PHYSICAL REVIEW E 78, 036215 共2008兲
Controlling chaos of a Bose-Einstein condensate loaded into a moving optical Fourier-synthesized lattice R. Chacón,1 D. Bote,2 and R. Carretero-González3,*
1
Departamento de Física Aplicada, Escuela de Ingenierías Industriales, Universidad de Extremadura, Apartado Postal 382, E-06071 Badajoz, Spain 2 Departamento de Matemáticas, Escuela de Ingenierías Industriales, Universidad de Extremadura, Apartado Postal 382, E-06071 Badajoz, Spain 3 Nonlinear Dynamical Systems Group,† Computational Science Research Center,‡ and Department of Mathematics and Statistics, San Diego State University, San Diego, California 92182-7720, USA 共Received 5 July 2008; published 12 September 2008兲 We study the chaotic properties of steady-state traveling-wave solutions of the particle number density of a Bose-Einstein condensate with an attractive interatomic interaction loaded into a traveling optical lattice of variable shape. We demonstrate theoretically and numerically that chaotic traveling steady states can be reliably suppressed by small changes of the traveling optical lattice shape while keeping the remaining parameters constant. We find that the regularization route as the optical lattice shape is continuously varied is fairly rich, including crisis phenomena and period-doubling bifurcations. The conditions for a possible experimental realization of the control method are discussed. DOI: 10.1103/PhysRevE.78.036215
PACS number共s兲: 05.45.Ac, 03.75.Lm, 03.75.Kk
II. ANALYTICAL TREATMENT
I. INTRODUCTION
The combination of a Bose-Einstein condensate 共BEC兲 共see, e.g., the reviews 关1,2兴兲 and an optical lattice provides a unique scenario for exploring new quantum phenomena and their classical manifestations. In particular, the presence of chaos in BECs has become a subject of great interest, partly because of its technological implications. As is well known, a cause of the quantum suppression of chaos is that the 共classical兲 chaos generally appears in nonlinear systems while the corresponding quantum Schrödinger equations are linear 关3兴. The possible existence of 共classical兲 chaos in BECs comes from the fact that the dynamics of dilute atomic BECs, close to zero temperature, can be well approximated by a nonlinear Schrödinger equation—the so-called Gross-Pitaevskii equation 共GPE兲 关1,2,4,5兴. Indeed, diverse manifestations of temporal 关6,7兴, spatial 关8,9兴, and spatiotemporal 关10,11兴 chaos in BECs have been reported recently, including the process of BEC collapse 关12,13兴 and open BECs 关14兴. It would therefore seem clear that a fundamental problem to address when considering applications of BECs is the prediction and control of chaos. A recent study in this respect, for example, showed the suppressive effects of dissipation and the velocity of a traveling optical lattice 关11兴. The aim of the present work is to show that the dissipative chaotic dynamics of a BEC loaded into a moving optical lattice exhibits great sensitivity to small changes of the lattice shape and can thus be reliably controlled by Fouriersynthesizing suitable lattice shapes. It is worth mentioning that this technique has been successfully used to control quantum transport of an atomic BEC 关15兴. Moving optical lattices 关16–18兴 and dissipative effects 关6,19,20兴 have also been studied recently.
