Exact solutions of the Gross-Pitaevskii equation in ...

JOURNAL OF MATHEMATICAL PHYSICS 48, 073520 共2007兲

Exact solutions of the Gross-Pitaevskii equation in periodic potential in the presence of external source E. Kengnea兲 Department of Mathematics and Computer Science, Faculty of Science, University of Dschang, P.O. Box 4509, Douala, Republic of Cameroon and Department of Mathematics and Statistics, Faculty of Science, University of Ottawa, 585 King Edward Avenue, Ottawa ON K1N 6N5, Canada

R. Vaillancourt Department of Mathematics and Statistics, Faculty of Science, University of Ottawa, 585 King Edward Avenue, Ottawa ON K1N 6N5, Canada 共Received 31 March 2007; accepted 15 May 2007; published online 30 July 2007兲

Exact periodic solutions, solitonlike solutions, singular solitary, and singular trigonometric wave solutions of the time-dependent Gross-Pitaevskii equation 共GPE兲 with elliptic function potential in the presence of external source are analyzed. A simple approach that applies equally to both attractive and repulsive timedependent GPE and allows one to find an extensive list of explicit periodic solutions of the GPE in terms of the Jacobian elliptic functions is developed. In the limit as the elliptic modulus tends to unity or to zero, the linear solutions, in either the Jacobian elliptic cosine or the Jacobian elliptic function of third order, give solitonlike solutions, while the rational solutions in these elliptic functions lead to singular solitary or trigonometric wave solutions. The stability of these solutions is investigated numerically. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2748618兴

I. INTRODUCTION

The cubic nonlinear Schrödinger equation with attractive or repulsive nonlinearity and a potential is often referred to as the Gross-Pitaevskii equation 共GPE兲 and is used as a mean-field model for the dynamics of a dilute-gas Bose-Einstein condensate 共BEC兲.1–8 In suitable units, the one-dimensional GPE is

i

冋

册

1 ⳵2 ⳵␺ = − + V共x兲 + g兩␺兩2 ␺ , 2 ⳵x2 ⳵t

共1兲

where ␺共x , t兲 represents the macroscopic wave function of the condensate and the real-valued function V共x兲 is an experimentally generated macroscopic potential. The parameter g encapsulates the strength of the atom-atom interactions and determines whether Eq. 共1兲 is repulsive 共g = 1, defocusing nonlinearity兲, or attractive 共g = −1, focusing nonlinearity兲. Note that for BEC applications, both signs of g are relevant. The GPE 共1兲 has attracted an increasing interest 共see Ref. 9 for a review兲. The main rigorous results concern both the derivation of this equation10 and the study of its stationary solutions.11 On the other hand, if the potential V共x兲 is not zero then very few rigorous results exist for the study of the solution of the time-dependent equation. A large class of periodic potentials is given by

a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

0022-2488/2007/48共7兲/073520/13/$23.00

48, 073520-1

© 2007 American Institute of Physics

Downloaded 30 Jul 2007 to 137.122.49.62. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

073520-2

J. Math. Phys. 48, 073520 共2007兲

E. Kengne and R. Vaillancourt

V共x兲 = − V0sn2共x,k兲,

共2兲

where sn共x , k兲 is the Jacobian elliptic sine function with elliptic modulus 0 艋 k 艋 1.12,13 In the limit as k → 1 − 0, V共x兲 becomes an array of well-separated hyperbolic secant potential barriers or wells, while in the limit as k → + 0, it becomes purely sinusoidal. Since sn共x , k兲 is periodic in x with period 4T共k兲 = 4兰0␲/2共1 − k2 sin2 ␣兲−1d␣, V共x兲 is periodic in x with period 2T共k兲. This period approaches infinity as k → 1. The goal of this paper is to find exact solutions of the GPE with a source14 i

冋

册

1 ⳵2 ⳵␺ = − + V共␰兲 + g兩␺兩2 ␺ + K exp共i␹共␰兲 − i␻t兲 2 ⳵x2 ⳵t

共3兲

with periodic potential of the form 共2兲, where K and ␻ are real constants related to the source amplitude and the chemical potential, respectively. Here ␹共␰兲 is a real function of ␰ = ␣共x − vt兲, ␣ and v being two real parameters. In the field of nonlinear optics, Eq. 共3兲 with K ⫽ 0 may describe the evolution of the local amplitude of an electromagnetic wave in the spatial domain, in a two-dimensional 共2D兲 waveguide 共then, t becomes the propagation distance and x becomes the retarded time兲, and the system is driven by an external plane pump wave.15–17 The sn共␰ , k兲 potential may describe a transverse modulation of the refractive index in the waveguide.15 If KV0 ⫽ 0, Eq. 共3兲 is not integrable and only small classes of explicit solutions can most likely exist. In the limit V0 → 0, Eq. 共3兲 is a cubic nonlinear Schrödinger equation with an external source. Some exact solutions of Eq. 共3兲 with constant potential are found in Ref. 18, while periodic solutions in the absence of source 共K = 0兲 are found in Ref. 19. In this paper we restrict our attention to traveling wave solutions of Eq. 共3兲 with K ⫽ 0. The rest of the paper is organized as follows. In Sec. II, we find some classes of exact solutions of Eq. 共3兲. The numerical simulations are carried out in Sec. III, and the results obtained in this work are summarized in Sec. IV. II. EXACT SOLUTIONS OF THE GPE IN PERIODIC POTENTIAL WITH SOURCE

