A Generalized Method and Exact Solutions in Bose Einstein Condensates in an Expulsive Parabolic Potential

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1 Commun. Theor. Phys. (Beijing, China 48 (007 pp c International Academic Publishers Vol. 48, No. 3, September 15, 007 A Generalized Method and Exact Solutions in Bose Einstein Condensates in an Expulsive Parabolic Potential LI Biao 1,, and CHEN Yong 1,,3 1 Nonlinear Science Center and Department of Mathematics, Ningbo University, Ningbo 31511, China Institute of Theoretical Computing, East China Normal University, Shanghai 0006, China 3 MM Key Lab, the Chinese Academy of Sciences, Beijing , China (Received September 1, 006; Revised November 0, 006 Abstract In the paper, a generalized sub-equation method is presented to construct some exact analytical solutions of nonlinear partial differential equations. Making use of the method, we present rich exact analytical solutions of the onedimensional nonlinear Schrödinger equation which describes the dynamics of solitons in Bose Einstein condensates with time-dependent atomic scattering length in an expulsive parabolic potential. The solutions obtained include not only non-traveling wave and coefficient function s soliton solutions, but also Jacobi elliptic function solutions and Weierstrass elliptic function solutions. Some plots are given to demonstrate the properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient. PACS numbers: 0.30.Jr, Yv, 0.70.Wz Key words: nonlinear Schrödinger equation, symbolic computation, soliton 1 Introduction The equation for the wavefunction of Bose Einstein condensates (BECs has the form of a multidimensional nonlinear Schrödinger (NLS equation with a trap potential. It is usually known as the Gross Pitaevskii (GP equation. In the quasi-one-dimensional case, the dynamics of the wavefunction can be modeled by a one-dimensional (1D NLS equation, leading to different forms of soliton solutions. Recently, with the experimental observation and theoretical studies of BECs, [1] there has been intense interest in the nonlinear excitations of the atomic matter waves, such as dark, [] bright solitons, [3 5] and vortices. [6] Recent experiments have demonstrated that the variation of the effective scattering length, even including its sign, can be achieved by utilizing the so-called Feshbach resonance. [7,8] In Ref. [9], it has been demonstrated that the variation of nonlinearity of the GP equation via Feshbach resonance provides a powerful tool for controlling the generation of bright and dark soliton trains starting from periodic waves. At the mean-field level, the GP equation governs the evolution of the macroscopic wave function of BECs. In the physically important case of the cigar-shaped BECs, it is reasonable to reduce the GP equation into a onedimensional NLS equation, [4,10 13] ψ(x, t i + ψ(x, t t x + a(t ψ(x, t ψ(x, t λ x ψ(x, t = 0. (1 In Eq. (1, time t and coordinate x are measured in units /ω and a, where a = ( h/mω 1/ and a 0 = ( h/mω 0 1/ are linear oscillator lengths in the transverse and cigar-axis directions, respectively. ω and ω 0 are corresponding harmonic oscillator frequencies, m is the atomic mass, and λ = ω 0 /ω 1. The Feshbachmanaged nonlinear coefficient reads a(t = a s (t /a B = g 0 exp(λt, where a B is the Bohr radius. [14] The normalized macroscopic wave function ψ(x, t is connected to the original order parameter Ψ(r, t as follows: 1 ( x Ψ(r, t = ψ, ω t πab a a exp ( iω t y + z. ( a In Ref. [14], Liang et al. presented a family of exact solutions of Eq. (1 by using the so-called Darboux transformation [15] and analyzed the dynamics of a bright soliton Eq. (1. In Ref. [16], Zhang et al. obtained two families of solitons of Eq. (1 and investigated the propagation of solitons under the Feshbach-managed nonlinear coefficient a(t = g 0 exp(λt. Recently, three families of exact analytical solutions of Eq. (1 were constructed by the generalized Riccati equation rational expansion method. [17] The motivation for the present study lies in the physical importance of the BECs and the need to have some exact solutions, especially solitons. To have some explicit analytical solutions of Eq. (1 may enable one to better understand the physical phenomena which it describes. The exact solutions, which are accurate and explicit, may help physicists and engineers to discuss and The project supported by the Natural Science Foundation of Zhejiang Province of China under Grant Nos and Y604056, the Doctoral Foundation of Ningbo City under Grant No. 005A61030, and the Postdoctoral Science Foundation of China under Grant No biaolee000@yahoo.com.cn

