Construction of modulated amplitude waves via averaging in collisionally inhomogeneous Bose-Einstein condensates
aa r X i v : . [ n li n . PS ] A p r Construction of modulated amplitude waves via averaging incollisionally inhomogeneous Bose-Einstein condensates
Qihuai Liu a,b, ∗ , Dingbian Qian a a School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin541002, China b School of Mathematical Sciences, Soochow University, Suzhou 215006, China
Abstract
We apply the averaging method to analyze spatio-temportal structures in nonlinear Schr¨odingerequations and thereby study the dynamics of quasi-one-dimensional collisionally inhomoge-neous Bose-Einstein condensates with the scattering length varying periodically in spatialand crossing zero. Infinitely many (positive measure set) modulated amplitude waves (pe-riodic and quasi-periodic), which are instable, can be proved to exist by adjusting the inte-gration constant c on some open interval. Finally, some numerical simulations support ourresults. Keywords:
Modulated amplitude waves; Gross-Pitaevskii equations; Collisionallyinhomogeneous Bose-Einstein condensates; Averaging method
PACS: : 05.45.-a, 03.75.Lm, 05.30.Jp, 05.45.Ac
1. Introduction
Since the experimental realization of Bose-Einstein condensates (BECs) in the mid-1990s[1, 2], the study of matter-wave patterns including existence and stability in BECs has drawna great deal of interest from experimentalists [3, 4] and theorists [5–8].In atomic physics, the Feshbach resonance of the scattering length of interatomic inter-actions is used for control of Bose-Einstein condensates [9, 10]. We consider the main modelof this paper for Feshbach resonance given by the perturbed Gross-Pitaevskii (GP) equationof the dimensionless form [11] i ∂ψ∂t = − ∂ ψ∂x + ˜ g ( x ) | ψ | ψ + ˜ V ( x ) ψ, (1.1)where the nonlinearity coefficient ˜ g ( x ) varies in space. In Eq. (1.1), ψ is the mean-fieldcondensate wave function (with density | ψ | measured in units of the peak 1D density n ), ∗ Corresponding author at: School of Mathematics and Computing Science, Guilin University of ElectronicTechnology, No. 2, Jinji Street, Guilin 541004, China. Tel./Fax.: +086 0773 3939803.
Email addresses: [email protected] (Qihuai Liu), [email protected] (Dingbian Qian)
Preprint submitted to Elsevier November 21, 2018 and t are normalized, respectively, to the healing length ξ = ~ / p n | g | m and ξ/c (where c = ~ p n | g | /m is the Bogoliubov speed of sound), and energy is measured in units of thechemical potential δ = g n . In the above expressions, g = 2 ~ ω ⊥ a , where ω ⊥ denotes theconfining frequency in the transverse direction, and a is a characteristic (constant) value ofthe scattering length relatively close to the Feshbach resonance. Finally, ˜ V ( x ) is the rescaledexternal trapping potential, and the x -dependent nonlinearity is given by ˜ g ( x ) = a ( x ) /a ,where a ( x ) is the spatially varying scattering length.In the past few years, GP equation (1.1) has been widely studied, such as the stability anddynamics of bright, dark solitary waves [12–15] and modulated amplitude waves (MAWs)[11].In order to study the dynamics of BECs with scattering length subjected to a spatiallyperiodic variation, Porter and Kevrekidis et al. [11] transform equation (1.1) into a new GPequation with a constant coefficient and an additional effective potential i ∂ψ∂t = − ∂ ψ∂x + | ψ | ψ + ˜ V ( x ) ψ + ˜ V eff ( x ) ψ, ˜ V eff ( x ) = 12 f ′′ f − ( f ′ ) f + f ′ f ∂∂x with f ( x ) = p ˜ g ( x ), then the