The traveling wave solutions of Eq. 共3兲 with potential 共2兲 are taken to be of the form ␺共x , t兲 = ␳共␰兲exp共i␹共␰兲 − i␻t兲. Inserting this expression for ␺共x , t兲 in Eq. 共3兲 and separating the real and imaginary parts of the equation yield

␹⬘ =

␳ 3␳ ⬙ = C 2 +

v

␣

+

C , ␳2

2K 3 v2 + 2␻ 4 2V0 2 2g ␳ − ␳ − 2 sn 共␰,k兲␳4 + 2 ␳6 , ␣2 ␣2 ␣ ␣

共4兲

共5兲

where C is a constant of integration. In order that the external phase be independent of ␺, we consider only solutions with C = 0. A. Solutions where ␳„␰… is linear in either sn„␰ , k…, cn„␰ , k…, or dn„␰ , k…

In this subsection, we find solutions for which ␳共␰兲 is a linear function of sn共␰ , k兲, cn共␰ , k兲, or dn共␰ , k兲, where cn共␰ , k兲 and dn共␰ , k兲 denote the Jacobian elliptic cosine function and the third Jacobian elliptic function, respectively. First, we discuss the solutions with sn共␰ , k兲. The quantities associated with these solutions will be denoted with a subscript 1, while the quantities associated with the cn共␰ , k兲 and dn共␰ , k兲 solutions will have subscripts 2 and 3, respectively. Substituting

␳共␰兲 = A1sn共␰,k兲 + B1

共6兲

in Eq. 共5兲 and equating the coefficients of equal powers of sn共x , k兲 result in relations among the solution parameters A1, and B1, and the equation parameters V0, g, K, ␣, and ␻. Under the conditions


073520-3

J. Math. Phys. 48, 073520 共2007兲

Exact solution of the GPE

␣2 = −

2V0 , 3k2

V0 ⬍ −

3k2共2␻ + v2兲 − V0共k2 + 1兲 27k3

K= ±

冑

2共k2 + 1兲V0 + 3共2␻ + v2兲k2 , 2g

3k2共2␻ + v2兲 , 2共k2 + 1兲

共7兲

we find A21 =

V0 , 3g

B21 =

2共k2 + 1兲V0 + 3共2␻ + v2兲k2 . 18gk2

共8兲

It follows from the first equalities in Eqs. 共7兲 and 共8兲 that V0 ⬍ 0 and g ⬍ 0. The condition g ⬍ 0 corresponds to the GPE with attractive nonlinearity. The cn共␰ , k兲 solutions are obtained by substituting

␳共␰兲 = A2cn共␰,k兲 + B2

共9兲

in Eq. 共5兲. Equating powers of cn共x , k兲 imposes the following constraints on the parameters

␣2 = −

2V0 , 3k2

K=

3k2共2␻ + v2兲 + 共8k2 − 1兲V0 B 2, 9k2

A22 = −

V0 , 3g

B22 =

V0 ⬎ −

3k2共2␻ + v2兲 , 2共k2 + 1兲

2共k2 + 1兲V0 + 3共2␻ + v2兲k2 . 18gk2

共10兲

共11兲

From the positivity of ␣2 and A22, we conclude that the cn共␰ , t兲 solutions exist only for V0 ⬍ 0 and g ⬎ 0. The condition g ⬎ 0 corresponds to a GPE with repulsive nonlinearity. Looking for dn共␰ , k兲 solutions, we substitute the ansatz

␳共␰兲 = A3dn共␰,k兲 + B3

共12兲

in Eq. 共5兲. Equating powers of dn共x , k兲 imposes the following relations between the parameters:

␣2 = −

2V0 , 3k2

K=

A23 = −

3k2共2␻ + v2兲 + V0共8 − k2兲 B 3, 9k2 V0 , 3k2g

B23 =

V0 ⬎ −

3k2共2␻ + v2兲 , 2共k2 + 1兲

2共k2 + 1兲V0 + 3共2␻ + v2兲k2 . 18gk2

共13兲

共14兲

As in the case of the cn共␰ , k兲 solutions, the dn共␰ , k兲 solutions exist only for V0 ⬍ 0 and g ⬎ 0. Relations 共13兲 with B3 given by Eq. 共14兲 are the constraint relationships that must be satisfied by the parameters V0, g, K, ␣, and ␻ of the equation. Because the expression of the source strength K given by the second equation in Eqs. 共7兲, 共10兲, and 共13兲 contains B j, we conclude that B j ⫽ 0, j = 1 , 2 , 3. The last condition in Eqs. 共7兲, 共10兲, and 共13兲 are obtained from the positivity of B2j . Remark: In the limit of the GPE in the absence of the external source 共K → 0兲, our solutions coincide with those obtained in page 056615-3 of Ref. 19. B. Solutions where ␳„␰… is a rational function in sn2„␰ , k…, cn2„␰ , k…, or dn2„␰ , k…

To find the solutions of this subsection, it is sufficient to find the solutions where ␳共␰兲 is a rational function in sn2共␰ , k兲, and then, using Jacobian elliptic function identities,12 one easily deduces the solutions in cn2共␰ , k兲 or dn2共␰ , k兲. For solutions in a rational function with respect with sn2共␰ , k兲, we consider the ansatz


073520-4

J. Math. Phys. 48, 073520 共2007兲


␳共␰兲 =

A + Bsn2共␰,k兲 . 1 + Dsn2共␰,k兲

共15兲

Substituting this ansatz in Eq. 共5兲 and equating the coefficients of equal powers of sn共␰ , k兲, after a straightforward but lengthy manipulation, we obtain the following relations: A=

KD , 共V0 + k2␣2兲

2K + 2gA3 + 2␣2AD − 共2␻ + v2兲A = 0,

D=

2共V0 + 6k2␣2兲 , v + 2␻ − 4␣2共k2 + 1兲 2

B = 0,

共16兲

3KD − AV0 − AD共2␻ + v2兲 − AD␣2共3D + 2k2 + 2兲 = 0. 共17兲

Thus, for the solutions given by Eqs. 共15兲 and 共16兲, the equation parameters V0, g, K, ␣, and ␻ must satisfy the constraints 共17兲. For example, by taking K = 1, k2 = 21 , ␣2 = 31 , and ␻ = 共4 − v2兲 / 2, Eqs. 共17兲 become 10V0 + 2V20 + 11= 0, g = g共V0兲 = 共9 + 2V0 − 2V20兲共6V0 + 1兲2 / 216共V0 + 1兲3. The first of these equations gives V0 = V±0 = 21 共−5 ± 冑3兲. Hence g共V−0 兲 ⬇ 2.6265 and g共V+0 兲 ⬇ −0.552 46. Then, V−0 corresponds to a GPE with repulsive nonlinearity, while V+0 corresponds to a GPE with attractive nonlinearity. For these values of the equation parameters, we have A共V±0 兲 = 共−9 ± 3冑3兲 ⫻共−14± 3冑3兲−1, D共V±0 兲 = 0.5共−3 ± 冑3兲, and B = 0. Physical solutions ␳共␰兲 must be bounded. Considering the fact that 0 艋 sn2共␰ , k兲艋 1 for every ␰ and every k, one obtains conditions, expressed in terms of the equation parameters, that determine physical solutions. These conditions are obtained if the denominator 1 + Dsn2共␰ , k兲 in Eq. 共15兲 does not change sign, i.e., 1 + Dsn2共␰ , k兲 is always positive. We then find that either 共V0 + 6k2␣2兲共v2 + 2␻ − 4␣2共k2 + 1兲兲 ⬎ 0 or 共V0 + 6k2␣2兲关v2 + 2␻ − 4␣2共k2 + 1兲兴 ⬍ 0