*http://www.rohan.sdsu.edu/⬃rcarrete/ †
http://nlds.sdsu.edu/ http://www.csrc.sdsu.edu/
‡
1539-3755/2008/78共3兲/036215共6兲
Let us consider the case of a quasi-one-dimensional 共1D兲 BEC that is tightly confined in two transverse directions 共the so-called cigar-shaped condensate兲 described by the following 1D GPE: ប共i + ␥兲
ប 2 2 =− + g0兩兩2 + V0sn2共 ;m兲 , 共1兲 t 2ma x2
where V0sn2共 ; m兲 is the periodic moving optical lattice; = x + ␦t / 共2k兲 is the space-time variable with ␦ being the frequency difference between the two Fourier-synthesized counter-propagating laser beams and k = 2 / the laser wave vector which determines the velocity of the traveling lattice as vL = ␦ / 共2k兲; ma is the atomic mass; g0 = 4ប2a / ma characterizes the attractive 共a ⬍ 0 being the s-wave scattering length兲 interatomic interaction strength; = 2K共m兲k / with K共m兲 being the complete elliptic integral of the first kind; sn共· ; m兲 is the Jacobian sine elliptic function of parameter m 苸 关0 , 1兴; and ប␥ / t is a dissipation term 关20–23兴. The static version of the elliptic optical lattice in Eq. 共1兲 has been studied extensively 关24–30兴. For the sake of simplicity, we shall here concentrate only on the traveling-wave solutions of Eq. 共1兲 in the form of a Bloch-like wave
= 共兲exp关i共␣x + t兲兴,
共2兲
where ␣ and  are real constants to be determined. Note that this choice implies that the traveling wave 共兲 has the same velocity as the elliptic optical lattice. After inserting Eq. 共2兲 into Eq. 共1兲 and normalizing the function by the factor k3/2 and the variable by the factor 2K共m兲 / , one straightforwardly obtains the ordinary differential equation
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©2008 The American Physical Society
PHYSICAL REVIEW E 78, 036215 共2008兲
CHACÓN, BOTE, AND CARRETERO-GONZÁLEZ
p x ; m
1 0.8 0.6
0.2 Error 0.1 0 0
0.4 0.2 0
0.2
0.4
x
0.6
0.8
1
FIG. 1. Potential function p共x ; m兲 关cf. Eq. 共5兲兴 for m = 0 共thick line兲, 0.995 共medium line兲, and 1 − 10−14 共thin line兲. The quantities plotted are dimensionless.
p x ; m, p(2)x ; m
d ˜ d + ␥v − 共 + ˜␣2兲 − g兩兩2 d d2
冉
冊
d ˜ + ˜V sn2关2K共m兲/ ;m兴 , + ␥ 0 d
冋
冉 冊册
d 2R v2 ˜ ;m + V0 p 2 − 4 d 2
R − gR3 = − ␥v
dR , d
p共x;m兲 ⬅ sn2关4K共m兲x;m兴.
共5兲
When m = 0, then p( / 共2兲 ; m) = sin2共兲 = sin2共k兲, i.e., one recovers the previously studied case of a pure trigonometric optical lattice 关11兴, while at the limiting value m = 1 the elliptic optical lattice becomes constant except on a set of points of Lebesgue measure zero, i.e., one recovers a case with no moving periodic lattice where spatially chaotic steady states are no longer possible 共see Fig. 1兲. Therefore, starting from a chaotic state at m = 0, one would expect to observe a regularization of the traveling-wave steady states as m → 1 while keeping the remaining parameters constant. To keep the analysis close to a possible experimental realization, we expand the elliptic optical lattice in the form ⬁
sn2共 ;m兲 = 兺 b j−1 sin2 j=1
冉 冊
j , 2K
共6兲
where the first Fourier coefficients are approximately given by 共see the Appendix for details兲
m
0.2 0.4 0 4
0.6 0.4 0.2 0
0.1
0.2
x
0.3
0.4
0.5
FIG. 2. 共Color online兲 Top panel: Relative error 关p共x ; m兲 − p共2兲共x ; m兲兴 / p共x ; m兲 with p as in Eq. 共5兲 and p共2兲 corresponding to its two-term Fourier expansion as in Eq. 共8兲. Shape parameter in the range m 苸 关0 , 0.9兴. Bottom panel: Functions p共x ; m = 0.9兲 共thick line兲 and p共2兲共x ; m = 0.9兲 共thin line兲, showing their proximity. The quantities plotted are dimensionless.