2 39 LI Biao and CHEN Yong Vol. 48 examine the sensitivity of the model to several physical parameters. To construct exact solutions to nonlinear partial differential equations (NLPDEs, a number of powerful methods have been presented. We can cite the Darboux transformation, [15] the inverse scattering transformation and Bäcklund transformation, [18] the formal variable separation approach, [19] the Jaccobi elliptic function method, [0] the tanh method, [1 4] various extended tanh methods [5 31] and generalized projective Riccati equations method [3,33] and so on. Recently, Fan [34] developed an algebraic method which further exceeds the applicability of tanh method in obtaining a series of exact solutions of nonlinear equations. More recently, Yan, [35] Chen et al. [36] and Yomba [37] have further developed this method and obtained some new and more general solutions for some NLPDEs. In this paper, on the basis of the work done in Refs. [34] [37], we present a generalized sub-equation method by a more general rational forms so that it can be used to obtain more general formal solutions. The solutions obtained include a series of non-traveling wave and coefficient functions solutions, namely: soliton-like solutions, Jacobi elliptic functions solutions, and Weierstrass elliptic function solutions. The paper is organized as follows. In Sec., a generalized sub-equation method is established. In Sec. 3, rich exact solutions of Eq. (1 are obtained by the generalized method and some plots are given to demonstrate the main properties of some exact solutions under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient. Finally, some conclusions are given. Method Now we establish the generalized sub-equation method as follows. Give a nonlinear partial differential equations with, say, two variables: {z, t}, p(u t, u z, u zt, u tt, u zz,... = 0. (3 Step 1 We assume that the solutions of Eq. (3 are the following forms: u(z, t = A 0 + M i=1 φi 1 (ξ[a i φ(ξ + B i φ (ξ] a 0 + N j=1 φi 1 (ξ[a j φ(ξ + b j φ (ξ], (4 where A 0 = A 0 (z, t, A i = A i (z, t, B i = B i (z, t (i = 1,..., M, a 0 = a 0 (z, t, a j = a j (z, t, b j = b j (z, t (j = 1,..., N, ξ = ξ(z, t are all differentiable functions of {z, t}, h i (i = 0, 1,, 3, 4 are constants and the new variable φ = φ(ξ satisfies ( dφ φ (ξ = dξ = h 0 + h 1 φ(ξ + h φ (ξ + h 3 φ 3 (ξ + h 4 φ 4 (ξ. (5 The parameters M and N can be determined by balancing the highest-order derivative term and the nonlinear terms of Eq. (3. M and N are usually positive integers, if not, some proper transformation u(z, t u m (z, t may be in order to satisfy this requirement. Step Substituting Eq. (4 along with Eq. (5 into Eq. (3, extracting the numerator of the obtained system, setting the coefficients of φ i (ξ(φ (ξ j (i = 0, 1,... ; j = 0, 1 to zero, we obtain a set of over-determined PDEs (or ODEs with regard to the differential functions A 0, A i, B i, (i = 1,,..., M, a 0, a j, b j, (j = 1,,..., N, ξ. Step 3 Solving the over-determined PDEs (or ODEs by a symbolic computation system Maple, we would end up the explicit expressions for A 0, A i, B i (i = 1,..., M, a 0, a j, b j (j = 1,..., N and ξ or the constraints among them. Step 4 By using the results obtained in the above steps and the various solutions of Eq. (5, we can derive rich solutions for Eq. (3. By considering the different values of h 0, h 1, h, h 3, and h 4, we can derive that equation (5 admits a series of fundamental solutions. For simplicity, we only list some solitary wave solutions, Jacobi and Weierstrass doubly periodic solutions as follows. (i Solitary Wave Solutions (a Bell shaped solitary wave solutions φ(ξ = h sech( h ξ, h 4 h 0 = h 1 = h 3 = 0, h > 0, h 4 < 0, (6 φ(ξ = h sech ( h h 3 ξ, h 0 = h 1 = h 4 = 0, h > 0. (7 (b Kink-shaped solitary wave solutions φ(ξ = h ( tanh h h 4 ξ, h 0 = h, 4h 4 h 1 = h 3 = 0, h < 0, h 4 > 0, (8 φ(ξ = 1 [ p + ( p q p 4qr tanh 4qr ] ξ, (9 where p 4qr > 0, pqr 0 and h 0 = r, h 1 = rp, h = rq + p, h 3 = pq, h 4 = q. (c Solitary wave solutions When h 0 = h 1 = 0, equation (5 has a solution as follows: φ(ξ = = 4h exp( h (C 1 + ξ (4h h 4 h 3 exp(c 1 h + h 3 exp( h (ξ + C 1 exp( h ξ 4h C 0 sech( h ξ C 0 sech( h ξ (1 + C 0 (4h h 4 h 3 tanh( h ξ + C 0 (4h h 4 h 3 1, (10 where C 0 = exp( h C 1 is an arbitrary constant.

