Modulation of localized solutions in quadratic-cubic nonlinear Schrödinger equation with inhomogeneous coefficients
Wesley B. Cardoso, Hugo L. C. Couto, Ardiley T. Avelar, Dionisio Bazeia
aa r X i v : . [ n li n . PS ] A p r Modulation of localized solutions in quadratic-cubic nonlinearSchrödinger equation with inhomogeneous coefficients
Wesley B. Cardoso a ,1 , Hugo L. C. Couto a , Ardiley T. Avelar a , Dionisio Bazeia b a Instituto de Física, Universidade Federal de Goiás, 74.690-900, Goiânia, Goiás, Brazil b Departamento de Física, Universidade Federal da Paraíba, 58051-970 João Pessoa, Paraíba, Brazil
Abstract
We study the presence of exact localized solutions in a quadratic-cubic nonlinear Schrödinger equationwith inhomogeneous nonlinearities. Using a specific ansatz , we transform the nonautonomous nonlinearequation into an autonomous one, which engenders composed states corresponding to solutions localizedin space, with an oscillating behavior in time. Direct numerical simulations are employed to verify thestability of the modulated solutions against small random perturbations.
Keywords:
Nonlinear Schrödinger equation; quadratic-cubic nonlinearity; solitons; inhomogeneousmedium.
1. Introduction
Localized solutions in nonlinear media such as the shape preserving solitons, and breathers, which arecharacterized by internal oscillations, have fundamental applications for energy transport in optical fibersand waveguides [1, 2], polaronic materials [3], biological molecules [4], etc. These solutions can propagatewithout losing their shape due to equilibrium between diffraction (in spatial domain) or dispersion (intemporal domain) and nonlinearity [5–8]. They also appear describing localized excitations in dilute Bose-Einstein condensates in specific conditions of balance between the particle dispersion and the nonlineareffect due to the two-body interaction [9, 10] and open possibilities for future applications in coherent atomoptics, atom interferometry, and atom transport.In many cases, the equations that describe the systems have the form of a nonlinear Schrödinger (NLS)equation [11]. Indeed, the NLS equation appears as a universal equation governing the evolution ofslowly varying packets of quasi-monochromatic waves in weakly nonlinear media featuring dispersion.One can exemplify this with: light propagation in nonlinear optical fibers and planar waveguides; small-amplitude gravity waves on the surface of deep inviscid water and Langmuir waves in hot plasmas; in-teraction of high-frequency molecular vibrations and low-frequency longitudinal deformations in a modelof long biological molecules; mean-field description of Bose-Einstein condensates (in this case it becomesthe Gross-Pitaevskii equation); models that describe nonlinear dissipative media (in this case it becomesthe Ginzburg-Landau equation); and so on. Regarding the form of the nonlinearities in the NLS equationone can find: quadratic [12], quadratic-cubic [13, 14], cubic (or Kerr-type) [1, 15, 16], cubic-quintic [17–20],quintic [21, 22], nonpolynomial [23–28], logarithm [29, 30], saturable [31–33], and other nonlinearities.The inclusion of variable coefficients in the NLS equation, which are due to inhomogeneities, producesa gigantic backdrop of possibilities of modulation of the localized solutions, although it breaks the inte-grability of the standard cubic NLS equation [34]. Physically, the variable coefficients can be achieved bychanging the system structure, as for example, in the magnetohydrodynamics the inhomogeneity of the Corresponding author. Tel.: +55 6235211014.E-mail address: [email protected] (W. B. Cardoso).
Preprint submitted to arXiv April 12, 2017 eal plasma environment can be achieved by fluctuations of the density, temperature, and magnetic fields[35]; the inhomogeneities in nonlinear fibers or crystals are due to variation in the geometry and/or varia-tion in the material parameters in the fabrication process of such systems [1]; in Bose-Einstein condensates(BECs) the variations in the potential and nonlinearities can be controlled by the application of externalfields, which also induce modulation pattern of the local nonlinearity through the Feshbach-resonancemechanism, i.e., field-induced changes of the scattering length characterizing binary collisions betweenatoms, which contributes to modify the nonlinearity in the BEC [36].The construction of analytical solitonic solutions is a hard task. However, by applying a similaritytransformation technique, which transforms the nonautonomous NLS equation into an autonomous one,one can build analytical solutions. This method has been applied in a number of works [27, 30, 37–61]. Inparticular, in Ref. [37] the authors presented localized nonlinear waves in systems with time- and space-modulated cubic nonlinearities. More general models, including cubic-quintic nonlinearities and modula-tions with dependence on both time and space coordinates were considered in Refs. [19, 46, 48, 49, 52, 61].Exact solutions to three-dimensional generalized NLS equations with varying potential and nonlinearitieswere studied in Refs. [38–40]. Also, the modulation of breathers and rogue waves were investigated in Refs.[40, 42] and Refs. [47, 50, 58], respectively. In Refs. [43, 51, 59], one investigated solitons of two-componentsystems modulated in space and time. Moreover, in Ref. [53] one studied solitons in a generalized model,using space- and time-variable coefficients in a NLS equation with higher-order terms. More recently, thedynamics of self-similar waves in asymmetric twin-core fibers with Airy-Bessel modulated nonlinearitywas also investigated in [57].These previous studies have motivated the investigations of other new possibilities, among them thequadratic-cubic nonlinear Schrödinger (QCNLS) equations with inhomogeneous coefficients, in the pres-ence of several distinct background potentials. From the general physical perspective, QCNLS equationshave received considerable attention in classical field theory [62] because the localized solutions are non-topological or lump-like structures [63] that appear in several contexts in physics such as q-balls, tachyonbranes, and galactic dark matter properties (see [64] and references therein). In BEC, these equations canarise as an approximate model of a relatively dense quasi-1D BEC with repulsive local interactions be-tween atoms [25, 65] plus a long-range dipole-dipole attraction between them [66]. Recently, the presenceof chaotic solitons in this system under nonlinearity management has been investigated in [14] via numer-ical simulations and variational approximation with rational and hyperbolic trial functions. In the currentwork, our goal is to find analytical solutions describing localized structures modulated by nonautonomousQCNLS equations, allowing us to investigate different patterns of modulations. To do this, we employ thesimilarity transformation technique and direct numerical simulations to check stability of the solutions.The problem is of current interest since we know that information on the stability of solitonic solutions isof great significance in the study of atomic Bose-Einstein condensates [10, 67].The work is organized as follows. In Sec. 2 we introduce the theoretical model and apply the similaritytransformation to get information on the pattern of the inhomogeneous terms of the QCNLS equation.Also, we present two different solutions of the autonomous QCNLS equation. The linear stability analysisis displayed in Sec. 3. Next, in the Sec. 4 we consider three different modulation patterns and show theresults of the stability tests that are obtained via direct numerical simulations. We summarize our resultsand suggest new investigations in Sec. 5.