b0 ⯝
42q共1 − q + q2 − q3 + q4兲 , mK2共1 − q − q5 + q6兲
b1 ⯝
共4兲
where we introduce the potential
0.4 0.2 x 0.3
0.8
共3兲 where g = 8ak ⬍ 0 共a ⬍ 0 since we are considering an attractive condensate兲, = , v = 2mavL / 共បk兲, ˜␣ = ␣ / k, ˜ = ប / Er, and ˜V0 = V0 / Er, with Er = ប2k2 / 共2ma兲 being the recoil energy. Following Ref. 关11兴, we express the complex function 共兲 in the form 共兲 = R共兲ei共兲, and consider the simple situation where the phase 共兲 has a linear dependence on the dimensionless space-time variable: d / d = −˜ / v = −共v / 2 + ˜␣兲 关36兴. Note that R2 represents the particle number density, whose chaotic dynamics we wish to suppress by reshaping the optical lattice potential. In this case, Eq. 共3兲 reduces to a damped, parametrically and anharmonically driven, Duffing equation for the amplitude R:
0.1
1
2
= − i 共v + 2˜␣兲
0.8 0.6
b2 ⯝
82q2共1 − q4兲 , mK2共1 − q3 − q5 + q8兲
冉
冊
4 2q 3 2共1 − q3兲2 1 + , mK2共1 − q3兲2 共1 − q兲共1 − q5兲
共7兲
over the range 0 艋 m ⱗ 0.99, and where K = K共m兲, and q = q共m兲 ⬅ exp关−K共1 − m兲 / K共m兲兴 is the nome 关31兴. It will be useful to define the truncated Fourier expansion of order k for the elliptic optical lattice potential as k
p 共 ;m兲 ⬅ 兺 b j−1 sin2 共k兲
j=1
冉 冊
j , 2K
共8兲
such that p( / 共2兲 ; m) = limk→⬁ p共k兲共 ; m兲. To obtain an analytical estimate of the chaotic threshold in parameter space, let us assume in the following that the dissipation term and the optical lattice potential in Eq. 共4兲 are small-amplitude perturbations of the underlying integrable two-well Duffing equation 关32兴, i.e., they satisfy the Melnikov method 共MM兲 requirements 关32,33兴, and that, in the limiting case m = 0, the perturbed Duffing equation 共4兲 exhibits homoclinic chaos. Figure 2 depicts a comparison between the full potential 共5兲 and its truncated Fourier expansion 共8兲, when only the two first terms of the approximation for the elliptic parameter m = 0.9 are retained. Figure 3 depicts the same comparison as in Fig. 2 but for a larger elliptic param-
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PHYSICAL REVIEW E 78, 036215 共2008兲
M⫾ n 共兲 ⬅ 0.2
Error
0.1 0 0
0.1
0.2 x
0 ⬅
0.8 0.6 0.4 m 0.3
0.4 0 4
p x ; m, p(3) x ; m
4 sinh共2/v兲 . sinh共4/v兲
共11兲
Next, we study the appearance of simple zeros of M ⫾ n 共兲 with the constraint 0 ⬎ 1 共i.e., there exists homoclinic chaos at m = 0兲 by considering the zeros of the quartic polynomial in z which arises from Eq. 共10兲 after the substitution z = sin共2兲. Solving this quartic equation, one straightforwardly obtains that a necessary condition for M ⫾ n 共兲 关and hence for M ⫾共兲兴 to present simple zeros is
1 0.8 0.6 0.4 0.2
20关1442b20b21 + 11524b41 + 2共b20 − 42b21兲2兴2 0
0.1
0.2
x
0.3
0.4
0.5
艋 4关482b21 + 20共b20 − 42b21兲2兴3 ,
FIG. 3. 共Color online兲 Top panel: Relative error 关p共x ; m兲 − p共3兲共x ; m兲兴 / p共x ; m兲, with p as in Eq. 共5兲 and p共3兲 corresponding to its three-term Fourier expansion as in Eq. 共8兲. Shape parameter in the range m 苸 关0 , 0.99兴. Bottom panel: Functions p共 ; m = 0.99兲 共thick line兲 and p共3兲共 ; m = 0.99兲, 共thin line兲 showing their proximity. The quantities plotted are dimensionless.
