Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

A good approximation of modulated amplitude wavesin Bose–Einstein condensates

1007-5704/$ - see front matter � 2014 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.cnsns.2013.12.034

⇑ Corresponding author at: School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541003, PR China+86 07733939803.

E-mail address: qhuailiu@gmail.com (Q. Liu).

Leilei Jia a, Qihuai Liu b,c,⇑, Zhongjun Ma b

a Department of Electrical Engineering, Guilin College of Aerospace Technology, Guilin 541002, PR Chinab School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541003, PR Chinac School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 November 2013Received in revised form 24 December 2013Accepted 30 December 2013Available online 9 January 2014

Keywords:Periodic solutionsModulated amplitude wavesGross–Pitaevskii equationHamiltonian system

In this paper, we present a perturbation method that utilizes Hamiltonian perturbationtheory and averaging to analyze spatio-temporal structures in Gross–Pitaevskii equationsand thereby investigate the dynamics of modulated amplitude waves (MAWs) in quasi-one-dimensional Bose–Einstein condensates with mean-field interactions. A good approx-imation for MAWs is obtained. We also explore dynamics of BECs with the nonresonantexternal potentials and scatter lengths varying periodically in detail using Hamiltonianperturbation theory and numerical simulations.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

The dynamics of Bose–Einstein condensates (BECs) is a fundamental phenomenon connected to superfluidity and super-conductivity in liquid helium [1]. In the mean-field approximation, and at sufficiently low temperatures, the dynamics ofmatter waves in BECs are accurately described by the Gross–Pitaevskii (GP) equation [2], namely a variant of the nonlinearSchrödinger (NLS) equation with an external potential and the nonlinearity coefficient

i@w@t¼ �1

2@2w@x2 þ gðxÞjwj2wþ VðxÞw; ð1:1Þ

where w is the mean-field condensate wave function (with density jwj2 measured in units of the peak 1D density n0), x and t

are normalized, respectively, to the healing length n ¼ �h=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0jg1jm

pand n=c (where c ¼ �h

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin0jg1j=m

pis the Bogoliubov speed

of sound), and energy is measured in units of the chemical potential d ¼ g1n0. In the above expressions, g1 ¼ 2�hx?a0, wherex? denotes the confining frequency in the transverse direction, and a0 is a characteristic value of the scattering length. Thenonlinearity coefficient gðxÞ is caused by the spatially varying scattering length, which has been crucial to many experimen-tal achievements, such as the formation of molecular condensates and probing the so-called BEC–BCS crossover. Of particularinterest are BECs in optical lattices (periodic potentials), which have already been used to study Josephson effects, squeezedstates, Landau–Zener tunneling and Bloch oscillations, and the transition between superfluidity and Mott insulation.

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2716 L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723

Finding exact solutions to the GP equation in nontrivial settings is a problem of great significance [3], and the literature ofthe exact solutions for the GP equation’ (1.1) consists of numerous papers, for some recent work we can refer to [4–8]. Mod-ulated amplitude waves (MAWs) with the uniformly propagating coherent structures

wðt; xÞ ¼ RðxÞ expði½HðxÞ � lt�Þ; ð1:2Þ

as a class of exact solutions, yielding a special interest, which generalize the Bloch modes occurring in linear system withperiodic potentials [9], have been extensively studied [10,11] in case of h independent of x (standing waves), wherewðt; xÞ is periodic and quasiperiodic with respect to the spatial variable x. Meanwhile, the MAWs in the complex Ginz-burg–Landau system have been studied by Brusch and co-authors [12].

The dynamics of solutions with coherent structure (1.2) for GP equation have been studied in recent paper [14,13] by theaveraged method and topology degree theory. In these papers, a complicated transform is constructed, which can change asingular system into a standard form of averaging. However, as stated in [13], this transformation is not exact symplectic,and the Hamiltonian structure has been destroyed which maybe lead to loss of dynamical information of the original system.Therefore, looking for an appropriate symplectic transformation is an important topic for discussion.