3 No. 3 A Generalized Method and Exact Solutions in Bose Einstein Condensates in an Expulsive Parabolic Potential 393 (ii Jacobi and Weierstrass Doubly Periodic Solutions When h 1 = h 3 = 0, we have the following solutions for Eq. (5, h m φ(ξ = ( h 4 (m 1 cn h m 1 ξ, h 4 < 0, h > 0, h 0 = h m (1 m h 4 (m 1, (11 h ( φ(ξ = h 4 ( m dn h m ξ, h 4 < 0, h > 0, h 0 = h (1 m h 4 ( m, (1 h m φ(ξ = ( h 4 (m + 1 sn h m + 1 ξ h, h 4 > 0, h < 0, h 0 = m h 4 (m + 1, (13 where 0 m 1 is a modulus; ( h 3 φ(ξ = ξ, g, g 3, h = h 4 = 0, h 3 > 0, (14 where g = 4h 1 /h 3 and g 3 = 4h 0 /h 3 are called invariants of Weierstrass elliptic function. When m 1, the Jacobi functions degenerate to the hyperbolic functions, i.e. sn ξ tanh ξ, cn ξ sech ξ, dn ξ sech ξ. When m 0, the Jacobi functions degenerate to the triangular functions, i.e. sn ξ sin ξ, cn ξ cos ξ, dn ξ 1. 3 Exact Solutions of Equation (1 in BECs We now investigate the NLS equation (1 with the generalized sub-equation method proposed in Sec.. Firstly, we select the solutions of (1 as the following special forms: ( A0 (t + A 1 (tφ(ξ + B 1 (tφ (ξ ψ(x, t = 1 + µφ(ξ + νφ (ξ exp[iθ(x, t], (15 Θ(x, t = η (tx + η 1 (tx + η 0 (t, (16 ξ = f(tx + g(t, (17 where A 0 (t, A 1 (t, B 1 (t, f(t, g(t, h 0 (t, h 1 (t, and h (t are differential functions of t to be determined, µ and ν are constants, and φ(ξ satisfies Eq. (5. Substituting Eqs. (15 (17 into Eq. (1, removing the exponential terms, collecting coefficients of monomials of φ(ξ, φ (ξ, and x of the resulting system, then separating each coefficients to the real part and imaginary part and setting each part to zero, yields an ordinary differential equation (ODE system with respect to differentiable functions a(t, A 0 (t, A 1 (t, B 1 (t, f(t, g(t, η 0 (t, η 1 (t, and η (t. Because the ODE system includes 116 equations, for simplification, we omit them in the paper. Solving the ODE system with symbolic computation system: Maple, we can obtain five sets of solutions. Then from Eqs. (6 (17 and the five sets of solutions from ODE system, the following five families of solutions to Eq. (1 can be derived. In the following paper, c i (i = 0, 1,... are arbitrary constants and η (t is satisfied by the following condition: η (t = 1 ( e λ(t+ C 0 1λ e λ(t+ C 0 or η (t = ± λ 4, (18 where C 0 is an arbitrary constant. Thus from Eqs. (15 (17 and Cases 1 5, five families of exact analytical solutions for BECs equation (1 are obtained as follows. Family 1 When h 1 = h 3 = µ = ν = A 0 (t = B 1 (t = 0 and under different parameters of h 0, h, and h 4, three solutions of Eq. (1 are obtained, ψ 11 = c exp h m η (tdt h 4 (m 1 ( h cn m 1 ξ exp[iθ(x, t], (19 where h 4 < 0, h > 0, h 0 = h m (1 m /h 4 (m 1 ; ( ψ 1 = c exp h η (tdt h 4 ( m ( h dn m ξ exp[iθ(x, t], (0 where h 4 < 0, h > 0, h 0 = h (1 m /h 4 ( m ; ψ 13 = c exp h m η (tdt h 4 (m + 1 ( sn h m + 1 ξ exp[iθ(x, t], (1 where h 4 > 0, h < 0, h 0 = h m /h 4 (m + 1. If further setting m 1, from the solutions {ψ 11, ψ 1 } and ψ 13, a bright soliton and a dark soliton can be derived as follows: ψ 14 = c exp ψ 15 = c exp h η (tdt sech ( h ξ exp[iθ(x, t], h 0 = 0, ( h 4 h ( h η (tdt tanh h 4 ξ exp[iθ(x, t], h 0 = h. (3 4h 4

4 394 LI Biao and CHEN Yong Vol. 48 In {ψ 11, ψ 1, ψ 13, ψ 14, ψ 15 }, ξ, Θ(x, t, and a(t are all determined by the following equations, ξ = c 3 exp 4 η (tdt x c 1 c 3 exp 8 η (tdt dt + c 4, Θ(x, t = η (tx + c 1 exp 4 η (tdt x + ( c 1 + h c 3 exp 8 η (tdt dt + c 5, a(t = h 4c 3 c exp 4 η (tdt. (4 Family When ν = B 1 (t = 0, and h 0, h 1, h 4, µ( c are arbitrary constants, a family of solution to (1 is obtained, where where where ξ = c 3 A 0(tx c 1 c 3 A 4 0(tdt + c 4, ψ (x, t = A 0 (t 1 + c φ(ξ exp[iθ(x, t], (5 1 + µφ(ξ [ c Θ(x, t = η (tx + c 1 A 0(tx + 3 c µ(µ + c µ + c h 0 c 3h 4 (µ + c + c 3(c + µ h 1 (µ + c [ h 1 µ a(t = (µ + c µ3 (µ + c h 0 + h ] 4 (µ + c c 3A 0(t, A 0 (t = c 0 exp η (tdt, c 1] A 4 0(tdt + c 5, h 3 = h 1 c µ 4(c µ h 0 + h 4, h = c µh 0 + (c + 6c µ + µ h 1 + 4h 4 µ + c (µ + c (µ + c. (6 If further setting µ = 0, h 0 = h 1 = 0, from Eqs. (10 and (5, we can derive the following solutions of Eq. (1, 4h C 0 sech( h ξ ψ 1 (x, t = c 0 exp η (tdt [1 + c φ(ξ] exp[iθ(x, t], φ(ξ = C 0 sech( h ξ tanh( h ξ 1, (7 Family 3 h 3 = 4h 4, h = 4h 4 c c, A 0 (t = c 0 exp η (tdt, Θ(x, t = η (tx + c 1 A 0(tx (c 1c + h 4 c 3 A 4 0(tdt c + c 5, ξ = c 3 A 0(tx c 1 c 3 A 4 0(tdt + c 4, a(t = A 0(th 4 c 3 c. (8 When ν = B 1 (t = 0, and h 0, h 1, h are arbitrary constants, a solution of Eq. (1 is derive as follows: A 0 (t = c 0 exp ψ 3 (x, t = A 0 (t 1 + c φ(ξ exp[iθ(x, t], (9 1 c φ(ξ η (tdt, ξ = c 3 (A 0 (t x c 3 c 1 A 4 0(tdt + c 4, ( Θ(x, t = η (tx + c 1 A 0(tx + 3c c 3 3h 0 c 3 h c 1 A 4 0(tdt + c 5, h 4 = c 4 h 0, h 3 = c h 1, a(t = 1 4 c 3 A 4 0(t(c h 1 + h + h 0 c. (30 If further setting h 0 = r, h 1 = rp, h = rc r + p, h 3 = pc r = c h 1, h 4 = c 4 r = c 4 h 0, from Eq. (9, the solution (9 is changed into the following forms, 1 + c φ(ξ ψ 31 (x, t = c 0 exp η (tdt exp[iθ(x, t], 1 c φ(ξ φ(ξ = 1 [ ( c r p + p 4c p r tanh 4c ] r ξ. (31 Family 4 When ν = A 0 (t = B 1 (t = 0, and h 0, h, h 4, µ( 0 are arbitrary constants, a solution of Eq. (1 is derived as follows: φ(ξ ψ 4 = c exp η (tdt exp[iθ(x, t], (3 1 + µφ(ξ