共18兲

and 关2共V0 + 6k2␣2兲 + v2 + 2␻ − 4␣2共k2 + 1兲兴关v2 + 2␻ − 4␣2共k2 + 1兲兴 ⬎ 0. For the above values of the equation parameters, 共V0 + 6k2␣2兲关v2 + 2␻ − 4␣2共k2 + 1兲兴 = −3 ± 冑3, which is negative for both signs “⫹” and “⫺.” Thus 关2共V±0 + 6k2␣2兲 + v2 + 2␻ − 4␣2共k2 + 1兲兴关v2 + 2␻ − 4␣2共k2 + 1兲兴 = 2共−1 ± 冑3兲, which is positive for the “⫹” sign and negative for the “⫺” sign. Thus only V0 = V+0 satisfies condition 共18兲 corresponding to a physical solution. We then obtain the physical solution ␳共␰兲 = 2共3冑3 − 9兲 / 共3冑3 − 14兲共2 + 共冑3 − 3兲sn2共␰ , 冑2 / 2兲兲. For the values of the equation parameters ␣2 = 1, v2 = 1, V0 = 1, the different values of A and B for physical solutions are shown in Fig. 1共a兲 as functions of the elliptic modulus 0 艋 k 艋 1; these values correspond to ␻, K, and g, shown in Fig. 1共b兲, as functions of the elliptic modulus 0 艋 k 艋 1. In Fig. 1共b兲, the red line corresponds to conditions 共18兲 and gives the maximal values of ␻ for which the physical solutions exist. This figure shows that for the equation parameters ␣2 = 1, v2 = 1, V0 = 1 and for any elliptic modulus 0 ⬍ k 艋 1, there exist a source amplitude K, an attractive nonlinearity g, and a “chemical potential” ␻ such that Eqs. 共15兲 and 共16兲 define a physical solution of the GPE 共3兲 with the corresponding set of parameters ␣, v, V0, K, g, and ␻. For example, for k = 0.7, Fig. 1共b兲 gives ␻ ⬇ −3.8349, g ⬇ −0.716 84, and K ⬇ −1.6028, while Fig. 1共a兲 gives A ⬇ 0.671 14 and D ⬇ −0.623 92. We then obtain from Eq. 共15兲 the physical solution ␳共␰兲 = 0.671 14/ 1 − 0.623 92 sn2共␰ , 0.7兲. This solution is plotted in Fig. 2 共solid line兲, where the dotted line indicates the corresponding potential V共␰兲. Because Eqs. 共17兲 and 共18兲 do not impose any condition on the sign of g, one concludes that solution 共15兲 with coefficients 共16兲 and constraints 共17兲 and 共18兲 is valid both for the GPE with attractive nonlinearity 共g ⬍ 0兲 and with repulsive nonlinearity 共g ⬎ 0兲.


073520-5


J. Math. Phys. 48, 073520 共2007兲

FIG. 1. 共Color online兲 Plot of the solution parameters A and D for the physical solutions as functions of the elliptic modulus 0 艋 k 艋 1 关curve 共a兲兴 for fixed values of ␣, v, and V0, and for various values of g, ␻, and K 关curve 共b兲兴. The red curve in 共b兲 corresponds to the maximal values of ␻.

We point out that for a given set of equation parameters, there can exist solutions of form 共15兲 and one of form 共6兲, 共9兲, or 共12兲. For example, if we take k = 0.3487, K = −5.6656, V0 = −0.182 39, g = −2.1741⫻ 10−3, ␣2 = 1, and ␻ = −v2 / 2 − 0.856 84 共for every v ⫽ 0兲, then conditions 共7兲 are satisfied and using Eqs. 共6兲 and 共8兲, we find ␳共␰兲 = ± 5.2881 sn共␰ , 0.3487兲 + 14.743. For this same set of equation parameters, conditions 共17兲 and 共18兲 are satisfied, and according to Eqs. 共15兲 and 共16兲, we have ␳共␰兲 = −5.4828/ 1 + 2.0667 sn2共␰ , 0.3487兲. These solutions are shown in Fig. 3.

FIG. 2. Plot of solutions of type 共15兲 for ␣ = v = V0 = 1, ␻ = −3.8349, g = −0.716 84, and K = −1.6028. V共␰兲 is indicated with a dotted line.


073520-6

J. Math. Phys. 48, 073520 共2007兲


FIG. 3. 共Color online兲 Plot of solutions: linear in sn共␰ , k兲共black lines兲 and rational in sn共␰ , k兲共solid red line兲 for k = 0.3487, K = −5.6656, V0 = −0.182 39, ␣2 = 1, g = −2.1741⫻ 10−3, and ␻ = −v2 / 2 − 0.856 84. The dotted red line is the plot of 30⫻ V共␰兲.

In this figure, the solutions linear in sn共␰ , k兲 are plotted in black 共solid line for the “⫹” sign and dotted line for the “⫺” sign兲, while the rational solution in sn共␰ , k兲 is plotted in the solid red line. The dotted red line is the plot of the potential V共␰兲. C. Trigonometric limit

In the limit k → + 0, the elliptic functions reduce to trigonometric functions and the potential V共␰兲 = −V0 sin2 ␰ is sinusoidal and thus describes the case of an optical lattice . The solutions which correspond to this limit are obtained from the solutions in terms of sn共␰ , k兲 and cn共␰ , k兲. Because the sn共␰ , k兲 and the cn共␰ , k兲 solutions of Sec. II A with k → + 0 give only the zero solution, the nontrivial solution corresponding to k → + 0 can be obtained only from the solutions of Sec. II B. By setting k = 0 in Eqs. 共15兲–共17兲, we obtain the rational trigonometric function solution