In this paper, we will investigate the dynamics of MAWs by the averaged method and Hamiltonian perturbed method. Wetransform the original system with the Hamiltonian form into a simple one by constructing a series of canonical transfor-mations via generating functions. The dynamics of the transformed system is easily studied. If the frequencies of the externalpotential VðxÞ and the nonlinearity coefficient gðxÞ are nonresonant to the ones of the unperturbed system, the existence ofinfinitely many periodic and quasiperiodic MAWs can be proved by the Moser twist theorem and Poincaré–Birkhoff twisttheorem. In this situation, a good approximation of order one for MAWs is obtained. In the other case, we can only determinethe existence of the two periodic solutions corresponding to the MAWs via the equilibria of the averaged system by the aver-aged method for given integration constant.

Inserting (1.2) into (1.1), we obtain the following two coupled nonlinear ordinary differential equations

R00 þ ~dR� c2

R3 þ e~gðxÞR3 þ e~VðxÞR ¼ 0; ð1:3Þ

H00 þ 2H0R0=R ¼ 0) H0ðxÞ ¼ c

R2 ; ð1:4Þ

where

~d :¼ 2l; e~gðxÞ :¼ �2gðxÞ; e~VðxÞ :¼ �2VðxÞ

and the integration constant c, determined by the velocity and number density, plays the role of ‘‘angular momentum’’.Inspecting Eq. (1.3) we know that for the simple case, i.e., c ¼ 0, it is just the parametrically driven Duffing equation with

the time variable replaced by the spatial coordinate, the MAWs in this system have been widely researched [10,15]. In thegeneral case, c – 0, the system with regularities becomes more complicated and the MAWs may be kept [13].

For notational convenience, we drop the tildes from ~d; ~g and ~V , and without loss of generality, we take the integrationconstant c ¼ 1, so that (1.3) is written in the form of a forced second-order ODE as

R00 þ dR� 1R3 þ egðxÞR3 þ eVðxÞR ¼ 0: ð1:5Þ

Similarly, for convenience we move the coordinate x ¼ 0 to x ¼ �c0 :¼ d1=4, then Eq. (1.5) is equivalent to the following pla-nar Hamiltonian system

R0 ¼ S

S0 ¼ �dðR� c0Þ þ 1ðR�c0Þ3

� egðxÞðR� c0Þ3 � eVðxÞðR� c0Þ:

(ð1:6Þ

In this paper, we consider the case that both VðxÞ and gðxÞ are periodic functions (periodic potential and scattering lengthsubjected to a spatially periodic modulation), and d > 0 corresponding to a positive chemical potential.

The rest of paper is organized as follows. In Section 2, at first we introduce the change of action-angle variables, thentransform equation (1.6) to a standard form of averaging. The existence of period solutions will be determined by the equi-libria of the averaged system. In Section 3 we construct a series of canonical transformations via generating functions totransform the original Hamiltonian into a simple form. A good approximation of order one for MAWs is obtained.

2. Construction of action and angle variables

We carry out the standard reduction for Eq. (1.6) to the action and angle variables [16]. In order to introduce action andangle variables, we consider the auxiliary autonomous system

R0 ¼ S; S0 ¼ �dðRþ c0Þ þ1

ðRþ c0Þ3: ð2:1Þ

L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723 2717

It is easy to verify that all solutions of this system are pffiffidp -periodic in x. For each h 2 ð

ffiffiffidp

;þ1Þ,

C :12

S2 þ 12

dðRþ c0Þ2 þ1

2ðRþ c0Þ2¼ h

is a simple closed curve in the half plane.Let

IðhÞ ¼ 12p

IC

RdS ¼ 1p

Z Rþ

R�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2h� dðRþ c0Þ2 � ðRþ c0Þ�2

qdR; ð2:2Þ

where

c0 ¼ d�1=4;

R� ¼ �c0 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid�1ðh�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffih2 � d

qÞ

r; for h 2 ð

ffiffiffidp

;þ1Þ:

