that disorder in the non-linearity may, or may notlead to Anderson-type localization, depending on the natureof the perturbation. Our results have direct impact on work inoptical lattices with impurities in the crystal, laser-generatedrandomness in the non-linearity, and many-body effects in thedynamics of Bose-Einstein condensates and their collectiveexcitations and transport. FIG. 6: Plots of the soliton evolution | ψ ( x, t ) | with of nonpe-riodic perturbation in the nonlinearity (see text for details). In (a) isdisplayed the solution and its profile (top panel) and (b) its height atposition x = 0 . Acknowledgments
The authors would like to thank CAPES, CNPQ, FUNAPE-GO, and FAPESP for partial financial support. [1] G. P. Agrawal,
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