2. The quadratic-cubic model and the analytical solutions
The model of interest in this work is described by the QCNLS equation with inhomogeneous coeffi-cients. It is given by i ψ t = − ψ xx + V ( x , t ) ψ + g ( t ) | ψ | ψ + g ( t ) | ψ | ψ , (1)where ψ = ψ ( x , t ) , ψ t = ∂ψ / ∂ t , ψ xx = ∂ ψ / ∂ x , V ( x , t ) is the background or trapping potential, and g ( t ) and g ( t ) represent the quadratic and cubic nonlinearity intensities, which are modulated in time,2espectively. This second order partial differential equation with quadratic and cubic time-dependent non-linearities is very hard to solve, but it describes very interesting physical systems such as cigar-shapedcondensates with repulsive interatomic interactions [25] plus a dipole-dipole attraction [66], i.e, Eq. (1) ap-pears as an effective 1D equation that governs the axial dynamics of mean-field cigar-shaped condensatesand accounts accurately the contribution from the transverse degrees of freedom. Hence, our goal is toconstruct explicit nontrivial solutions of this QCNLS with potentials depending on the spatial coordinateand on time, with nonlinearities depending on time. To achieve this, we use the following ansatz [42] ψ = ρ ( t ) e i η ( x , t ) Φ [ ζ ( x , t ) , τ ( t )] (2)that connects the nonautonomous QCNLS to an autonomous QCNLS equation i Φ τ = − Φ ζζ + G | Φ | Φ + G | Φ | Φ , (3)with constant coefficients G and G , which is easier to solve. Note that the ansatz (2) transfer all the spaceand time dependence of the coefficients of (1) to the external parameters, the amplitude ρ ( t ) and the phase η ( x , t ) , and now the new coordinates ζ ( x , t ) and τ ( t ) describe the space and time evolution in relation to aframe which is moving with the localized solution.By inserting (2) into (1) one gets the Eq. (3) provided that the set of conditions ρ t + ρη xx =
0, (4) ζ t + ζ x η x =
0, (5) τ t − ζ x =
0, (6) ζ xx =
0. (7)are satisfied in order to connect the external parameters with the internal ones. These conditions are tightlyrelated to each other, but they allow that we go on: for instance, from Eq. (7) one derives ζ ( x , t ) = a ( t ) x + b ( t ) , where a ( t ) is the inverse of the width of the localized solution (it is positive definite) and − b ( t ) / a ( t ) is the position of its center of mass, which implies in Eq. (6) the following condition τ ( t ) = R a dt . Then,one can obtain the amplitude and phase of the ansatz (2), given by ρ ( t ) = √ a , (8) η ( x , t ) = − a t x a − b t xa + ǫ ( t ) , (9)respectively.In addition, the trapping potential and nonlinear terms must have the form V = − η t − η x , (10) g = G ζ x ρ , (11) g = G ζ x ρ = G g G ρ , (12)which, by using the Eqs. (8) and (9), can be rewritten as V ( x , t ) = α ( t ) x + β ( t ) x + δ ( t ) , (13) g ( t ) = G a ( t ) , (14) g ( t ) = G a ( t ) , (15)3here α ( t ) = (cid:16) aa tt − a t (cid:17) /2 a , (16) β ( t ) = ( ab tt − a t b t ) / a , (17) δ ( t ) = − (cid:16) a ǫ t + b t (cid:17) /2 a . (18)Note that the modulation in the present model is completely defined by setting the functions a ( t ) , b ( t ) , and ǫ ( t ) . Physically, this can be done by setting appropriately the patterns of linear and nonlinear coefficients( V , g , and g , respectively) in the system. For example, in a BEC the harmonic potential and nonlinearitiesmay vary in time due to the application of a modulated laser beam that controls the interactions optically.Next, we go further on the subject and consider an interesting possibility, with the solution Φ = ϕ ( ζ ) e − i µτ of Eq. (3), such that µϕ = − ϕ ζζ + G ϕ + G ϕ , (19)where µ is a constant and ϕ >
0. In this case, we can get two distinct solutions, the first one having theform ϕ = A + B ζ , (20)with A = − G /3 G (assuming A > B = − G /9 G and µ =
0. Note that we need B > G < G >
0. The presence of a negative G implies thatthe system engenders focusing cubic nonlinearity, and since G is positive, one is dealing with a defocusingquadratic nonlinearity. Also, one can obtain a different solution ϕ = A ′ + B ′ cosh ( ζ ) , (21)where A ′ = − ( G ) , B ′ = − q G − G / ( G ) and µ = − B ′ that present nonsingular solutions, viz., B ′ > G < G < G /9; B ′ < − G > G <
0. Both cases are of current interest since they can present self-focusing or self-defocusingnonlinearities (competitive or not), which can correspond to different types of materials constituting thenonlinear fiber or the crystal [1].