By calculating the definite integral (2.2), we have

IðhÞ ¼ h

2ffiffiffidp : ð2:3Þ

Then for every ðR; SÞ 2 ð�c0;þ1Þ � R, the action and angle variables can be defined by

hðR; SÞ ¼

R RR�

2ffiffidpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2hðR;SÞ�dðsþc0Þ2�ðsþc0Þ�2p ds; if y P 0

2p�R R

R�2ffiffidpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2hðR;SÞ�dðsþc0Þ2�ðsþc0Þ�2p ds; if y < 0

8><>:

¼p2 � arcsin hðR;SÞ�dðRþc0Þ2ffiffiffiffiffiffiffiffi

h2�dp ; if S P 0

3p2 þ arcsin hðR;SÞ�dðRþc0Þ2ffiffiffiffiffiffiffiffi

h2�dp ; if S < 0

8>><>>:

ð2:4Þ

and

IðR; SÞ ¼ 12ffiffiffidp hðR; SÞ; ð2:5Þ

with

hðR; SÞ ¼ 12

S2 þ d2ðRþ c0Þ2 þ

1

2ðRþ c0Þ2; ð2:6Þ

where the generating function is found to be

GðR; IÞ ¼

R RR�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hðIÞ � dðnþ c0Þ2 � ðnþ c0Þ�2

qdn; if S P 0

2pI �R R

R�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2hðIÞ � dðnþ c0Þ2 � ðnþ c0Þ�2

qdn; if S < 0:

8><>:

with hðIÞ the inverse function of IðhÞ defined by (2.3).In view of (2.4) and together with (2.5) and (2.6), we have

Rðh; IÞ ¼ 1d1=4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2I �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4I2 � 1

pcos h

q� c0;

Sðh; IÞ ¼ d1=2ð4I2 � 1Þ ð2I þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4I2 � 1

pcos hÞ

4I2 sin2 hþ cos2 h

!12

sin h: ð2:7Þ

We have examined the Jacobian determinant

det@ðR; SÞ@ðh; IÞ

� �¼ 1;

which yields dR ^ dS ¼ dh ^ dI, i.e., we have defined a canonical transformation.The Hamiltonian induced by (1.6) may be written as

2718 L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723

HðR; S; xÞ ¼ 12

S2 þ d2ðRþ c0Þ2 þ

1

2ðRþ c0Þ2þ 1

2eVðxÞðRþ c0Þ2 þ

14egðxÞðRþ c0Þ3 ð2:8Þ

¼ hðR; SÞ þ 12eVðxÞðRþ c0Þ2 þ

14egðxÞðRþ c0Þ4; ð2:9Þ

where R; S are canonically conjugate variables and hðR; SÞ is defined by (2.6). In the action and angle variables coordinatesdefined by (2.7), the canonically transformed Hamiltonian becomes

Hðh; I; xÞ ¼ hðRðh; IÞ; Sðh; IÞÞ þ 12eVðxÞvðh; IÞ þ 1

4gðxÞev2ðh; IÞ ¼ H0ðIÞ þ

12eVðxÞvðh; IÞ þ 1

4egðxÞv2ðh; IÞ; ð2:10Þ

where the unperturbed component of the Hamiltonian is H0ðIÞ ¼ 2ffiffiffidp

I and

vðh; IÞ ¼ 1d1=2 ð2I �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4I2 � 1

pcos hÞ:

Note that when e ¼ 0, the unperturbed Hamiltonian system with the Hamiltonian H0ðIÞ is an isochronous system, i.e., eachsolution of the unperturbed Hamiltonian system is a periodic solution with the least period s ¼ p=

ffiffiffidp

in the half-plane.