␳共␰兲 = −

2K , V0共1 − 2 sin2 ␰兲

共19兲

where the parameters ␣ , g , V0, and K satisfy the conditions v2 + 2␻ − 4␣2 + V0 = 0 and K2g − ␣2V20 = 0, meaning that g ⬎ 0. The singularity of solution 共19兲 follows from the fact that the physical conditions 共18兲 are not satisfied when k → + 0. In a 2D waveguide this singular solution implies simultaneous onset of chaos at a periodic set of points; however, such a solution is unstable, as a small perturbation would result in the occurrence of collapse at one particular point, rather than simultaneously at all points belonging to the periodic lattice. D. Zero potential function limit

In the limit V0 → 0, the potential V共␰兲 = 0, and Eq. 共3兲 becomes a cubic nonlinear Schrödinger equation with zero potential function in the presence of an external source; particular solutions of such equation can be obtained by passing in the limit V0 → 0 in the solutions obtained in Sec. II B 共because ␣ ⫽ 0, we cannot pass in the limit V0 → 0 in the solutions of Sec. II A兲. If the parameters ␣, ␻, g, K, and v satisfy the conditions 共v2 + 2␻兲2 − 16␣4共k4 − k2 + 1兲 = 0,

共20兲

4096gK2 − 共11共v2 + 2␻兲 + 20␣2共k2 + 1兲兲共v2 + 2␻ − 4␣2共k2 + 1兲兲2 = 0,

共21兲

and either


073520-7

J. Math. Phys. 48, 073520 共2007兲


v2 + 2␻ − 4␣2共k2 + 1兲 ⬎ 0 or v2 + 2␻ − 4␣2共k2 + 1兲 ⬍ 0 and v2 + 2␻ + 4␣2共2k2 − 1兲 ⬍ 0,

共22兲 then passing in the limit V0 → 0 in Eqs. 共15兲 and 共16兲, we obtain the following solution in the case of a cubic nonlinear Schrödinger equation with an external source:

␳共␰兲 =

16K . v + 2␻ − 4␣ 共k + 1兲 + 16k2␣2sn2共␰,k兲 2

2

2

共23兲

Regrouping conditions 共20兲–共22兲, we obtain for the existence of solution 共23兲 the following relations: K2 =

␣6 冑 4 2 共 k − k + 1 − k2 − 1兲2共5k2 + 20 − 11冑k4 − k2 + 1兲; 64g

v2 + 2␻ = 4␣2冑k4 − k2 + 1,

which means that g ⬎ 0 and ␻ ⬎ −v2 / 2. Inserting Eq. 共23兲 into the expression for ␺共x , t兲 and using Eq. 共4兲 lead to the solution

␺共x,t兲 =

16K exp关iv共x − vt兲 − i␻t兴 , v2 + 2␻ − 4␣2共k2 + 1兲 + 16k2␣2sn2共␰,k兲

0艋k艋1

of the cubic nonlinear Schrödinger equation without potential and in the presence of an external source. It is important to notice that the results obtained in this subsection do not generalize those contained in Ref. 14, what follows from the fact that constant B in expression 共15兲 is zero. This condition B = 0 follows from the presence of the variable potential function in Eq. 共3兲. E. Solitonlike solutions

Solitonlike solutions are obtained from the solutions in cn共␰ , k兲 and dn共␰ , k兲 of Sec. II A and from the solutions of Sec. II B in the limit as k → 1 − 0. In the limit as k → 1 − 0, the Jacobian elliptic cosine function and the third Jacobian elliptic function reduce to the hyperbolic secant functions and the potential V共␰兲 becomes an array of well-separated hyperbolic secant potential barriers or wells, V共␰兲 = −V0 tanh2 ␰. Then Eqs. 共9兲 and 共12兲 give

␳共␰兲 = A j sech共␰兲 + B j,

j = 2,3,

共24兲

under conditions 共10兲, 共11兲, 共13兲, and 共14兲, respectively, with k = 1. Equations 共15兲 and 共16兲 give ␳共␰兲 = 2共V0 + 6␣2兲K / 共V0 + ␣2兲1 / 共v2 + 2␻ − 8␣2兲 + 2共V0 + 6␣2兲tanh2共␰兲 under conditions 共17兲 and 共18兲 in which we set k = 1. Using Eq. 共16兲, we solve Eq. 共17兲 and find that the set of parameters V0 = 23 共4␣2 − v2 − 2␻兲, g = 4␣2共14␣2 − 3v2 − 6␻兲2 / 27K2, D = −3, and A = −6K / 共14␣2 − 3共v2 + 2␻兲兲 do not satisfy the physical conditions 共18兲. This set corresponds to the singular hyperbolic solution