5 No. 3 A Generalized Method and Exact Solutions in Bose Einstein Condensates in an Expulsive Parabolic Potential 395 where ξ = c 3 exp 4 η (tdt x c 1 c 3 ( Θ(x, t = η (tx + c 1 exp 4 α = µ4 h 0 µh 3 + h 4 c c 3 exp 4 exp 8 η (tdt dt + c 4, η (tdt x 4c 3µ 3 h 0 + c 1µ c 3h 3 µ exp 8 η (tdt dt + c 5, η (tdt, h 1 = 4µh 0, h = 8h 0µ 3 + h 3. (33 µ If further setting h = h 4 = 0, h 3 = 8h 0 µ 3 > 0 in Eq. (3, a Weierstrass elliptic function solution is obtained from Eq. (14, ( h3 ψ 4 = c exp η (tdt ξ, 1 µ, µ µ ( h3 ξ, exp[iθ(x, t]. (34 1 µ, µ 3 Family 5 When ν = A 0 (t = A 1 (t = 0, and h 0, h, h 4, µ( 0 are arbitrary constants, a solution of Eq. (1 is derived as follows: φ ξ ψ 5 = c 0 exp η (tdt exp[iθ(x, t], ( µφ(ξ where h 0, h, h 4, µ 0 are arbitrary constants and ξ = c B1(tx c 1 c B1(tdt 4 + c 3, B 1 (t = c 0 exp η (tdt, Θ(x, t = η (tx + c 1 B1(tx c µ h 1c h 4 + c 1µ B1(tdt + c 4, µ a(t = c µ B 1 (t, h 1 = (µ h 4h 4 µ 3, h 3 = 4h 4 µ. (36 If setting h 0 = r, h 1 = rp, h = rp + p, h 3 = pq, h 4 = q, h 3 = 4h 4 /µ µ = q/p, from Eq. (9, the solution Eq. (35 is changed into the following forms, p p 4qr [ ( p ψ 51 = c 0 exp η (tdt coth 4qr ( p ξ tanh 4qr ] ξ exp[iθ(x, t]. (37 4q Remark From the solutions ( and (3, it is not difficult to verify that the dark soliton (11 and bright soliton (1 obtained in Ref. [16] can be recovered. Therefore the solutions of Eq. (1 obtained in Ref. [16] are special cases of the solutions (19 (1. But to our knowledge, the other solutions have not been reported earlier. If we select η (t = λ/4 in Eqs. ( and (3, then the scattering length a(t = (h 4 c 3/c exp(λt = g 0 exp(λt, i.e., the Feshbach-managed nonlinear coefficient [1] (because of λ 1, the nonlinear coefficient a(t = g 0 exp(λt can be expressed as a(t = g 0 [1+λt+o (λt]. Under this condition, we can obtain well-known dark and bright solitons. [14,16] If we select η (t = (1/4( e λ(t+ C 0 1λ/(1+ e λ(t+ C 0, then the scattering length a(t = (h 4 c 3/c sech (λ(t+ C 0 = g 1 sech (λ(t + C 0, i.e., the hyperbolic secant function coefficient. Fig. 1 The evolution plots of the solutions (19 under the following parameters: η (t = λ/4, λ = 0.01, h = 1, h 4 = 1, c 1 = 0.1, c = 1, c 3 = 0, c 4 = c 5 = 0. (a m = 0.9, (b m = 1.