3. Averaged system and resonant equilibria

Generally, averaged method involves two steps: transforming to standard form; solving the averaged equation. With acanonical change of coordinates from ðh; IÞ to the new variables ð/;qÞ via the global generating function

Fðh;q; xÞ ¼ qhþ Gðh;q; xÞ ¼ qh� 2qffiffiffidp

x; ð3:1Þ

the Hamiltonian (2.10) can be changed into the standard form of averaging. The transformation defined by (3.1) is given by

/ ¼ hþ @G@qðh;q; xÞ ¼ h� 2

ffiffiffidp

x ð3:2Þ

and

I ¼ qþ @G@hðh;q; xÞ ¼ q: ð3:3Þ

The transformed Hamiltonian takes the form

Hð/;q; xÞ ¼ Hðh; I; xÞ þ @G@xðh;q; xÞ ¼ 1

2eVðxÞvð/þ 2

ffiffiffidp

x;qÞ þ 14egv2ð/þ 2

ffiffiffidp

x;qÞ; ð3:4Þ

The vector field associated with the new Hamiltonian (3.4) is found to be

q0 ¼ � @H@/¼ e

2d1=2 ðVðxÞ þ gvðq;/þ 2ffiffiffidp

xÞÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4q2 � 1

psinð/þ 2

ffiffiffidp

xÞ; ð3:5Þ

/0 ¼ @H@q¼ e

d1=2 ðVðxÞ þ gvðq;/þ 2ffiffiffidp

xÞÞ 1� 2qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4q2 � 1

p cosð/þ 2ffiffiffidp

xÞ" #

; ð3:6Þ

where the prime denotes the differentiation with respect to x, the spatial variable.To achieve some analytical understanding of the dynamics of MAWs, we average (3.5) and (3.6) in the physically relevant

case of the OL potential [11,17],

VðxÞ ¼ V0 cosð2j1xÞ ðV0 > 0Þ ð3:7Þ

and the x-dependent nonlinearity coefficient [10]

gðxÞ ¼ a0 þ g0 sin2ð2j2xÞ; ð3:8Þ

where V0; g0 and j1;j2 are, respectively, the amplitude and wave number of the modulation. The nonlinearity coefficient iscaused by the scattering length subjected to a spatially periodic modulation. In experiments, this modulation can be inducedby a periodically patterned configuration of the external (magnetic, optical, or electric) field that controls the Feshbach res-onance. Here, we take ji ¼

ffiffiffidp

; i ¼ 1;2, which implies the modulated frequencies of g and V is equal to the ones of the unper-turbed system.

In case of the x-dependent nonlinearity coefficient (3.8), taking the integral mean value over the interval 0; pffiffidp

h iwith re-

spect to the spatial variable x, the averaged equations for (3.5) and (3.6) have the form

�q0 ¼ � effiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�q2 � 1

p8d

ð2V0

ffiffiffidpþ g0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�q2 � 1

pcos �/Þ sin �/; ð3:9Þ

Fig. 1.domainreferred

L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723 2719

�/0 ¼ e�q4d

6ð2a0 þ g0Þ �4V0

ffiffiffidp

cos �/ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4�q2 � 1

p � g0 cos 2�/

" #: ð3:10Þ

One can find two kinds of equilibria

ð�/; �qÞ ¼ 0;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidV2

0

9ð2a0 þ g0Þ2 þ

14

s !; if 2a0 þ g0 > 0; ð3:11Þ

ð�/; �qÞ ¼ p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffidV2

0

9ð2a0 þ g0Þ2 þ

14

s !; if 2a0 þ g0 < 0; ð3:12Þ

ð�/; �qÞ ¼ 0;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4dV2

0

g20

þ 14

s0@

1A; if 2a0 þ g0 ¼ 0 & g0 < 0; ð3:13Þ

ð�/; �qÞ ¼ p;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4dV2

0

g20

þ 14

s0@

1A; if 2a0 þ g0 ¼ 0 & g0 > 0 ð3:14Þ

and

ð�/; �qÞ ¼ arccosg0ð24a0 þ 11g0Þ

6V0

ffiffiffidp ;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi� 3dV2

0

ð24a0 þ 11g0Þg0þ 1

4

s0@

1A; ð3:15Þ

if

�1 <g0ð24a0 þ 11g0Þ

6V0

ffiffiffidp < 0:

The number of the equilibria is determined by a0 and g0. In Fig. 1, there are two equilibria in the green domain where thenonlinear coefficient gðxÞ crosses zero.