3. Analysis of linear stability
To analyze the linear stability of our analytical solutions of the autonomous QCNLS equation, we per-turbed it by normal modes as Φ ( ζ , τ ) = n ϕ ( ζ ) + [ v ( ζ ) + w ( ζ )] e λτ + [ v ∗ ( ζ ) + w ∗ ( ζ )] e λ ∗ τ o e − i µτ , (22)where v ( ζ ) , w ( ζ ) ≪ λ is the eigenvalue of this normal mode. In-serting this perturbed solution in (3) and linearizing, we obtain the following linear-stability eigenvalueproblem: L Ψ = λ Ψ , (23)where L = i (cid:18) ∇ + F ∇ + F (cid:19) , Ψ = (cid:18) vw (cid:19) , (24)and F = µ − G ϕ − G ϕ , F = µ − G ϕ − G ϕ ,4here we have assumed ϕ real and positive. Here we use the Fourier collocation method to computeeigenvalues of the linear-stability operator L , in which one expands the eigenfunction Ψ into a Fourierseries and turns Eq. (23) into a matrix eigenvalue problem for the Fourier coefficients of the eigenfunction Ψ . One can find examples of application of this method in Ref. [34], and here we investigate the four newdistinct possibilities which we describe below.
4. Analytical results and numerical simulations
We now examine the modulation of the above solutions and their stability by numerical simulations.The numerical method is based on the 4 th order split-step Crank-Nicholson algorithm in which the evolu-tion equation is split into several pieces (linear and nonlinear terms), which are integrated separately. Tothis end, we use the steps ∆ x = ∆ t = ψ = ψ [ + v ( x )] , (25)where ψ = ψ ( x , 0 ) is the analytical solution obtained via ansatz (2) and v ∈ [ − ] is a real randomnumber with zero mean (white noise) evaluated at each point of discretization grid in x -coordinate. Also,to ensure the stability of the method we also checked the norm (power) and the energy of the solutiondefined by P = R ∞ − ∞ | ψ | dx and E = Z ∞ − ∞ dx (cid:26) | ψ x | + V | ψ | + g | ψ | + g | ψ | (cid:27) , (26)respectively.In order to focus on the practical use of the above results, in the following we present some specificexamples of typical potentials and nonlinearities given by Eqs. (13)-(15) that can be found in experimentalsetups. Here, for pedagogical purpose, four distinct cases are addressed. In the first, presented in Subsec.4.1, we consider the system without modulation, in order to give us a landmark about the stability of suchsolutions. In Subsec. 4.2 we include a potential which is asymmetric in space and periodically modulatedin time, which is found by setting α ( t ) = δ ( t ) = β ( t ) =
0. This type of potential is interestingbecause we can see how the center of mass of the solutions behave under a periodically oscillating uni-form field. Another potential, the harmonically symmetric in space and periodically modulated in timepotential, which is obtained with β ( t ) = δ ( t ) = α ( t ) =
0, is studied in Subsec. 4.3. Here the mainmotivation is to investigate how the solutions behave under the effect of the squeezing and anti-squeezingproduced by the oscilatting harmonic potential. Finally, a more general case, mixing the two previouscases, is considered in Subsec. 4.4.Indeed, all patterns of potential and nonlinearities addressed here are feasible in several scenarios, forexample, in nonlinear fiber optics and BECs [36, 68, 69]: the first case can be attained by the action of aperiodic heterogeneity obtained in the fiber construction, and the second one may be driven by externalpotentials and by using the Feshbach-resonance management.
In this case we assume that V = a = b =
0, and ǫ =
0, for simplicity. Then, one gets ρ = η = τ = t ,and ζ = x . Also, the quadratic and cubic nonlinearities present a constant behavior ( g = G and g = G ).Then, the solution has a constant amplitude modulation ρ = η = | ψ | for the cases given by Eqs. (20) and(21), in the absence of modulation (The values chosen by us for the nonlinearities are such that the normof the solution approaches 1). Note that we choose three ranges of values for the nonlinearities, namely, g < g < g < g > igure 1: (Color online) Localized solutions | ψ | obtained from the ansatz (2) without modulation ( α = β = δ = a = b =
0, and ǫ = G = G = − G = − G = − G = − G = G = G = − −2 0 2x 10 −3 −202 Re( λ ) I m ( λ ) (a) −1 0 1x 10 −3 −202 Re( λ ) I m ( λ ) (b) −1 0 1x 10 −3 −202 Re( λ ) I m ( λ ) (c) −1 0 1x 10 −3 −202 Re( λ ) I m ( λ ) (d) Figure 2: (Color online) Stability spectrum for solitary waves shown in Fig. 1 obtained via the linear-stability eigenvalue problem(23). The parameters used here are the same of Fig. 1. and g > g < ( λ ) =
0, one gets a linearly unstablesolution (cf. Eq. (22)). In our example, the Lorentzian-type solution (Eq. (20)) is prone to be unstable whilethe others solutions are linearly stable. From now on, we will call the examples for those specific choicesof nonlinearities presented in Figs. 1(a)-(d) by cases A, B, C, and D, respectively.