transformed equation was investigated. For weak underlyinginhomogeneity, the effective potential takes a form resembling a superlattice, and the am-plitude dynamics of the solutions of the constant-coefficient GP equation obey a nonlineargeneralization of the Ince equation. In the small-amplitude limit, they use averaging toconstruct analytical solutions for modulated amplitude waves (MAWs), whose stability wassubsequently examined using both numerical simulations of the original GP equation andfixed-point computations with the MAWs as numerically exact solutions. However, men-tioned in their paper, the transformation ˜ ψ = p ˜ g ( x ) ψ applies only in the case when ˜ g ( x )does not cross zero. A natural question is that for general periodic function ˜ g ( x ), whetherthe similar results upon the dynamics can be obtained. On the other hand, the phasesof MAWs considered in [11] are trivial, which are corresponding to standing waves. Thus,another question is that whether the MAWs with nontrivial phases can exist.With these questions discussed above, in this paper we investigate the existence andstability of MAWs with nontrivial phases in collisionally inhomogeneous BECs modeledby GP equation (1.1) for general small periodic function ˜ g ( x ). The method is based onaveraging, and we use the averaging principle to replace a GP equation by the correspondingaveraged system. Along this paper, we assume that g ( x ) and V ( x ) are analytic and periodicfunctions with the least positive period T = π/ √ δ .The rest paper is organized as follows. In Section 2, we introduce modulated amplitudewaves involving periodic and quasi-periodic, and an averaging theorem is obtained in Section3. In Section 4 we investigate the existence and stability of equilibrium points for the aver-aged system and thereby study the periodic orbits and a numerical simulation is presentedas prescribed parameters in Section 5. Finally, we summarize our results in Section 6.2 . Coherent structure we consider uniformly propagating coherent structures with the ansatz ψ ( t, x ) = R ( x ) exp( i [Θ( x ) − µt ]) , (2.1)where R ( x ) ∈ R gives the amplitude dynamics of the condensate wave function, θ ( x ) deter-mines the phase dynamics, and the “chemical potential” µ , defined as the energy it takesto add one more particle to the system, is proportional to the number of atoms trapped inthe condensate. When the (temporally periodic) coherent structure (2.1) is also spatiallyperiodic, it is called a modulated amplitude wave (MAW) [16, 17]. Similarly, a solutionof the equation (1.1) with the (temporally periodic) coherent structure (2.1) is called a quasi-periodic modulated amplitude wave (QMAW) if it is also spatially quasi-periodic.Inserting (2.1) into (1.1), we obtain the following two couple nonlinear ordinary differ-ential equations R ′′ + δR − c R + εg ( x ) R + εV ( x ) R = 0 , (2.2)Θ ′′ + 2Θ ′ R ′ /R = 0 ⇒ Θ ′ ( x ) = cR , (2.3)where εg ( x ) := − ˜ g ( x ) , εV ( x ) := − ˜ V ( x )and the integration constant c , determined by the velocity and number density, plays therole of “angular momentum” [18].In case of c = 0, the phase of the condensate wave function (standing wave) is trivial andconstant. In the general case, c = 0, the system (2.2) becomes more complicated and thephase is no longer constant [19]. Even the amplitude R ( x ), a solution of (2.2), is T -periodic,the corresponding condensate wave function ψ ( x, t ) may be not periodic, but quasi-periodic,with respect to the spatial variable x [20].