Owing to the averaged theorem, the original systems (3.5) and (3.6) have a p=ffiffiffidp

-periodic solution ðqeðxÞ; /eðxÞÞ withrespect to the spatial variable x near the equilibrium ð�q; �/Þ such that ðqeðxÞ; /eðxÞÞ ! ð�q; �/Þ, as e! 0. In terms of the ori-ginal coordinates, together with (3.2) and (3.3), we have

R2ðx; /e;qeÞ ¼ ReðxÞ þ c0ð Þ2 ¼ 2qeðxÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4q2

e ðxÞ � 1q

cos½/eðxÞ þ 2ffiffiffidp

x�

¼ 4q2e ðxÞ sin2½/eðxÞ þ 2

ffiffiffidp

x� þ cos2½/eðxÞ þ 2ffiffiffidp

x�2qeðxÞ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4q2

e ðxÞ � 1p

cos½/eðxÞ þ 2ffiffiffidp

x�;

hðx; /e;qeÞ ¼Z x

0

1

ReðxÞ þ c0ð Þ2dx

1.0 0.5 0.5g0

0.4

0.2

0.2

0.4

a0

Existence domain of equilibria for parameter a0 and g0. The number of the equilibria is determined by a0 and g0. There are two equilibria in the green, and in the other domain, there exists only one equilibrium. (For interpretation of the references to colour in this figure caption, the reader is

to the web version of this article.)

2720 L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723

and weðt; xÞ ¼ Rðx; /e;qeÞ exp½iðhðx; /e;qeÞ � dt=2Þ�. Note that we have only obtained a rough result

weðt; xÞ ¼ Rðx; �/; �qÞ exp½iðhðx; �/; �qÞ � dt=2Þ� þ OðeÞ

near the equilibrium ð�/; �qÞ via one-order averaging. In the following section, we will go into further discussion.

4. Canonical transformation and Hamiltonian perturbed analysis

In this section, we will construct a series of canonical transformation to transform the Hamiltonian system into a simpleform, whose dynamics are easily studied. This method is technical in KAM iteration. Now we define a canonical transforma-tion implicitly given by

# ¼ hþ e@W@.ðh;.; x; eÞ

I ¼ .þ e@W@hðh;.; x; eÞ; ð4:1Þ

where Wðh;.; x; eÞ is the generating function to be determined later.Under the transformation (4.1), the Hamiltonian (2.9), denoted still by H, becomes

Hð#;.;xÞ¼2ffiffiffidp

.þe12

VðxÞw1ð.;hÞþ14

gðxÞw21ð.;hÞ

�þ2

ffiffiffidp @W

@hþ@W@x

�

þe2 1d@W@h

ffiffiffidp

2VðxÞ

"þ1

2gðxÞ.�2.ðdVðxÞþgðxÞ.Þcoshffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4.2�1p �1

2gðxÞ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2�1

qcoshþ2gðxÞ.cos2 h

�( )þOðe3Þ; ð4:2Þ

where

w1ð.; hÞ ¼1ffiffiffidp ð.�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

qcos hÞ:

We firstly consider in detail VðxÞ and gðxÞ defined as in (3.7) and (3.8) with j1 :ffiffiffidp

– 1 : 1 and j2 :ffiffiffidp

– f1 : 1;1 : 2g, i.e.,this situation is the nonresonant case. Now we have

12

VðxÞw1ð.;hÞþ14

gðxÞw21ð.;hÞ¼

2a0þg0

16d½ð12.2�1Þþð4.2�1Þcos2h�8.