6 396 LI Biao and CHEN Yong Vol. 48 Fig. Fig. 1. The evolution plots of the solutions (19 with η (t = λ/4, other parameters are the same as the parameters in Fig. 3 The evolution plots of the solutions (19 with η (t = (1/4(e λ(t+ C 0 1λ/(1 + e λ(t+ C 0, other parameters are the same as the parameters in Fig. 1. Fig. 4 The evolution plots of the solutions (1 with η (t = λ/4, λ = 0.01, h = 1, h 4 = 1, c 1 = 0.1, c = 1, c 3 = 0, c 4 = c 5 = 0. (a m = 0.9, (b m = 1. In order to understand the significance and main features of these solutions obtained in the paper, the solutions (19 and (1 were chosen to investigate by using direct computer simulations. From Figs. 1 6, we can see that when 0 < m < 1, the solutions (19 and (1 show the periodic property of Jacobi elliptic functions; and when m = 1 the solutions (19 and (1 show the soliton property. (Note: in the figures of this paper: M = ψ(x, t. Two figures (Fig. 1(b and Fig. 4(b are plotted to show the dynamics of the Feshbach resonance managed soliton in the expulsive parabolic potential given by Eqs. (19 and (1 under η (t = λ/4, m = 0.9 and some other special

7 No. 3 A Generalized Method and Exact Solutions in Bose Einstein Condensates in an Expulsive Parabolic Potential 397 parameters. From them, we can see that the amplitude of the solitary wave increases and the width gets compressed during their propagation. When η (t = λ/4, m = 0.9, the propagation of the solitons (19 and (1 is contrary to the case of η (t = λ/4, m = 0.9. When η (t = (1/4( e λ(t+ C 0 1λ/(1 + e λ(t+ C 0, from Figs. 3(b and 6(b, the height of the solitary wave decreases exponentially and its width gets enlarged during its propagation. Fig. 5 Fig. 1. The evolution plots of the solutions (1 with η (t = λ/4, other parameters are the same as the parameters in Fig. 6 The evolution plots of the solutions (1 with η (t = (1/4( e λ(t+ C 0 1λ/(1 + e λ(t+ C 0, other parameters are the same as the parameters in Fig Summary In this paper, to construct some exact analytical solutions of nonlinear partial differential equations, a generalized sub-equation method with symbolic computation is presented. Making use of the method and symbolic computation, we present rich exact analytical solutions of the one-dimensional nonlinear Schrödinger equation in Bose Einstein condensates with the time-dependent interatomic interaction in an expulsive parabolic potential, which include nontravelling wave and coefficient function s soliton solutions, Jacobi elliptic functions solutions and Weierstrass elliptic function solutions. The dynamics of some exact solutions are simulated by computer simulation under the Feshbachmanaged nonlinear coefficient and the hyperbolic secant function coefficient. The method proposed here can be applied to construct some exact solutions for other PDE and coupled ones. References [1] F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71 ( [] T. Busch and J.R. Anglin, Phys. Rev. Lett. 84 (000 98; C.K. Law, C.M. Chan, P.T. Leung, and M.C. Chu, Phys. Rev. Lett. 85 ( ; B.P. Anderson, P.C. Haljan, et al., Phys. Rev. Lett. 86 (001 96; B. Wu, J. Liu, and Q. Niu, Phys. Rev. Lett. 88 ( [3] K.E. Strecker, G.B. Partridge, A.G. Truscott, and R.G. Hulet, Nature (London 417 ( [4] V.M. Perez-Garcia, H. Michinel, and H. Herrero, Phys. Rev. A 57 ( ; G. Fibich and X.P. Wang, Physica D (Amsterdam 175 (003 96; K.D. Moll, A.L. Gaeta, and G. Fibich, Phys. Rev. Lett. 90 ( [5] P.G. Kevrekidis, G. Theocharis, D.J. Frantzeskakis, and

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