3. Averaging theorem
Rewrite equation (2.2) in the planar equivalent form R ′ = SS ′ = − δR + c R − εg ( x ) R − εV ( x ) R. (3.1)Generally, averaging method involves two steps: transforming to standard form; solving theaveraging equation. In order to proceed we need to transform (3.1) to a standard form forthe method of averaging. 3 emma 3.1. Under the transformation
Ψ : T × (cid:0) q c δ , + ∞ (cid:1) → (0 , + ∞ ) × R defined by R = ρ s cos ( √ δx + θ ) + c δρ sin ( √ δx + θ ) S = ρ √ δ (cid:18) c δρ − (cid:19) cos( √ δx + θ ) sin( √ δx + θ ) s cos ( √ δx + θ ) + c δρ sin ( √ δx + θ ) , system (3.1) changes into a new system ρ ′ = ε (cid:26) g ( x ) √ δ ρ (cid:20)
14 (1 + c δρ ) sin 2( √ δx + θ ) + 18 (1 − c δρ ) sin 4( √ δx + θ ) (cid:21) + ρ √ δ V ( x ) sin 2( √ δx + θ ) (cid:27) θ ′ = ε (cid:26) g ( x )( δρ + c )8 δ ρ (cid:0) √ δx + θ ) (cid:1) + g ( x )( δ ρ + c )2 δ ρ ( δρ − c ) cos 2( √ δx + θ )+ 12 δ V ( x ) (cid:18) δρ + c δρ − c cos 2( √ δx + θ ) (cid:19)(cid:27) (3.2) with the new coordinates ( θ, ρ ) in the half-plane T × (cid:0) q c δ , + ∞ (cid:1) . The transformation Ψ arises from the variation of constant by using the solutions of theunperturbed system ( ε = 0), and ρ plays the role of “energy”. When taking the integrationconstant c = 0, the transformation Ψ is the usual change of polar coordinates in the halfplane. The proof the Lemma 3.1 follows from the basic computation (maybe lengthy), andit can be found in [20].Now write the T -periodic functions g ( x ) , V ( x ) as the Fourier series g ( x ) = g + ∞ X k =1 ( α k sin 2 k √ δx + β k cos 2 k √ δx ) , (3.3) V ( x ) = v + ∞ X k =1 ( a k sin 2 k √ δx + b k cos 2 k √ δx ) . (3.4)After inserting (3.3) and (3.4) into (3.2) and then multiplying the right side of (3.2) by 1 /T T , we obtain the averaged system ρ ′ = ε (cid:26) δ ¯ ρ + c δ √ δ ¯ ρ A sin(2¯ θ + φ ) + ¯ ρ √ δ B sin(2¯ θ + φ ) (cid:27) := εF (¯ θ, ¯ ρ ) θ ′ = ε (cid:26) g ( δ ¯ ρ + c )8 δ ¯ ρ + ( δ ¯ ρ + c )4 δ ¯ ρ ( δ ¯ ρ − c ) A cos(2¯ θ + φ )+ v δ + 14 δ δ ¯ ρ + c δ ¯ ρ − c B cos(2¯ θ + φ ) (cid:27) := εF (¯ θ, ¯ ρ ) , (3.5)where A = q α + β , B = q a + b ,φ = arctan α β , ( φ = π · sign( φ ) , if β = 0) ,φ = arctan a b , ( φ = π · sign( a ) , if b = 0) . Theorem 3.1. [ Averaging theorem ]
There exists a c r , r ≥ , change of variables ρ = ¯ ρ + εw (¯ θ, ¯ ρ, x, ε ) , θ = ¯ θ + εw (¯ θ, ¯ ρ, x, ε ) with w , w T -periodic functions of x , transforming (3.2) into ( ¯ ρ ′ = εF (¯ θ, ¯ ρ ) + ε g (¯ θ, ¯ ρ, x, ε )¯ θ ′ = εF (¯ θ, ¯ ρ ) + ε g (¯ θ, ¯ ρ, x, ε ) (3.6) with g , g T -periodic functions of x . Moreover, (i) If ( θ ε ( x ) , ρ ε ( x )) and (¯ θ ( x ) , ¯ ρ ( x )) are solutions of the original system (3.2) and averagedsystem (3.5) respectively, with the initial value such that | ρ ε (0) − ¯ ρ (0) | + | θ ε (0) − ¯ θ (0) | = O ( ε ) , then | ρ ε ( x ) − ρ ( x ) | + | θ ε ( x ) − θ ( x ) | = O ( ε ) , for times x of order /ε . (ii) If P is an equilibrium point of (3.5) such that the corresponding Jacobian matrixhas no eigenvalue equal to zero, then (3.2) admits