eter m = 0.99 when retaining three terms in the expansion 共8兲. It is clear that using only the first two terms in the expansion is enough to accurately capture the main deviation of the Jacobian elliptic potential from a purely trigonometric one for values of the elliptic parameter up to m ⯝ 0.9. Therefore, for the sake of simplicity, we only retain the two first terms of the expansion 共6兲 and, after some simple algebraic manipulation, the application of the MM to Eqs. 共4兲–共7兲 yield the Melnikov function
冉 冊 冉 冊
8b1˜V0 4 + csch sin共4兲, g v
共9兲
where the plus 共minus兲 sign corresponds to the right 共left兲 homoclinic orbit of the unperturbed Duffing equation. Since b0共m = 0兲 = 1, b1共m = 0兲 = 0 关cf. Eq. 共7兲兴, the hypothesis of homoclinic chaos for a single-humped 共pure兲 trigonometric optical lattice 共m = 0兲 implies that 2b0˜V0 csch共2 / v兲 ⬎ ␥v4 / 6, which is a necessary condition for M ⫾共兲 to present simple zeros at m = 0. Thus, to analyze the suppressive effect of an elliptic optical lattice 共m ⬎ 0兲 we shall consider in the following the normalized Melnikov function M⫾ n 共兲 = 1 + 0 sin共2兲 + 0共b0 − 1兲sin共2兲 + 0b1 sin共4兲, 共10兲
共12兲
where the equality provides the boundary function in the parameter space, and hence one obtains the chaotic threshold function U共m , v兲 such that ˜V0 / ␥ 艌 U共m , v兲 provides a necessary condition for the perturbed Duffing equations 共4兲–共7兲 to exhibit homoclinic chaos 关34兴. Recalling that the Melnikov function 共9兲 is approximately valid over the range 0 艋 m ⱗ 0.9 and that b1 Ⰶ b0 over this range 关cf. Eq. 共7兲兴, one can drop the terms proportional to any power of b1 in Eq. 共12兲 to finally obtain an approximate necessary condition for the perturbed Duffing equations 共4兲–共7兲 to exhibit homoclinic chaos: ˜ ˜V 0 ˜ 共m, v兲 ⬅ U0共m = 0, v兲 , 艌U ␥ b0共m兲
共13兲
4 ˜ 共m = 0, v兲 ⬅ v sinh共2/v兲 U 0 12
共14兲
where
2 ␥v4 2b0˜V0 + csch sin共2兲 M ⫾共 兲 = 6g g v
where
12˜V0 csch共2/v兲 , ␥v4
⬅
0.2
M ⫾共 兲 , ␥v4/共6g兲
is the chaotic threshold function associated with a pure trigonometric optical lattice 共m = 0兲 关11兴. Remarkably, the simplicity of condition 共13兲 does not imply, however, a significant loss of accuracy with respect to the condition ˜V0 / ␥ 艌 U共m , v兲. Indeed, Fig. 4 indicates that ˜ the differences between the chaotic threshold functions U and U are noticeable only for values of the shape parameter very close to 1. The top panel of Fig. 5 shows a plot of the ˜ 共m , v兲. One sees that U ˜ 共m dimensionless function U = const, v兲 presents a single minimum as a function of the dimensionless lattice velocity 共see, Fig. 5, bottom panel兲, ˜ 共m , v = const兲 presents a monotonically increasing while U behavior as a function of the shape parameter 共see Fig. 5, middle panel兲, as expected. Therefore, if one considers fixing the parameters 共V0, ␥, k, vL, and hence v兲 for the particle number density of the BEC to exhibit chaos at m = 0, then as m is increased a window of regular dynamics will appear, provided the initial chaotic state is sufficiently near the cha-
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0.04 Error 0.02
8 U’ 7 6 5
2.5 2
0 0.2
0.4 m
1.5 0.6
0.8 0 8
2.5 2 0.2
v
0.4 m 0.6
v
1.5 00.88
1
1
5.2 U’ m
U’ m , U m
6 5.8
5 4.8 4.6
5.6 5.4
0
0.2
0.4 m 0.6
0.8
5.2 0.92
0.94 m
0.96
0.98
5.4 5.2 U’ v
0.9
FIG. 4. 共Color online兲 Top panel: Relative error 关U共m , v兲 ˜ 共m , v兲兴 / U共m , v兲 关cf. Eqs. 共12兲 and 共13兲兴 for the parameters in the −U ranges m 苸 关0 , 0.99兴 and v 苸 关1 , 2.5兴. Bottom panel: Functions ˜ 共m , v = 2兲 共thin line兲 in the range m U共m , v = 2兲 共thick line兲 and U 苸 关0.9, 0.99兴. The quantities plotted are dimensionless.