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2�1

qcosh�þ V0ffiffiffi

dp .cosð2j1xÞ

� 116d

g0ð12.2�1Þcosð4j2xÞ� V0

4ffiffiffidp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2�1

q½cosð2j1xþhÞþcosð2j1x�hÞ�þg0.

4d

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2�1

q½cosð4j2xþhÞþcosð4j2x�hÞ��g0ð4.2�1Þ

32d½cosð4j2xþ2hÞþcosð4j2x�2hÞ�:

In order to kill the h-dependence and x-dependence term of order e in the Hamiltonian (4.2), the generating function Wdefining the corresponding canonical transformation is found to be

Wðh;.; x; eÞ ¼ 2a0 þ g0

16d� 1

4ffiffiffidp ð4.2 � 1Þ sin 2hþ 4ffiffiffi

dp .

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

qsin h

� �� V0.

2j1

ffiffiffidp sin 2j1xþ g0

64j2dð12.2 � 1Þ

� sin 4j2xþ V0

4ffiffiffidp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

q sinð2j1xþ hÞ2ðj1 þ

ffiffiffidpÞþ sinð2j1x� hÞ

2ðj1 �ffiffiffidpÞ

" #� g0.

4d

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

q sinð4j2xþ hÞ2ð2j2 þ

ffiffiffidpÞþ sinð4j2x� hÞ

2ð2j2 �ffiffiffidpÞ

" #þ g0ð4.2 � 1Þ

32dsinð4j2xþ 2hÞ

4ðj2 þffiffiffidpÞþ sinð4j2x� 2hÞ

4ðj2 �ffiffiffidpÞ

" #: ð4:3Þ

Then the transformed Hamiltonian function (4.2) is as follows

Hð#;.; xÞ ¼ 2ffiffiffidp

.þ e2a0 þ g0

16dð12.2 � 1Þ þ Oðe2Þ: ð4:4Þ

The equations of motion defined by (4.4) are

.0 ¼ Oðe2Þ;

#0 ¼ 2ffiffiffidpþ e

32dð2a0 þ g0Þ.þOðe2Þ:

By integrating the equations from x ¼ 0 to x ¼ p=ffiffiffidp

, the Poincaré mapping is

.1 ¼ .0 þOðe2Þ; ð4:5Þ

L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723 2721

#1 ¼ #0 þ 2pþ e3p

2d3=2 ð2a0 þ g0Þ.0 þOðe2Þ: ð4:6Þ

Applying the Moser twist theorem [18], if 2a0 þ g0 – 0, then there is an invariant curve of the Poincaré mapping in anypunctured disk ð1=2 < a 6 . 6 bÞ for all e sufficiently small. In the original coordinates ðR; SÞ, the invariant curve is corre-sponding to a quasi-periodic solution. In addition, we also can obtain infinitely many periodic solutions directly by Poin-caré–Birkhoff twist theorem [19], and Moser twist theorem implies the existence of invariant curves.

We say that the frequencies x1;x2; . . . ;xn are rationally independent, if for all k 2 Zn n f0g one hashx; ki :¼

Pni¼1xiki – 0. We remark that if j1;j2;

ffiffiffidp

are rationally independent, one can transform the Hamiltonian (4.2)up to arbitrary finite order.

In the following, we will investigate the dynamics of equations in case of resonance. In order to compare the results inprevious section, we take VðxÞ and gðxÞ as in (3.7) and (3.8) with j1 : j2 :

ffiffiffidp¼ 1 : 1 : 1. We can see that the resonant fre-

quencies correspond to vanishing denominators of Wð.; h; x; eÞ defined by (4.3) and the resonant term includingcosð2j1 � hÞ and cosð4j2 � 2hÞ can not be removed. Thus, an appropriate choice for the generating function yields the fol-lowing Hamiltonian

Hð#;.; xÞ ¼ 2ffiffiffidp

.þ e2a0 þ g0

16dð12.2 � 1Þ � V0

4ffiffiffidp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

qcosð2

ffiffiffidp

x� #Þ � g0ð4.2 � 1Þ32d

cosð4ffiffiffidp

x� 2#Þ� �

þOðe2Þ; ð4:7Þ

where the generating function Wðh;.; x; eÞ is

Wðh;.; x; eÞ ¼ 2a0 þ g0

16d� 1

4ffiffiffidp ð4.2 � 1Þ sin 2hþ 4ffiffiffi

dp .