a T -periodic solution ( θ ε ( t ) , ρ ε ( t )) suchthat | ( ρ ε ( t ) , θ ε ( t )) − P | = O ( ε ) , for sufficiently small ε ; if P is a hyperbolic equilibriumpoint of (3.5) , then there exists ǫ > such that, for all < ε < ε , system (3.2) possessesa hyperbolic periodic orbits γ ε ( x ) = P + O ( ε ) of the same stability type as P . If ( θ ε ( x ) , ρ ε ( x )) ∈ W s ( γ ε ( x )) is a solution of system (3.2) lying in the stable manifoldof the hyperbolic periodic orbit γ ε ( x ) = P + O ( ε ) , (¯ θ ( x ) , ¯ ρ ( x )) ∈ W s ( P ) is a solution ofsystem (3.5) lying in the stable manifold of the hyperbolic equilibrium point P and | ( θ ε (0)) , ρ ε (0) − (¯ θ (0) , ¯ ρ (0)) | = O ( ε ) , then | ( θ ε ( x ) , ρ ε ( x )) − (¯ θ ( x ) , ¯ ρ ( x )) | = O ( ε ) , for x ∈ [0 , + ∞ ) . Similar results apply to solutions lying in the instable manifold on theinterval x ∈ ( −∞ , . (iv) If P is an equilibrium point of system (3.5) and there exists a neighborhood U ( r ; P ) of P (with radius r and center P ) such that there is not another equilibrium point in theclosure of U and deg( F, U, P ) = 0 with F = ( F , F ) . Then for | ε | > sufficiently small, there exist a T -periodic solution ϕ ε ( x ) of system (3.2) such that ϕ ε ( · , ε ) → P as ε → . Proof.
The proof of (i)-(iii) follows directly from [21] or [22]. The proof of (iv) is based aframework of coincidence degree theory. Without loss of generality, we assume P = 0. Wedefine homotpoy operator H : C ([0 , T ] , R ) × [0 , → L ([0 , T ] , R ) by H (¯ θ, ¯ ρ, λ ) := λεF (¯ θ, ¯ ρ ) + (1 − λ ) ε G (¯ θ, ¯ ρ, x, ε ) , where G := ( g , g ). According to [23, Ch. VI] , H is L -compact on Ω × [0 , T ], where Ω is abounded open set of C ([0 , T ] , R ) defined byΩ := { (¯ θ, ¯ ρ ) ∈ C ([0 , T ] , R ) : k (¯ θ, ¯ ρ ) k < r } . We remark that (¯ θ, ¯ ρ ) ∈ Ω is a T -periodic solution of system (3.6) if and only if (¯ θ, ¯ ρ ) isa solution of L (¯ θ, ¯ ρ ) = H (¯ θ, ¯ ρ,
0) in Ω. Since there is not another equilibrium point in theclosure of U , we let M = min (¯ θ, ¯ ρ ) ∈ ∂U | F (¯ θ, ¯ ρ ) − F (¯ θ, ¯ ρ ) | > ,M ( ε ) = ε max {| g (¯ θ, ¯ ρ, x, ε ) − g (¯ θ, ¯ ρ, x, ε ) | : (¯ θ, ¯ ρ, x, ε ) ∈ U × [0 , T ] } with M ( ε ) → ε →
0. We also assume M ( ε ) >
0, otherwise P is a solution of system(3.6) and the result is proved.First, we claim that, for each ε ∈ ( − ε , ∪ (0 , ε ) with ε = max ε ∈ [ − , { M /M ( ε ) } ,there exists no solution (¯ θ, ¯ ρ ) ∈ ∂ Ω for the operator equation L (¯ θ, ¯ ρ ) = H (¯ θ, ¯ ρ, λ ) , λ ∈ (0 , . (3.7)6n fact, if (¯ θ, ¯ ρ ) ∈ ∂ Ω is a solution of (3.7), then there exists ξ ∈ [0 , T ] such that¯ θ ′ ( ξ ) + ¯ ρ ′ ( ξ ) = 0 , k (¯ θ, ¯ ρ ) k = max x ∈ [0 ,T ] q ¯ θ ( x ) + ¯ ρ ( x ) = q ¯ θ ( ξ ) + ¯ ρ ( ξ )and ¯ θ ′ ( ξ ) ¯ ρ ′ ( ξ ) ≤ . Thus, it follows that0 = | ¯ θ ′ ( ξ ) + ¯ ρ ′ ( ξ ) |≥ | ε | · | F (¯ θ, ¯ ρ ) − F (¯ θ, ¯ ρ ) | − ε | g (¯ θ, ¯ ρ, x, ε ) − g (¯ θ, ¯ ρ, x, ε ) |≥ | ε | M − εM ( ε ) > , which is a contradiction.Without loss of generality, we suppose that L (¯ θ, ¯ ρ ) = H (¯ θ, ¯ ρ, λ ) , (¯ θ, ¯ ρ ) ∈ ∂ Ω (3.8)holds for λ ∈ [0 , θ, ¯ ρ ) ∈ ∂ Ω. Thus, we can apply thehomotopy property of the coincidence degree and obtain | D L ( L − H ( · , , Ω) | = | D L ( L − H ( · , , Ω) | = | deg( F, Ω ∩ R , | 6 = 0 . Hence, by the existence property of the coincidence degree, there is (¯ θ, ¯ ρ ) ∈ Ω such that L (¯ θ, ¯ ρ ) = H (¯ θ, ¯ ρ, . Then (¯ θ, ¯ ρ ) is a T -periodic solution of (3.6). Thus, owing to the changeof variables in this theorem, there is a T -periodic solution ( θ, ρ ) for system (3.5).We remark that the proof of part (iv) of Theorem 3.1 does not need the smoothnesscondition upon F . So, it is convenient to deal with the existence of periodic solutions fornonlinear systems with loss of smoothness by Theorem 3.1.
4. Periodic orbits and stability
To study the dynamics of MAWs or QMAWs for system (1.1), we must investigate thebehavior of the periodic orbits for system (3.1) including the existence and stability. Ac-cording to the method of averaging, the equilibrium point of the averaged system determinesthe properties of the periodic orbit of the corresponding perturbed system. For example,the equilibrium point with its eigenvalue of linearization nonzero implies that there exists atleast one periodic orbit; in addition, if the equilibrium point is hyperbolic, then the periodicorbit has the same type of stability as the equilibrium point, for sufficiently small parameter ε . 7n order to find periodic orbits of system (3.2), it is sufficient to find equilibrium pointsof the averaged system (3.5). In the following, we will discuss the existence and stability ofthe equilibrium points for the averaged system (3.5). For simplification, we assume that φ = π + φ , φ = − π/ , g = A/ , v = − B/ . Recalling system (3.5), together with the assumption, we have the averaged system ρ ′ = ε (cid:26) A √ δ ¯ ρ (cid:16) ¯ ρ − BA ¯ ρ + c δ (cid:17) sin(2¯ θ + φ ) (cid:27) θ ′ = ε ( δ ¯ ρ · ( δ ¯ ρ − c δ ) (cid:18) A ρ − c δ ) − B ¯ ρ ( ¯ ρ − c δ )+ h A (cid:0) ¯ ρ + c δ (cid:1) − B ¯ ρ (cid:0) ¯ ρ + c δ (cid:1)i cos(2¯ θ + φ ) (cid:19)(cid:27) . (4.1)Notice that there exists a constant c > c ∈ (0 , c ), the equation¯ ρ − BA ¯ ρ + c δ = 0 (4.2)has at least two real roots ρ , = ± s BA + r B A − c δ ∈ ( −∞ , − p c /δ ) ∪ ( p c /δ, + ∞ ) . Moreover, equation (4.2) implies that A ρ − c δ ) − B ¯ ρ ( ¯ ρ − c δ ) = 0 . Thus, we can find four equilibrium points as follows P , : (cid:16) ± s BA + r B A − c δ , kπ + π − φ (cid:17) ,P , : (cid:16) ± s BA + r B A − c δ , kπ − π − φ (cid:17) , k ∈ Z . The eigenvalues of the equilibrium points P , and P , are given by λ (1)1 , = εA √ δ r B A − c δ > , λ (2)1 , = − εA ( ρ , − c δ )4 δρ , < λ (1)3 , = − λ (1)1 , , λ (2)3 , = − λ (2)1 , , respectively. So, the equilibrium points P , and P , arehyperbolic, and as a consequence persist as periodic orbits for system (3.2); in addition,these periodic orbits are instable. 8f ¯ θ = kπ − φ /