␳共␰兲 = −

1 6K . 14␣2 − 3v2 − 6␻ 1 − 3 tanh2共␰兲

共25兲

This solution corresponds to the repulsive case, i.e., g ⬎ 0. The singularity of the pulse profile may correspond to the beam power exceeding the material breakdown due to self-focusing.20–23 We have thus found various types of solitonlike solutions, both dark and bright, depending on the values of the equation parameters. Note that, according to Eqs. 共11兲 and 共14兲, the coefficients A j and B j appearing in solutions 共24兲 are given in terms of the equation parameters. III. NUMERICAL SIMULATION

In this section, we discuss the numerical solutions of Eq. 共3兲 with initial conditions chosen from the exact solutions given in the previous section. It is worth pointing out that the numerical


073520-8

J. Math. Phys. 48, 073520 共2007兲


FIG. 4. 共Color online兲 Unstable solutions from Eq. 共6兲. Here, g = −1, V0 = −1, k = 冑 3 , K = 108 , ␣ = 1, v = 1, and ␻ = 6 . The 冑3 冑2 computed coefficients 共8兲 give A1 = 3 and B1 = 6 . The bottom figures show the dynamics of the exact solution 共left兲 and the potential 共right兲 for t 苸关0 , 0.04兴. 2

13冑3

1

techniques based on the fast Fourier transform 共FFT兲 are expensive as they require the FFT of the external source. Hence, we have used the Crank-Nicolson finite difference method24 to solve the GPE with source 共3兲, which is quite handy, and unconditionally stable. For each experiment, a small amount of white noise was added to the initial conditions as a perturbation: ␺共x , t = 0兲 = ␳共␣x兲关1 + ⑀兴exp共i␹共␣x兲兲 = R共x兲. The fixed boundary condition is used at the left boundary point where we impose to the numerical solution to coincide with the exact solution at the initial time t = 0, while no condition is imposed at the right boundary point. The fixed boundary conditions are preferred to be used here because they are easiest to implement25 and our simulations need minimal computational time. In our simulations, we are interested in propagating the wave function ␺共x , t兲 of the GPE 共3兲 forward in time, starting with the initial state ␺共x , t = 0兲 = R共x兲. We then represent the GPE with source 共3兲 numerically as a difference equation, with position x broken up into a grid of points x j spaced by a distance h, and time t as discrete steps of size ␶. The wave function is defined on this grid as a vector of length M : ␺nj = ␺共x j , tn兲 = ␺共x0 + 共j − 1兲h , 共n − 1兲␶兲, where j = 1, 2, . . ., M, M and n being two independent integers. The GPE with source 共3兲 can be written in this discrete representation as


073520-9

J. Math. Phys. 48, 073520 共2007兲


FIG. 5. 共Color online兲 Unstable solutions from Eq. 共9兲. Here, g = 1, V0 = −1, k = 冑 3 , K = − 54冑2 , v = 1, ␣ = 1, and ␻ = 2 . For 1 1 these parameters, the computed coefficients 共11兲 give A2 = 冑3 and B2 = 3冑2 . The bottom figures show the dynamics of the exact solution 共left兲 and the potential 共right兲 for t 苸关0 , 0.08兴. 2

i

冋

1

1

册

n+1 ␺n+1 − ␺nj + ␺n+1 1 ␺nj+1 − 2␺nj + ␺nj−1 ␺n+1 j j+1 − 2␺ j j−1 =− + + 关Vnj + g兩␺nj 兩2兴␺nj + f nj , 4 ␶ h2 h2

共26兲

where f共x , t兲 = K exp共i␹共␰兲 − i␻t兲 is the external source in Eq. 共3兲 and ␰ = ␣共x − vt兲. It is convenient to write Eq. 共26兲 in the form

␺n+1 j−1 +

冉

冊

2共− ␶ + 2ih2兲 n+1 2ih2 n ␺ j + ␺n+1 = − ␺ + 2 1 + + 2h2Vnj + 2h2g兩␺nj 兩2 ␺nj − ␺nj+1 + 4h2 f nj , j−1 j+1 ␶ ␶

which can be reduced to the matrix equation

␺n+1 = Q−1关Hn␺n + Fn兴,

␺1 = 共R共x1兲,R共x2兲, . . . ,R共xM 兲兲T ,

共27兲

where Q is a tridiagonal matrix with elements qii = 2共2ih − ␶兲 / ␶, qii+1 = qi−1,i = 1, i = 1 , 2 , . . . , M, Hn n n n is a tridiagonal matrix with elements hkk = 2共1 + 2共ih2 / ␶兲 + 2h2Vnk + 2h2g兩␺nk 兩2兲, hkk+1 = hk−1k = −1, n T n T n n n n n n n ␺ = 共␺1 , ␺2 , . . . , ␺M 兲 , and F = 共f 1 , f 2 , . . . , f M 兲 . Thus Q and H are two M ⫻ M matrices. Equation 共27兲 is the implicit discrete representation of the GPE with source 共3兲. The explicit form of the discrete representation of Eq. 共3兲 is numerically unstable if the time step ␶ is too large, but the implicit discrete representation 共27兲 is unconditionally stable. For each class of solutions of Eq. 共3兲关either linear in sn共x , k兲, cn共x , k兲, or dn共x , k兲, or rational in sn2共x , k兲, or solitary wavelike兴, we 2