We now analyze a potential with linear modulation in x -coordinate and periodic modulation in t . So,we choose a = b = − sin ( ω t ) , such that, α = β = ω sin ( ω t ) , and δ = ǫ ( t ) . In this case, the amplitude and phase of the solution will be given by ρ = η = ω x cos ( ω t ) − ω [ cos ( ω t ) sin ( ω t ) + ω t ] , respectively. Also, one gets ζ = x − sin ( ω t ) , τ = t , g = G and g = G (constant nonlinearities), and a seesaw potential with the form V = ω x sin ( ω t ) .Note that the amplitude of the above potential depends on the square of oscillation frequency of the tem-poral modulation.We display in Fig. 3 the analytical profiles ( | ψ | ) of the localized solutions modulated by the seesawpotential. Note that in Figs. 3(a)-3(d) we contemplate the same cases shown in Figs. 1(a)-1(d), respectively,now with modulation of a seesaw potential. We stress that in the present case, the linear stability analysisemployed in the previous case do not work anymore. Then, we analyze the stability of the solutions bydirect numerical simulations of the perturbed profiles (Eq. (25)). Based on the analytical solutions, weexpect stable solutions when the variance in x is approximately “constant”, i.e, var ( x ) = h x i − h x i , with h•i = R ∞ − ∞ •| ψ ( x , t ) | dx . Indeed, due to the perturbations it will only suffer small random variations.In Figs. 4(a)-4(d) we display the variance of x versus t . Note that we do not present the 3D profiles toavoid problems of graphical resolution because the number of oscillations up to t = ω , varying by the step 0.05 into the range [
0, 1.0 ] .6 igure 3: (Color online) Localized solutions | ψ | obtained from the ansatz (2) considering α = δ = β = ω sin ( ω t ) (herewith ω =
1, for simplicity). The profiles shown in (a)-(d) corresponds to those non-modulated cases presented in Figs. 1(a)-1(d),respectively, but now with modulation. The values of the nonlinearities are the same used in Fig. 1. (a) (b) (c) (d)
Figure 4: (Color online) Stability tests via direct numerical simulations, using the input state given by (25) with different modulationfrequencies, viz. ω = x versus t ( var ( x ) ) corresponding to the casesdisplayed in Figs. 3(a)-3(d), respectively. Here we assume a quadratic modulation in x -coordinate with a periodic modulation in t -coordinate.Thus, we use a = + γ cos ( ω t ) (with γ <
1) and b = ǫ =
0, getting α = γω ( γ cos ( ω t ) − cos ( ω t ) − γ ) ( γ cos ( ω t ) + ) , (27) β =
0, and δ =
0. So, the amplitude and phase of the solution, the external potential and the modulatednonlinearities will be given by ρ = p + γ cos ( ω t ) , η = γω x sin ( ω t ) / { [ γ cos ( ω t ) + ] } , V = α ( t ) x (with α given by Eq. (27)), g = G [ + γ cos ( ω t )] , and g = G [ + γ cos ( ω t )] , respectively.Analytical profiles of the modulated solutions are shown in Fig. 5. Note that a breathing pattern isobtained since we have nonlinearities varying harmonically while the potential presents an attractive-to-expulsive harmonic change in its profile. In Figs. 6(a)-(d) we show the time evolution of the variance of x obtained by direct numerical simulations of Eq. (1). Now, differently from the variance predicted for theseesaw potential (Subsec. 4.2), here this parameter will oscillate around a constant value, which reflects thebreathing pattern of the solutions.We found different regions of stability/instability for each case. Interestingly, we observe in the case Athat the Lorentzian solution becomes stable for ω ∈ [ ] . This behavior is observed in the results shownin Fig. 6(a), where one can see that the variance of the curve for ω = ≤ ω ≤ ω ∈ [ ] and ω = ω ∈ [ ] and ω = ω = In this case, we consider a mixed potential with the form V = α ( t ) x + β ( t ) x , and we take a = + γ cos ( ω t ) and b = − sin ( ω t ) . We choose ǫ in a way such that δ =
0. The temporal modulation functions7 igure 5: (Color online) Localized solutions | ψ | obtained from the ansatz (2) considering a = + γ cos ( ω t ) , b =
0, and ǫ =