otic threshold associated with the single-humped trigonometric optical lattice 关cf. Eq. 共14兲兴. It is worth noting that very similar quantitative predictions are obtained for a moving periodic optical lattice given by V0cn2共 ; m兲, where cn共· ; m兲 is the Jacobian elliptic cosine function of parameter m. Mathematically, this is because of the fundamental relationship cn2共· ; m兲 + sn2共· ; m兲 = 1 together with the fact that the periodic lattice acts as a parametric excitation in Eq. 共4兲 共and hence the Melnikov integral corresponding to the unity term of the fundamental relationship vanishes兲. Physically, one has indeed that in the limiting case m = 1 the elliptic optical lattice cn2共 ; m兲 vanishes except on a set of points of Lebesgue measure zero, i.e., one again recovers a case with no moving periodic lattice where chaotic dynamics is no longer possible. We next compare the MM analytical predictions with numerical results 共bifurcation diagrams兲, but with the added caveat that one cannot expect too good a quantitative agreement between the two kinds of findings because the MM is a perturbative method generally related to transient chaos, while bifurcation diagrams provide information solely concerning steady chaos. A typical example is shown in Fig. 6 where the particle number density R2 = 兩兩2 = 兩兩2 is plotted vs the shape parameter m for the experimental parameters 关18兴 ma = 23m p with m p the proton mass, 0 = 589 nm, ␥ = 0.05, ˜V = 2, and v = 3 ⫻ 10−2 m / s such that v = 2.03 and g = 0 L −0.75. Typically, the BEC traveling-wave steady state goes from a chaotic state which propagates in the direction of the motion of the optical lattice to a steady-state equilibrium
5 4.8 4.6 4.4 1
1.4
v
1.8
2.2
FIG. 5. 共Color online兲 Plots of the chaotic threshold function ˜ 共m , v兲 关see Eqs. 共13兲 and 共14兲兴. Top panel: U ˜ vs m and v. Middle U ˜ panel: U vs m for v = 1.5 共thick line兲, 1.9 共medium line兲, and 2.03 ˜ vs v for m = 0 共thick line兲, 0.8 共medium 共thin line兲. Bottom panel: U line兲, and 0.9 共thin line兲. The quantities plotted are dimensionless.
associated with a static and uniform optical lattice as the shape parameter increases from 0 to 1. The progression of the steady states is characterized by the particle number density undergoing an inverse period-doubling route as the shape parameter is increased, which is preceded by diverse crises 共see Fig. 6, bottom panel兲. We found similar regularization routes for other sets of experimentally realizable parameters, and would therefore emphasize that the present reshaping-induced control method could be implemented in experiments. Indeed, the two-term-approximation potential p共2兲共 ; m兲 = b0 sin2共2 / 兲 + b1 sin2共4 / 兲 关cf. Eq. 共8兲兴 used in the above theoretical analysis can be obtained from the two Fourier-synthesized counterpropagating laser beams
036215-4
E1共r,t兲 =
b1/2 0 共m兲 Re共e−i共t−kx+/2兲兲 2 +
b1/2 1 共m兲 Re共e−i共t−2kx+/2兲兲, 2
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PHYSICAL REVIEW E 78, 036215 共2008兲
the traveling optical lattice shape while keeping the remaining parameters constant. Finally, we provided an explicit expression for the two Fourier-synthesized counterpropagating laser beams for possible experimental realization of the control method. APPENDIX: DERIVATION OF FORMULAS (6)
Using the Fourier expansion of sn共· ; m兲 关31兴, one has ⬁
⬁
42 sn 共 ;m兲 = 兺 兺 an共m兲al共m兲 mK2共m兲 n=0 l=0 2
冉
⫻sin FIG. 6. Bifurcation diagrams of particle number density R2 as a function of the shape parameter m for parameter values ␥ = 0.05, ˜V = 2, v = 2.03, and g = −0.75. Bottom panel shows the detail corre0 sponding to the range where crises appear. The quantities plotted are dimensionless.