073520-10

J. Math. Phys. 48, 073520 共2007兲


FIG. 6. 共Color online兲 Unstable solutions from Eq. 共12兲. Here, g = 1; V0 = −1, k = 冑 3 , ␣ = −1, v = 1; K = 27冑2 , and ␻ = 2 . For 1 −1 these parameters, the computed coefficients 共14兲 give A3 = 冑2 and B3 = 3冑2 . The bottom figures show the dynamics of the exact solution 共left兲 and the potential 共right兲 for t 苸关0 , 0.108兴. 2

5

1

apply the implicit discrete representation 共27兲 to find the perturbed numerical solutions with perturbation ⑀ = 0.01. Figures 4–8 show that the amplitude of the wave increases with time t and that all the solutions found in the previous section are unstable. In these figures, the three columns give, from left to right, the dynamics of the numerical solution, the error 兩兩␺共x , T兲兩2 − 兩␺0共x , T兲兩2兩 at the final time T, and the error 兩兩␺共x , t兲兩2 − 兩␺0共x , t兲兩2兩 for t 苸关0 , T兴, where ␺0共x , t兲 is the exact solution, ␺共x , t兲 is the perturbed numerical solution, and T is the final time, shown in the third column. The bottom figure shows 共left兲 the dynamics of the exact solution and 共right兲 the potential V共x , t兲 = −V0 sn2共v共x − vt兲 , k兲. Instability begins to develop around t = t0. Before instability occurs, 兩␺共x , t兲兩2 is equal to its initial condition, up to effects due to the added noise and numerical round-offs. We consider solution 共6兲 with coefficients 共8兲 under conditions 共7兲. It is seen from Fig. 4 that this solution is unstable. The figure shows the dynamics of the perturbed solution and the error for the time intervals 关0,0.04兴, 关0,1.2兴, and 关0,5兴共from the second bottom to the top兲. The first bottom figure shows the dynamics of the exact solution, while the second one shows the potential, for t 苸关0 , 0.04兴. The instability onset occurs between t = 0.04 and t = 1.2. As one can observe from the second column of this figure, instability begins to develop around t = 1. Solution 共9兲 with coefficients 共11兲 under conditions 共10兲 is observed to be numerically unstable. Its dynamics is shown in Fig. 5 for the time intervals 关0,0.08兴, 关0,1.1兴, and 关0,3兴共from the


073520-11


J. Math. Phys. 48, 073520 共2007兲

FIG. 7. 共Color online兲 Unstable solutions from Eq. 共15兲. Here, g = −0.716 84, V0 = 1, k = 0.7, ␣ = v = 1, K = −1.6028, and ␻ = −3.8349. For these parameters, the computed coefficients 共17兲 give A = 0.67114 and B = −0.623 92. The bottom figures show the dynamics of the exact solution 共left兲 and the potential 共right兲 for t 苸关0 , 0.025兴.

second bottom to the top兲. This figure has the same format as Fig. 4. The instability onset occurs between t = 0.08 and t = 1.1. From the error column 共second column兲, it is seen that instability begins to develop around t = 0.1. Solution 共12兲 with coefficients 共14兲 under conditions 共13兲 is observed to be numerically unstable, as one can see from its dynamics shown in Fig. 6 for the time intervals 关0,0.108兴, 关0,3.8兴, and 关0,4.2兴共from the second bottom to the top兲. It is seen from this figure that the onset on instability occurs between t = 0.108 and t = 3.8, and instability begins to develop around t = 2. The dynamics of solution 共15兲 with coefficients 共16兲 under conditions 共17兲 and 共18兲 is shown in Fig. 7 for the time intervals 关0,0.025兴, 关0,0.1兴, and 关0,0.5兴共from the second bottom to the top兲. This figure shows that the instability onset occurs between t = 0.025 and t = 0.1, while instability begins to develop around t = 0.09. Figure 8 shows the dynamics of the solitonlike solution 共24兲 obtained from solution 共12兲 for the time intervals 关0,0.085兴, 关0,2.2兴, and 关0,2.8兴共from the second bottom to the top兲. The instability onset occurs between t = 0.085 and t = 2.2, while instability begins to develop around t = 2. As conclusion on the numerical simulation, we numerically find that all the solutions found in Sec. II are unstable. It is important to note that the instability of a solution does not exclude its