0. Theprofiles shown in (a)-(d) corresponds to the non-modulated cases presented in Figs. 1(a)-1(d), respectively, but now with modulation.The values of the nonlinearities are the same used in Fig. 1 plus ω = γ = (a) (b) (c) (d) Figure 6: (Color online) Stability tests via direct numerical simulations, using the input state given by (25) with different modulationfrequencies, viz. ω = x versus t ( var ( x ) ) corresponding to the casesdisplayed in Figs. 5(a)-5(d), respectively. for the potential will then be written in the form: β = ω sin ( ω t )[ − γ cos ( ω t )][ + γ cos ( ω t )] , (28)with α given by Eq. (27). With the above choices, it is possible to check that ρ = p + γ cos ( ω t ) and η = ω x [ γ x sin ( ω t ) + ( ω t )] / [ γ cos ( ω t ) + ] + ǫ ( t ) for the amplitude and phase of the solution, re-spectively, and that g = G [ + γ cos ( ω t )] and g = G [ + γ cos ( ω t )] . The modulated coordinatestake the forms ζ = [ + γ cos ( ω t )] x − sin ( ω t ) and τ = { γ [ + γ cos ( ω t )] sin ( ω t ) + ω t ( + γ ) } /2 ω .In Figs. 7 (a)-(d) we show the analytical profiles of the modulated solutions by the mixed potential.Indeed, as expected we observe the solution oscillating around the center of the trap, in a way similar tothe case of the seesaw potential, plus a breathing pattern, as in the case of the flying-bird potential. Similarlyto the case of flying-bird potential, due to the breathing pattern of the mixed potential the variance of x willalso oscillate around a constant value. This pattern was observed in the stable solutions obtained by thenumerical simulations.The temporal evolution of variance in x of the solutions are displayed in Figs. 8(a)-8(d). Specifically, forthe Lorentzian solution (case A, displayed in Fig. 8(a)) the solution stabilizes for ω ≥ ω ∈ [ ] in case B, for ω ∈ [ ] and for ω = ω ∈ [ ] Figure 7: (Color online) Localized solutions | ψ | obtained from the ansatz (2) considering a = + γ cos ( ω t ) , b = − sin ( ω t ) , and ǫ in a such way that allow us to get δ =
0. The profiles shown in (a)-(d) corresponds to the non-modulated cases presented in Figs.1(a)-1(d), respectively, but now with modulation. The values of the nonlinearities are the same used in Fig. 1 plus ω = γ = a) (b) (c) (d) Figure 8: (Color online) Stability tests via direct numerical simulations, using the input state given by (25) with different modulationfrequencies, viz. ω = x versus t ( var ( x ) ) corresponding to the casesdisplayed in Figs. 7 (a)-7(d), respectively. and for ω ∈ [ ] in case D. In fact, one can see in the example shown in Fig. 8(a) the increase in thevalue of the variance in x for ω = ω = var ( x ) with ω =
5. Summary
In this work, we investigated the NLS equation in the presence of quadratic and cubic nonlinearities thatare modulated in time, under the action of a background potential modulated in space and time. We havestudied two distinct solutions, constructed from Eqs. (20) and (21), and we dealt with several possibilities,with the nonlinearities being focusing or defocusing, and with the potential having four distinct features,as they appear in the subsections 4.1, 4.2, 4.3, and 4.4. The study presented several analytical solutions,showing how they can be stable or unstable, when one varies parameters that control each one of the fourspecific problems considered in the work.The main results of the current study are exemplified in the eight figures that are depicted above. Thefour odd numbered figures illustrate solutions with vanishing potential, and with potential of the seesaw,flying-bird, or mixed type, respectively. Also, the four even figures describe the corresponding stability,which we investigated numerically. Interestingly, we see that the frequency of modulation is an importantparameter to control stability of the solutions for both the self-focusing and the self-defocusing cubic non-linearity. Among all the interesting results, we stress that the stability is not guaranteed for certain types ofmodulations, so the pattern of modulation can work to stabilize or destabilize the solutions.The several investigations illustrate how to stabilize unstable solutions and how to accomplish thispossibility in a diversity of scenarios. Thus, from the experimental point of view one can, for example,stabilize an unstable solution, like the one shown in Fig. 2a, with an appropriate choice of the pattern ofmodulation, and with a specific choice in the frequency of modulation, which is the key parameter here.In this sense, the above results encourage us to study other models, with other types of nonlinearities andsolutions. We hope to report on this in the near future.
Acknowledgments
We thank the Brazilian agencies CAPES, CNPq and the Instituto Nacional de Ciência e Tecnologia-Informação Quântica (INCT-IQ) for partial support.
ReferencesReferences [1] Agrawal GP. Nonlinear Fiber Optics. Academic Press. Academic Press; 2013. Available from: https://books.google.com.br/books?id=xNvw-GDVn84Chttps://books.google.com.br/books?id=b5S0JqHMoxAC .