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

qsin h

� �� V0.

4dsin 2

ffiffiffidp

xþ g0

64ffiffiffidp

dð12.2 � 1Þ sin 4

ffiffiffidp

x

þ V0

4ffiffiffidp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

q sinð2ffiffiffidp

xþ hÞ4ffiffiffidp þ g0ð4.2 � 1Þ

32dsinð4

ffiffiffidp

xþ 2hÞ8ffiffiffidp � g0.

4d

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4.2 � 1

q sinð4ffiffiffidp

xþ hÞ6ffiffiffidp þ sinð4

ffiffiffidp

x� hÞ2ffiffiffidp

" #:

The spatial dependence in Hamiltonian (4.7) can be removed by making a canonical change of coordinates from ð#;.Þ toðH; JÞ via the global generating function,

Wð#; J; xÞ ¼ J#� 2ffiffiffidp

Jx: ð4:8Þ

The transformation defined by (4.8) is given by

H ¼ #� 2ffiffiffidp

x; . ¼ J: ð4:9Þ

The transformed Hamiltonian takes the form, up to Oðe2Þ,

~HðJ;HÞ ¼ Hð.; #Þ þ @W@xðJ; #; xÞ ¼ e

2a0 þ g0

16dð12J2 � 1Þ � V0

4ffiffiffidp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4J2 � 1

qcos H� g0ð4J2 � 1Þ

32dcos 2H

" #: ð4:10Þ

It is interesting to point that the Hamilton system with the Hamiltonian (4.10) is exactly the previous averaged Eqs. (3.9) and(3.10). Here we denote its equilibria by ðH0; J0Þ ¼ ð�/; �qÞ.

By focusing our analysis in ðJ;HÞ space and restricting J to a neighborhood of J0 2 ð1=2;þ1Þwith J0 the first component ofthe equilibrium ðJ0;H0Þ, let J ¼ J0 þ~J with ~J ¼ OðeÞ � 1. Using the expansion of power series at J ¼ J0, the Hamiltonian (4.10)becomes

Hð~J;HÞ ¼ �eV0

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4J2

0 � 1q

cos Hþ e~JJ0 a0 þg0

2� V0 cos Hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4J20 � 1

q0B@

1CAþOððe~JÞ2Þ; ð4:11Þ

where the ð~J;HÞ-independence constant is omitted since it does not affect the dynamics.The Hamiltonian system with the Hamiltonian (4.11) takes the form

~J0 ¼ �eV0

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4J2

0 � 1q

sin HþOððe~JÞ2Þ ¼ �eV2

0

2j2a0 þ g0jsin HþOððe~JÞ2Þ; ð4:12Þ

H0 ¼ eJ0 a0 þg0

2� V0 cos Hffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4J20 � 1

q0B@

1CAþOððe2~JÞÞ ¼ eJ0 a0 þ

g0

2

� �1� sign a0 þ

g0

2

� �cos H

� �þOððe2~JÞÞ; ð4:13Þ

Fig. 2. Spatial profile of the coherent structure corresponding to a MAW near the equilibrium ð0; 0:5163Þ for the parameter valuesa0 ¼ 0:15; g0 ¼ 2; V0 ¼ 1:6; j1 ¼ j2 ¼ d ¼ 1.