E2共r,t兲 =
b1/2 0 共m兲 2 +
2
冊
with q = q共m兲 where an共m兲 ⬅ qn+1/2共1 − q2n+1兲−1 ⬅ exp关−K共1 − m兲 / K共m兲兴 being the nome 关31兴. After applying the trigonometric relationships 2 sin ␣ sin  = cos共␣ − 兲 − cos共␣ + 兲 and cos ␣ = 1 − 2 sin2共␣ / 2兲 to Eq. 共A1兲, one straightforwardly obtains Eq. 共6兲 with b0 =
Re共e−i共t+␦t+kx−/2兲兲
b1/2 1 共m兲
冊 冉
共2n + 1兲 共2l + 1兲 sin , 共A1兲 2K共m兲 2K共m兲
42 兵a2共m兲 − 2a1共m兲关a0共m兲 + a2共m兲兴 mK2共m兲 0 + O关a1共m兲a2共m兲兴其,
Re共e−i共t+2␦t+2kx−/2兲兲,
b1 =
where is the common polarization 关35兴. b2 =
III. SUMMARY
42 兵2a0共m兲关a1共m兲 − a2共m兲兴 + O关a0共m兲a2共m兲兴其, mK2共m兲
42 兵a2共m兲 + 2a0共m兲a2共m兲 + O关a0共m兲a2共m兲,a21共m兲兴其. mK2共m兲 1
In summary, we have discussed a reshaping-induced method to suppress the existence of spatially chaotic steadystate traveling waves of a BEC with attractive interatomic interaction loaded into a traveling optical lattice of variable shape. We demonstrated theoretically and numerically that traveling spatially chaotic steady states of the particle number density can be reliably suppressed by small changes of
Since an共m兲 = csch关共n + 1 / 2兲K共1 − m兲 / K共m兲兴 / 2, one sees that limn→⬁an共m兲 = 0, ∀m苸 关0,1关. Thus, for the purposes of the present work, Eq. 共A2兲 can be approximated by Eq. 共7兲 and the remaining coefficients bn with n ⬎ 2 are negligible over the range 0 艋 m ⱗ 0.99.
关1兴 F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 共1999兲. 关2兴 A. J. Leggett, Rev. Mod. Phys. 73, 307 共2001兲. 关3兴 M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics 共Springer-Verlag, New York, 1990兲. 关4兴 Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment, edited by P. G. Kevrekidis, D. J. Frantzeskakis, and R. Carretero-González, Springer Series on Atomic, Optical, and Plasma Physics, Vol. 45 共Springer, Berlin, 2008兲. 关5兴 R. Carretero-González, D. J. Frantzeskakis, and P. G. Kevrekidis, Nonlinearity, 21, R139 共2008兲. 关6兴 F. Kh. Abdullaev and R. A. Kraenkel, Phys. Rev. A 62, 023613 共2000兲. 关7兴 C. Lee, W. Hai, L. Shi, X. Zhu, and K. Gao, Phys. Rev. A 64, 053604 共2001兲.