073520-12


J. Math. Phys. 48, 073520 共2007兲

FIG. 8. 共Color online兲 Unstable soliton like solutions 共24兲. Here, g = 1, V0 = −1, k = 1, ␣ = v = 冑 3 , K = − 27 , and ␻ = 3 . These 1 1 parameters give A2 = 冑3 and B2 = 3 . The bottom figures show the dynamics of the exact solution 共left兲 and the potential 共right兲 for t 苸关0 , 0.085兴. 2

1

2

observation in experiments. If the time scale over which the onset of instability occurs far exceeds the duration of the experiments, then a formally unstable solution is as relevant for the experiment as a stable one. IV. CONCLUSION

We have considered the time-dependent GPE with an elliptic time-dependent potential in the presence of a source term. Exact periodic solutions of this equation were found in terms of the Jacobian elliptic function. Some exact solutions of linear type contain soliton solutions, while the exact solutions of rational type contain singular solitary waves solutions, as well as the singular trigonometric solutions, implying simultaneous onset of chaos at a periodic set of points. The stability of these exact solutions was investigated numerically by means of the implicit CrankNicholson scheme. Using random-phase perturbations, we found that all exact solutions in Sec. II are unstable. However, the time scale for the onset of instability for different classes of solutions varies significantly. The soliton solutions becomes unstable more quickly. Our procedure is applicable both for the attractive and repulsive cases. Because of their exact nature, these will provide a better starting point for the treatment of general externally driven GPE. Considering the utility of this equation in the theory of BEC, the field of nonlinear optics, and other branches of physics, our results may find practical applications.


073520-13


J. Math. Phys. 48, 073520 共2007兲

ACKNOWLEDGMENTS

The authors benefited greatly from discussions with Professor B. A. Malomed. This research was partially supported by the Programme “Soutien et renforcement de l’excellence universitaire” de l’Agence Universitaire de la Francophonie, projets “Professeurs-chercheurs invités du Sud,” the Natural Sciences and Engineering Research Council of Canada, and the Centre de Recherche Mathématiques of the Université de Montréal. E. P. Gross, Nuovo Cimento 20, 454 共1961兲. L. P. Pitaevskii, Sov. Phys. JETP 13, 45 共1961兲. 3 L. D. Carr, M. A. Leung, and W. P. Reinhardt, J. Phys. B 33, 3983 共2000兲. 4 V. M. Pérez-García, H. Michinel, and H. Herrero, Phys. Rev. A 57, 3837 共1998兲. 5 E. A. Ostrovskaya, Y. S. Kivshar, M. Lisak, B. Hall, F. Cattani, and D. Anderson, Phys. Rev. A 61, 031601 共2000兲. 6 S. Giovanazzi, A. Smerzi, and S. Fantoni, Phys. Rev. Lett. 84, 4521 共2000兲. 7 D. Ananikian and T. Bergeman, Bull. Am. Phys. Soc. 49, 78 共2004兲. 8 L. Pezzé, A. Smerzi, G. P. Berman, A. R. Bishop, and K. A. Collins, New J. Phys. 7, 85 共2005兲. 9 F. Dalfono, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 共1999兲. 10 E. H. Lieb, R. Seiringa, and J. Yngvason, Phys. Rev. A 61, 043602 共2000兲. 11 S. Albeverio, F. Gesztesy, R. Hoegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics 共Springer-Verlag, New York, 1998兲. 12 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions 共National Bureau of Standards, Washington, DC, 1964兲. 13 K. Berg-Sorensen and K. Molmer, Phys. Rev. A 58, 1480 共1998兲. 14 T. Paul, K. Richter, and P. Schlagheck, Phys. Rev. Lett. 94, 020404 共2005兲. 15 R. Hao, R. Yang, P. Nie, L. Li, and G. Zhou, Phys. Scr. 74, 132 共2006兲. 16 N. Borisov, P. Stubbe, and L. Gorbunov, Phys. Plasmas 6, 268 共1999兲. 17 S. V. Nazarenko, A. C. Newell, and V. E. Zakharov, Phys. Plasmas 1, 2827 共1994兲. 18 T. S. Raju, C. N. Kumar, and P. K. Panigrahi, J. Phys. A 38, L271 共2005兲. 19 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. 20 G. Fibich and A. L. Gaeta, Opt. Lett. 25, 335 共2000兲. 21 L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1095 共1980兲. 22 R. W. Boyd, Nonlinear Optics 共Academic, Boston, MA, 1992兲. 23 M. Wenstein, Commun. Math. Phys. 87, 567 共1983兲. 24 W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vallerling, Numerical Recipes in Fortran 共Cambridge University Press, Cambridge 1992兲. 25 S. Z. Pilinsky, Z. Sipus, and L. Sumichrast, Proceedings of the 17th International Conference on Applied Electromagnetics and Communication 共ICECOM’03兲, Dubrovnik, Croatia, 274 共2003兲 1 2