2] Flach S, Willis CR. Discrete breathers. Phys Rep. 1998 mar;295(5):181–264. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0370157397000689 .[3] Andersen JD, Kenkre VM. Self-trapping and time evolution in some spatially extended quantum nonlinear systems: Exact solu-tions. Phys Rev B. 1993 may;47(17):11134–11142. Available from: http://link.aps.org/doi/10.1103/PhysRevB.47.11134 .[4] Peyrard M, Dauxois T, Hoyet H, Willis CR. Biomolecular dynamics of DNA: statistical mechanics and dynamical mod-els. Phys D Nonlinear Phenom. 1993 sep;68(1):104–115. Available from: http://linkinghub.elsevier.com/retrieve/pii/016727899390035Y .[5] Scott A. Nonlinear Science: Emergence and Dynamics of Coherent Structures. Oxford texts in applied and engineering mathe-matics. Oxford University Press; 2003. Available from: https://books.google.com.br/books?id=PkxK9AVQ6SgC .[6] Drazin PG, Johnson RS. Solitons: An Introduction. Cambridge Computer Science Texts. Cambridge University Press; 1989.Available from: https://books.google.com.br/books?id=HPmbIDk2u-gC .[7] Remoissenet M. Waves Called Solitons: Concepts and Experiments. Springer Berlin Heidelberg; 2013. Available from: https://books.google.com.br/books?id=qULtCAAAQBAJ .[8] Eilenberger G. Solitons: Mathematical Methods for Physicists. Springer Series in Solid-State Sciences. Springer Berlin Heidel-berg; 2012. Available from: https://books.google.com.br/books?id=YKvrCAAAQBAJ .[9] Pethick CJ, Smith H. Bose-Einstein Condensation in Dilute Gases. Cambridge University Press; 2002.[10] Pitaevskii LP, Stringari S. Bose-Einstein Condensation. International Series of Monographs on Physics. Clarendon Press; 2003.Available from: https://books.google.com.br/books?id=rIobbOxC4j4C .[11] Malomed BA. Nonlinear Schrödinger Equations. In: Encycl. Nonlinear Sci. Taylor and Francis; 2006. p. 639–643. Available from: https://books.google.com.br/books?id=KC7gZmIEAiwC .[12] Buryak A. Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys Rep. 2002nov;370(2):63–235. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0370157302001965 .[13] Hayata K, Koshiba M. Prediction of unique solitary-wave polaritons in quadratic-cubic nonlinear dispersive media. J Opt SocAm B. 1994 dec;11(12):2581. Available from: .[14] Fujioka J, Cortés E, Pérez-Pascual R, Rodríguez RF, Espinosa A, Malomed BA. Chaotic solitons in the quadratic-cubic nonlinearSchrödinger equation under nonlinearity management. Chaos An Interdiscip J Nonlinear Sci. 2011;21(3):033120. Available from: http://scitation.aip.org/content/aip/journal/chaos/21/3/10.1063/1.3629985 .[15] Dalfovo F, Giorgini S, Pitaevskii LP, Stringari S. Theory of Bose-Einstein condensation in trapped gases. Rev Mod Phys. 1999apr;71(3):463–512. Available from: http://link.aps.org/doi/10.1103/RevModPhys.71.463 .[16] Aranson IS, Kramer L. The world of the complex Ginzburg-Landau equation. Rev Mod Phys. 2002 feb;74(1):99–143. Availablefrom: http://link.aps.org/doi/10.1103/RevModPhys.74.99 .[17] Akhmediev NN, Afanasjev VV, Soto-Crespo JM. Singularities and special soliton solutions of the cubic-quintic complexGinzburg-Landau equation. Phys Rev E. 1996 jan;53(1):1190–1201. Available from: http://link.aps.org/doi/10.1103/PhysRevE.53.1190 .[18] Mihalache D, Mazilu D, Crasovan LC, Malomed BA, Lederer F. Three-dimensional spinning solitons in the cubic-quintic non-linear medium. Phys Rev E. 2000 jun;61(6):7142–7145. Available from: http://link.aps.org/doi/10.1103/PhysRevE.61.7142 .[19] Avelar AT, Bazeia D, Cardoso WB. Solitons with cubic and quintic nonlinearities modulated in space and time. Phys Rev E. 2009feb;79(2):025602. Available from: http://link.aps.org/doi/10.1103/PhysRevE.79.025602 .[20] Cardoso WB, Avelar AT, Bazeia D. One-dimensional reduction of the three-dimenstional Gross-Pitaevskii equation with two- andthree-body interactions. Phys Rev E. 2011 mar;83(3):036604. Available from: http://link.aps.org/doi/10.1103/PhysRevE.83.036604 .[21] Alfimov GL, Konotop VV, Pacciani P. Stationary localized modes of the quintic nonlinear Schrödinger equation with a periodicpotential. Phys Rev A. 2007 feb;75(2):023624. Available from: http://link.aps.org/doi/10.1103/PhysRevA.75.023624 .[22] Reyna AS, de Araújo CB. Nonlinearity management of photonic composites and observation of spatial-modulation instabilitydue to quintic nonlinearity. Phys Rev A. 2014 jun;89(6):063803. Available from: http://link.aps.org/doi/10.1103/PhysRevA.89.063803 .[23] Salasnich L, Parola A, Reatto L. Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates.Phys Rev A. 2002 apr;65(4):043614. Available from: http://link.aps.org/doi/10.1103/PhysRevA.65.043614 .[24] Salasnich L, Parola A, Reatto L. Condensate bright solitons under transverse confinement. Phys Rev A. 2002 oct;66(4):043603.Available from: http://link.aps.org/doi/10.1103/PhysRevA.66.043603 .[25] Mateo AM, Delgado V. Effective mean-field equations for cigar-shaped and disk-shaped Bose-Einstein condensates. Phys RevA. 2008 jan;77(1):013617. Available from: http://link.aps.org/doi/10.1103/PhysRevA.77.013617 .