1.1

1.05

1

0.95

0.9

1.75

1.5

1.25

1

0.75

0.5

0.25

-10 -5 0 5 100

0.5

-10 -5 0 5 10

1

1.5

2

-10 -5 0 5 10 -10

0.6

0.8

1

1.2

1.4

-5 0 5 10

2 2

22

Fig. 3. Dynamics evolution of MAW of GP equation (1.2) for a0 ¼ 0:15; g0 ¼ 2; V0 ¼ 1:6; j1 ¼ j2 ¼ d ¼ 1; e ¼ 0:08. The left panel shows the space–timecontour plot of the square modulus (density) jwj2 of the solution, and the right panels show four snapshots (spatial profiles) of the spatio-temporalevolution. The dynamics illustrate an apparent stability of the solution up through at least t ¼ 240.

2722 L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723

where signðxÞ is a sign function defined by

signðxÞ ¼�1; if x < 0;1; if x > 0;

�

which yields

d~JdH¼ � V2

0

2J0 a0 þ g02

� j a0 þ g0

2

� j

cotH2þOððe~JÞÞ

readily solved. Near the equilibrium H0 ¼ 0 or p, in view of (4.13) and H ¼ H0 þOðeÞ, we know that H0ðxÞ ¼ Oðe2Þ, whichyields ~J0ðxÞ ¼ Oðe2Þ.

When transformed back to the original variables, in view of (4.1) and (4.9), we have

hðxÞ ¼ H0 þ 2ffiffiffidp

x� e@W@.ðJ0; h; x; eÞ ¼ H0 þ 2

ffiffiffidp

x� e@W@.ðJ0;H0 þ 2

ffiffiffidp

x; x; eÞ þ Oðe2Þ;

IðxÞ ¼ J0 þ e@W@hðJ0; h; x; eÞ ¼ J0 þ e

@W@hðJ0;H0 þ 2

ffiffiffidp

x; x; eÞ þ Oðe2Þ:

L. Jia et al. / Commun Nonlinear Sci Numer Simulat 19 (2014) 2715–2723 2723

We examine the dynamical evolution of the periodic solution ðIðxÞ; hðxÞÞ near the equilibrium ðJ0;0Þ witha0 ¼ 0:15; g0 ¼ 2;V0 ¼ 1:6; d ¼ 1, where we obtain

IðxÞ ¼ 0:5163þ eð0:1291þ 0:0257 cos 2xþ 0:0017 cos 4xÞ;

hðxÞ ¼ 2xþ eð0:4034 sin 2x� 0:0787 sin 4xÞ mod ð2pÞ:

In this situation, we plot the spatial profile of R2ðx; 0; 0:5163Þ near the equilibrium ð0; 0:5163Þ, see Fig. 2.Returning to the original GP equation, we examine its dynamics with direct simulations of Eq. (1.2) with the initial value

wð0; xÞ near the solution ðRðx; 0;0:5163Þ; Sðx; 0;0:5163ÞÞ corresponding to the equilibrium ðH0; J0Þ ¼ ð0;0:5163Þ. The result ofsimulations for a0 ¼ 0:15; g0 ¼ 2; V0 ¼ 1:6 are shown in Fig. 3. The dynamics illustrate an apparent stability of the solutionup through at least t ¼ 240.

5. Conclusions

We have studied the dynamics of MAWs by the averaged method and Hamiltonian perturbed method. We transform theoriginal system with the Hamiltonian form into a simple one by constructing a series of canonical transformations via gen-erating functions. When the frequencies of the external potential VðxÞ and the nonlinearity coefficient gðxÞ are nonresonantto the ones of the unperturbed system, we have proved the existence of infinitely many periodic and quasiperiodic MAWsusing the Moser twist theorem and Poincaré–Birkhoff twist theorem. In this situation, a good approximation of order one forMAWs is obtained. In the other case, we can only determine the existence of the two periodic solutions corresponding to theMAWs via the equilibria of the averaged system by the averaged method for given integration constant.

6. Acknowledgements

This work is supported in partly by NNSF of China (10871142) and Doctoral Fund of Ministry of Education of China(20070285002). The authors acknowledge the support by the National Natural Science Foundation of China (Grant Nos.11301106, 11226130, 11162004, 11261013) and Guangxi Natural Science Foundation (Grant No. 2013GXNSFAA019006).

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