2, to find the equilibrium points, one will solve the following algebraicequation A ρ − c δ ) − B ¯ ρ ( ¯ ρ − c δ ) + A (cid:0) ¯ ρ + c δ (cid:1) − B ¯ ρ (cid:0) ¯ ρ + c δ (cid:1) = 0 . (4.3)Equation (4.3) has at least two roots ρ , = ± r B A + o ( c ) ∈ ( −∞ , − p c /δ ) ∪ ( p c /δ, + ∞ ) , for sufficiently small integration constant c > (cid:1) (cid:1) (cid:1) (cid:1) / d c / d c - (cid:1) fp -- r (cid:1) (cid:1) (cid:1) (cid:1) (cid:0) f- (cid:2) fp - (cid:3) fp - (cid:1) fp -- q (cid:1) Figure 1: The phase portrait associated with the averaged system (4.1). All the equilibrium points persistas periodic orbits for system (3.2).
If ¯ θ = kπ + π/ − φ /
2, the algebraic equation f ( ρ ) := ρ − BA ρ + 3 c δ = 0needs to be solved. Note that f (cid:16) ± r c δ (cid:17) = 4 c δ − BA (cid:0) c δ (cid:1) / < f ( ±∞ ) = + ∞ , (4.4)9or sufficiently small positive constant c . By the mean value theorem, equation (4.4) hastwo roots ρ , such that ρ , = ± r c BδA + o ( c ) ∈ ( −∞ , − p c /δ ) ∪ ( p c /δ, + ∞ ) , for sufficiently small c . As a consequence, four equilibrium points of the averaged system(4.1) are obtained as follows P , : (cid:16) ± r B A + o ( c ) , kπ − φ (cid:17) ,P , : (cid:16) ± r c BδA + o ( c ) , kπ + π − φ (cid:17) , k ∈ Z . The eigenvalues of the linearization at the equilibrium points P , and P , are given by λ (1 , , = ± iε s B ( B A − c δ )3 δ ( ρ , − c δ )and λ (1 , , = ± ε s A c δ / ρ , ( ρ , − c δ ) (cid:16) BA ρ , − ρ , − c δ (cid:17) , respectively. The equilibrium points P , imply that that two instable periodic orbit ofsystem (3.2) exist; while the equilibrium points P , are nonlinear centers, and also persistas periodic orbits for system (3.2). The phase portrait for system (4.1) is given in Figure1. Since the periodic orbits corresponding to the equilibrium points P , are not hyperbolic,one can not conclude their stability. This question is left open for further study.In summary, for the equilibrium P i ( i = 1 , · · · , , , ε > < ε ≤ ε , system (3.2) possesses a unique hyperbolic periodic orbits γ ε ( x ) = P i + O ( ε ),which is instable.We also remark that, if ( θ ε ( x ) , ρ ε ( x )) ∈ W s ( γ ε ( x )) is a solution of system (3.2) lying inthe stable manifold of the hyperbolic periodic orbit γ ε ( x ) = P i + O ( ε ), (¯ θ ( x ) , ¯ ρ ( x )) ∈ W s ( P )is a solution of system (4.1) lying in the stable manifold of the hyperbolic equilibrium P i ( i =1 , · · · , , ,
8) and | ( θ ε (0) , ρ ε (0)) − (¯ θ (0) , ¯ ρ )(0) | = O ( ε ), then | ( θ ε ( x ) , ρ ε ( x )) − (¯ θ ( x ) , ¯ ρ )( x ) | = O ( ε ), for all x ∈ [0 , + ∞ ). Similar results apply to solutions lying in the instable manifoldon the interval x ∈ ( −∞ , c > c also can be taken on aopen interval (0 , ¯ c ), for some positive constant ¯ c . By continuous dependence of solutionswith respect to the parameters, there is a connected set C of T -periodic solutions for system(3.2) and then for system (3.1). Since ψ ( t, x ) = R ( x )exp i [Θ( x ) − µt ]= R ( x ) (cid:0) cos[ ¯Θ( x ) + νx − µt ] + i sin[ ¯Θ( x ) + νx − µt ] (cid:1) , ν = 1 T Z x + Tx cR ( ξ ) d ξ and ¯Θ( x ) = Θ( x ) − ν is a T -periodic function with zero mean value, whether ψ ( t, x ) is aMAW or QMAW depends on the choosing of the integration constant c . Precisely, If 2 π/ν and T are rationally related, then ψ ( x, t ) is a MAW; if 2 π/ν and T are rationally irrelevant,then ψ ( x, t ) is not periodic but quasi-periodic, which is corresponding to a QMAW with thefrequency ω = h π/ν, T i .