关8兴 V. M. Eguiluz, E. Hernández-García, O. Piro, and S. Balle, Phys. Rev. E 60, 6571 共1999兲. 关9兴 G. Chong, W. Hai, and Q. Xie, Phys. Rev. E 71, 016202 共2005兲. 关10兴 A. D. Martin, C. S. Adams, and S. A. Gardiner, Phys. Rev. Lett. 98, 020402 共2007兲. 关11兴 G. Chong, W. Hai, and Q. Xie, Phys. Rev. E 70, 036213 共2004兲. 关12兴 V. S. Filho, A. Gammal, T. Frederico, and L. Tomio, Phys. Rev. A 62, 033605 共2000兲. 关13兴 H. Saito and M. Ueda, Phys. Rev. Lett. 86, 1406 共2001兲. 关14兴 P. Coullet and N. Vandenberghe, Phys. Rev. E 64, 025202共R兲 共2001兲. 关15兴 T. Salger, C. Geckeler, S. Kling, and M. Weitz, Phys. Rev. Lett. 99, 190405 共2007兲. 关16兴 P. A. Ruprecht, M. Edwards, K. Burnett, and C. W. Clark,
共A2兲
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CHACÓN, BOTE, AND CARRETERO-GONZÁLEZ Phys. Rev. A 54, 4178 共1996兲. 关17兴 P. Öhberg and S. Stenholm, J. Phys. B 32, 1959 共1999兲. 关18兴 J. H. Denschlag, J. E. Simsarian, H. Häffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. L. Rolston, and W. D. Phillips, J. Phys. B 35, 3095 共2002兲. 关19兴 I. Marino, S. Raghavan, S. Fantoni, S. R. Shenoy, and A. Smerzi, Phys. Rev. A 60, 487 共1999兲. 关20兴 A. Aftalion, Q. Du, and Y. Pomeau, Phys. Rev. Lett. 91, 090407 共2003兲. 关21兴 L. P. Pitaevskii, Sov. Phys. JETP 35, 282 共1959兲. 关22兴 M. Tsubota, K. Kasamatsu, and M. Ueda, Phys. Rev. A 65, 023603 共2002兲. 关23兴 K. Kasamatsu, M. Tsubota, and M. Ueda, Phys. Rev. A 67, 033610 共2003兲. 关24兴 J. C. Bronski, L. D. Carr, B. Deconinck, and J. N. Kutz, Phys. Rev. Lett. 86, 1402 共2001兲. 关25兴 L. D. Carr, J. N. Kutz, and W. P. Reinhardt, Phys. Rev. E 63, 066604 共2001兲. 关26兴 J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, Phys. Rev. E 63, 036612 共2001兲. 关27兴 J. C. Bronski, L. D. Carr, R. Carretero-González, B. Deconinck, J. N. Kutz, and K. Promislow, Phys. Rev. E 64, 056615
共2001兲. 关28兴 B. Deconinck, J. N. Kutz, M. S. Paterson, and B. W. Warner, J. Phys. A 36, 5431 共2003兲. 关29兴 N. A. Kostov, V. Z. Enolskii, V. S. Gerdjikov, V. V. Konotop, and M. Salerno, Phys. Rev. E 70, 056617 共2004兲. 关30兴 V. S. Gerdjikov, B. B. Baizakov, M. Salerno, and N. A. Kostov, Phys. Rev. E 73, 046606 共2006兲. 关31兴 See, e.g., L. M. Milne-Thomson, in Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun 共Dover, New York, 1972兲. 关32兴 See, e.g., J. Guckenheimer and P. J. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 共Springer, Berlin, 1983兲. 关33兴 V. K. Melnikov, Trans. Mosc. Math. Soc. 12, 1 共1963兲. 关34兴 An explicit expression for U共m , v兲 is available by using MATHEMATICA but its size and algebraic complexity prevent us from showing it easily. 关35兴 G. Grynberg and C. Robilliard, Phys. Rep. 355, 335 共2001兲. 关36兴 Note that more complicated 共兲 dependences are also possible; however, they are likely to yield a system of differential equations for the amplitude 关the equivalent of Eq. 共4兲兴 that cannot be analytically tractable using Melnikov’s method.
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