[26] Adhikari SK, Salasnich L. Effective nonlinear Schrödinger equations for cigar-shaped and disc-shaped Fermi superfluidsat unitarity. New J Phys. 2009 feb;11(2):023011. Available from: http://stacks.iop.org/1367-2630/11/i=2/a=023011?key=crossref.206d6a141e90aff3fffa0cee8a11aa07 .[27] Cardoso WB, Zeng J, Avelar AT, Bazeia D, Malomed BA. Bright solitons from the nonpolynomial Schrödinger equation withinhomogeneous defocusing nonlinearities. Phys Rev E. 2013 aug;88(2):025201. Available from: http://link.aps.org/doi/10.1103/PhysRevE.88.025201 .[28] Couto HLC, Cardoso WB. Dynamics of the soliton-sound interaction in the quasi-one-dimensional Munoz-Mateo–Delgadoequation. J Phys B At Mol Opt Phys. 2015 jan;48(2):025301. Available from: http://stacks.iop.org/0953-4075/48/i=2/a=025301?key=crossref.94b619bdc9e8159b1e066778aa3ab2d4 .[29] Biswas A, Milovi´c D. Optical solitons with log-law nonlinearity. Commun Nonlinear Sci Numer Simul. 2010 dec;15(12):3763–3767. Available from: http://linkinghub.elsevier.com/retrieve/pii/S1007570410000523 .[30] Calaça L, Avelar AT, Bazeia D, Cardoso WB. Modulation of localized solutions for the Schrödinger equation with logarithm onlinearity. Commun Nonlinear Sci Numer Simul. 2014 sep;19(9):2928–2934. Available from: http://linkinghub.elsevier.com/retrieve/pii/S1007570414000550 .[31] Soto-Crespo JM, Wright EM, Akhmediev NN. Recurrence and azimuthal-symmetry breaking of a cylindrical Gaussian beamin a saturable self-focusing medium. Phys Rev A. 1992 mar;45(5):3168–3175. Available from: http://link.aps.org/doi/10.1103/PhysRevA.45.3168 .[32] Stepi´c M, Kip D, Hadžievski L, Maluckov A. One-dimensional bright discrete solitons in media with saturable nonlinearity.Phys Rev E. 2004 jun;69(6):066618. Available from: http://link.aps.org/doi/10.1103/PhysRevE.69.066618 .[33] Melvin TRO, Champneys AR, Kevrekidis PG, Cuevas J. Radiationless Traveling Waves in Saturable Nonlinear SchrödingerLattices. Phys Rev Lett. 2006 sep;97(12):124101. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.97.124101 .[34] Yang J. Nonlinear Waves in Integrable and Nonintegrable Systems. Society for Industrial and Applied Mathematics; 2010.Available from: http://epubs.siam.org/doi/book/10.1137/1.9780898719680 .[35] Wang L, Li M, Qi FH, Xu T. Modulational instability, nonautonomous breathers and rogue waves for a variable-coefficientderivative nonlinear Schrödinger equation in the inhomogeneous plasmas. Phys Plasmas. 2015 mar;22(3):032308. Availablefrom: http://scitation.aip.org/content/aip/journal/pop/22/3/10.1063/1.4915516 .[36] Theis M, Thalhammer G, Winkler K, Hellwig M, Ruff G, Grimm R, et al. Tuning the Scattering Length with an OpticallyInduced Feshbach Resonance. Phys Rev Lett. 2004 sep;93(12):123001. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.93.123001 .[37] Belmonte-Beitia J, Pérez-García VM, Vekslerchik V, Konotop VV. Localized Nonlinear Waves in Systems with Time- and Space-Modulated Nonlinearities. Phys Rev Lett. 2008 apr;100(16):164102. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.100.164102 .[38] Yan Z, Konotop VV. Exact solutions to three-dimensional generalized nonlinear Schrödinger equations with varying potentialand nonlinearities. Phys Rev E. 2009 sep;80(3):036607. Available from: http://link.aps.org/doi/10.1103/PhysRevE.80.036607 .[39] Yan Z, Hang C. Analytical three-dimensional bright solitons and soliton pairs in Bose-Einstein condensates with time-spacemodulation. Phys Rev A. 2009 dec;80(6):063626. Available from: http://link.aps.org/doi/10.1103/PhysRevA.80.063626 .[40] Avelar AT, Bazeia D, Cardoso WB. Modulation of breathers in the three-dimensional nonlinear Gross-Pitaevskii equation. PhysRev E. 2010 nov;82(5):057601. Available from: http://link.aps.org/doi/10.1103/PhysRevE.82.057601 .[41] Cardoso WB, Avelar AT, Bazeia D. Bright and dark solitons in a periodically attractive and expulsive potential with nonlinearitiesmodulated in space and time. Nonlinear Anal Real World Appl. 2010 oct;11(5):4269–4274. Available from: http://linkinghub.elsevier.com/retrieve/pii/S1468121810000805 .[42] Cardoso WB, Avelar AT, Bazeia D. Modulation of breathers in cigar-shaped Bose–Einstein condensates. Phys Lett A. 2010jun;374(26):2640–2645. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0375960110004895 .[43] Cardoso WB, Avelar AT, Bazeia D, Hussein MS. Solitons of two-component Bose-Einstein condensates modulated in spaceand time. Phys Lett A. 2010 may;374(23):2356–2360. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0375960110004068 .[44] Serkin VN, Hasegawa A, Belyaeva TL. Nonautonomous matter-wave solitons near the Feshbach resonance. Phys Rev A. 2010feb;81(2):023610. Available from: http://link.aps.org/doi/10.1103/PhysRevA.81.023610 .[45] Serkin VN, Hasegawa A, Belyaeva TL. Solitary waves in nonautonomous nonlinear and dispersive systems: nonautonomoussolitons. J Mod Opt. 2010 aug;57(14-15):1456–1472. Available from: .[46] Zhang JF, Tian Q, Wang YY, Dai CQ, Wu L. Self-similar optical pulses in competing cubic-quintic nonlinear media with dis-tributed coefficients. Phys Rev A. 2010 feb;81(2):023832. Available from: http://link.aps.org/doi/10.1103/PhysRevA.81.023832 .[47] Yan Z. Nonautonomous "rogons" in the inhomogeneous nonlinear Schrödinger equation with variable coefficients. Phys Lett A.2010 jan;374(4):672–679. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0375960109014625 .[48] He JR, Li HM. Analytical solitary-wave solutions of the generalized nonautonomous cubic-quintic nonlinear Schrödinger equa-tion with different external potentials. Phys Rev E. 2011 jun;83(6):066607. Available from: http://link.aps.org/doi/10.1103/PhysRevE.83.066607 .[49] He Jd, Zhang Jf, Zhang My, Dai Cq. Analytical nonautonomous soliton solutions for the cubic–quintic nonlinear Schrödingerequation with distributed coefficients. Opt Commun. 2012 mar;285(5):755–760. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0030401811012302 .[50] Dai CQ, Wang YY, Tian Q, Zhang JF. The management and containment of self-similar rogue waves in the inhomogeneousnonlinear Schrödinger equation. Ann Phys (N Y). 2012 feb;327(2):512–521. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0003491611001898 .[51] Cardoso WB, Avelar AT, Bazeia D. Modulation of localized solutions in a system of two coupled nonlinear Schrödinger equa-tions. Phys Rev E. 2012 aug;86(2):27601. Available from: http://link.aps.org/doi/10.1103/PhysRevE.86.027601 .[52] Arroyo Meza LE, de Souza Dutra A, Hott MB. Wide localized solitons in systems with time- and space-modulated nonlinearities.Phys Rev E. 2012 aug;86(2):026605. Available from: http://link.aps.org/doi/10.1103/PhysRevE.86.026605 .[53] Yomba E, Zakeri GA. Solitons in a generalized space- and time-variable coefficients nonlinear Schrödinger equation with higher-order terms. Phys Lett A. 2013 dec;377(42):2995–3004. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0375960113008098 .[54] He JR, Yi L, Li HM. Localized nonlinear waves in combined time-dependent magnetic–optical potentials with spatiotemporallymodulated nonlinearities. Phys Lett A. 2013 nov;377(34-36):2034–2040. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0375960113006014 .[55] Zhong WP, Beli´c MR, Huang T. Periodic soliton solutions of the nonlinear Schrödinger equation with variable nonlinearity and xternal parabolic potential. Opt - Int J Light Electron Opt. 2013 aug;124(16):2397–2400. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0030402612006948 .[56] He JR, Yi L. Formations of n-order two-soliton bound states in Bose–Einstein condensates with spatiotemporally modulatednonlinearities. Phys Lett A. 2014 mar;378(16-17):1085–1090. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0375960114001303 .[57] Soloman Raju T. Dynamics of self-similar waves in asymmetric twin-core fibers with Airy–Bessel modulated nonlinearity. OptCommun. 2015 jul;346:74–79. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0030401815001157 .[58] Temgoua DDE, Kofane TC. Nonparaxial rogue waves in optical Kerr media. Phys Rev E. 2015 jun;91(6):063201. Available from: http://link.aps.org/doi/10.1103/PhysRevE.91.063201 .[59] Yang Y, Yan Z, Mihalache D. Controlling temporal solitary waves in the generalized inhomogeneous coupled nonlinearSchrödinger equations with varying source terms. J Math Phys. 2015 may;56(5):053508. Available from: http://scitation.aip.org/content/aip/journal/jmp/56/5/10.1063/1.4921641 .[60] Kumar De K, Goyal A, Raju TS, Kumar CN, Panigrahi PK. Riccati parameterized self-similar waves in two-dimensional graded-index waveguide. Opt Commun. 2015 apr;341:15–21. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0030401814011444 .[61] Meza LEA, Dutra AdS, Hott MB, Roy P. Wide localized solutions of the parity-time-symmetric nonautonomous nonlinearSchrödinger equation. Phys Rev E. 2015 jan;91(1):013205. Available from: http://link.aps.org/doi/10.1103/PhysRevE.91.013205 .[62] Rajaraman R. Solitons and Instantons: An Introd. to Solitons and Instantons in Quantum Field Theory. North-Holland; 1987.Available from: https://books.google.com.br/books?id=r-HCoAEACAAJ .[63] Avelar AT, Bazeia D, Cardoso WB, Losano L. Lump-like structures in scalar-field models in dimensions. Phys Lett A. 2009dec;374(2):222–227. Available from: http://linkinghub.elsevier.com/retrieve/pii/S037596010901353X .[64] Avelar AT, Bazeia D, Losano L, Menezes R. New lump-like structures in scalar-field models. Eur Phys J C. 2008 may;55(1):133–143. Available from: .[65] Muñoz Mateo A, Delgado V. Effective one-dimensional dynamics of elongated Bose-Einstein condensates. Ann Phys (N Y). 2009mar;324(3):709–724. Available from: http://linkinghub.elsevier.com/retrieve/pii/S0003491608001516 .[66] Sinha S, Santos L. Cold Dipolar Gases in Quasi-One-Dimensional Geometries. Phys Rev Lett. 2007 oct;99(14):140406. Availablefrom: http://link.aps.org/doi/10.1103/PhysRevLett.99.140406 .[67] Kartashov YV, Malomed BA, Torner L. Solitons in nonlinear lattices. Rev Mod Phys. 2011 apr;83(1):247–305. Available from: http://link.aps.org/doi/10.1103/RevModPhys.83.247 .[68] Malomed BA. Soliton Management in Periodic Systems. Springer US; 2006. Available from: https://books.google.com.br/books?id=sr2txF_x6AgC .[69] Kevrekidis PG, Theocharis G, Frantzeskakis DJ, Malomed BA. Feshbach Resonance Management for Bose-Einstein Condensates.Phys Rev Lett. 2003 jun;90(23):230401. Available from: http://link.aps.org/doi/10.1103/PhysRevLett.90.230401 ..