5. Numerical simulation
To demonstrate the process of averaging to BECs, a specific example of numerical com-putation is given in the following. We take g ( x ) = 12 −
32 sin 2 x,V ( x ) = 2 sin 2 x − δ = 1 , c = 0 . , ε = 0 .
01. Obviously, g ( x ) crosses zero. We can findequilibrium points for system (4.1) in the ( θ, ρ )-coordinates as follows P , : ( kπ, ± . , P , : ( kπ + π , ± . ,P , : ( kπ + 3 π , ± . , P , : ( kπ + π , ± . .P , are nonlinear centers with eigenvalues of the linearization λ (1 , = λ (1 , = ± iε .Using the transformation Ψ, these equilibrium points in the ( R, S )-coordinates with x = 0 are given by˜ P , : ( ± . , , ˜ P , : ( ± . , , ˜ P , : ( ± . , ± . , ˜ P , : ( ± . , ∓ . . We plot the solutions of system (3.1) starting from ˜ P i , i = 1 , , · · · ,
8, according to theaveraged theorem, which are a good approximation to the periodic orbits, see Figure 2.
6. Conclusion
In conclusion, we have presented first-order averaging theorem in the periodic case fordynamics of MAWs (or QMAWs) in collisionally inhomogeneous BECs. The transformedsystem is non-Hamiltonian, and we indicate how the averaging theorems can be used toprove the existence and stability of periodic solutions. The questions as mentioned in theintroduction have been answered. When the sufficiently small scattering length a ( x ) varies11 − − − R x Figure 2: The solutions portrait of system (3.1) with initial value ( R (0) , R ′ (0)) = ˜ P i , i = 1 , , · · · ,
8. Here,we take g ( x ) = − sin 2 x, V ( x ) = 2 sin 2 x − δ = 1 , c = 0 . , ε = 0 .
01. These solutionsare a good approximation to the periodic orbits for x of order 1 /ε . The solutions with solid lines are instable. periodically in spatial variable x and crosses zero, infinitely many (positive measure set)MAWs and QMAWs can be proved to exist by adjusting the integration constant c on someopen interval.A numerical approximation of periodic orbits is given for some prescribed parameters.We remark that, expanding at each equilibrium point and combining with multiple scaleperturbed theory, such as work in [6, 24–26], there may be a better approximation for eachcontinuation periodic orbit. However, we emphasis on the theory frame of averaging to studydynamics of collisionally inhomogeneous BECs.In the end, it should be remember that the asymptotic approximation are valid for small ε , but how small is usually a difficult problem. However, one advantage of averaging isobvious that it is set up for an easy return to the original variables. Acknowledgements
This work is supported by the National Natural Science Foundation of China (10871142)and Doctoral Fund of Ministry of Education of China (20070285002).
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