Exact Solutions of the Two-Dimensional Cubic-Quintic Nonlinear Schrödinger Equation with Spatially Modulated Nonlinearities

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1 Commun. Theor. Phys. 59 (2013) Vol. 59, No. 3, March 15, 2013 Exact Solutions of the Two-Dimensional Cubic-Quintic Nonlinear Schrödinger Equation with Spatially Modulated Nonlinearities SONG Xiang (Ý ) and LI Hua-Mei (ÓÜÛ) Department of Physics, Zhejiang Normal University, Jinhua , China (Received September 21, 2012; revised manuscript received December 7, 2012) Abstract Applying the similarity transformation, we construct the exact vortex solutions for topological charge S 1 and the approximate fundamental soliton solutions for S = 0 of the two-dimensional cubic-quintic nonlinear Schrödinger equation with spatially modulated nonlinearities and harmonic potential. The linear stability analysis and numerical simulation are used to exam the stability of these solutions. In different profiles of cubic-quintic nonlinearities, some stable solutions for S 0 and the lowest radial quantum number n = 1 are found. However, the solutions for n 2 are all unstable. PACS numbers: Lm, y, Yv Key words: vortex, fundamental soliton, harmonic potential 1 Introduction The nonlinear Schrödinger equation (NLSE), as a universal mathematical model, is applied to solve various nonlinear physical phenomena, especially in nonlinear optics [1] and Bose-Einstein condensates (BECs). [2] In the one-dimensional setting with the constant or variable coefficients, a great deal of analytical and numerical soliton solutions [3] for bright and dark types have been constructed. In Ref. [4], many special kinds of explicit soliton solutions, such as periodic, resonant, and quasiperiodically oscillating soliton solutions, of the NLSE with spatiotemporally inhomogeneous nonlinearities have been reported. Beitia et al. [5] and Tang et al. [6] showed that localized spatially dependent nonlinearities can support bound states with an arbitrary number solitons, respectively. Wang et al. [7] investigated localized waves in the BECs with spatially inhomogeneous two- and three-body interactions in various external potentials. However, as we know, most of recently developed investigations turned into the direction of higher dimensional settings, [8 9] especially for two-dimensional (2D) vortex solutions. [10 13] In 2D geometry, vortex solutions are localized solutions of nonlinear physical equations, which are characterized by a phase singularity at the pivotal point. However, vortices in self-attractive BECs are unstable against collapse. The same problem impedes creation of vortices in Kerrtype focusing media. In BECs, the harmonic [11 12] and periodic optical lattice potentials [14 15] are used to stabilize vortices. In nonlinear optics, the important scheme supporting stable vortices is the localized bulk medium with competing nonlinearity, e.g., cubic-quintic (CQ) nonlinear medium. [15] Many references [15 17] shown that the constant-coefficient NLSE with the attractive (or focusing) cubic and repulsive (or defocusing) quintic nonlinear terms is the well-known model to allow for the existence of stable structures, such as Dong et al. [16] investigated the existence and stability of numerically found vortex solutions. In the past decade, with the development of the means for nonlinearity management, for instance, the magnetic [18] or optical [19] Feshbach-resonance technique, the 2D cubic NLSE with the inhomogeneous nonlinearity has been discussed in many literature. [11 13] Recently, Wu et al. [11] and Wang et al. [12] reported stable exact 2D vortex solutions in harmonic potential with variable cubic nonlinearity coefficient, respectively. However, analytical vortex solutions of 2D cubic-quintic nonlinear Schrödinger equation (CQNLSE) with the inhomogeneous nonlinearities have been the subject of relatively fewer studies. In this paper, we study the 2D CQNLSE with spatially modulated nonlinearities and an external potential which includes both the two- and three-body interactions of the condensate in BECs. [20] The equation is given by iψ t = 2 ψ + g(r) ψ 2 ψ + G(r) ψ 4 ψ + V (r)ψ, (1) where ψ is the macroscopic wave function, 2 is the 2D Laplacian, V (r) is an external potential, g(r) and G(r) are the inhomogeneous cubic and quintic nonlinear coefficients, respectively, corresponding to the two-body and three-body interactions. The signs of g(r) and G(r) are negative or positive in the radial coordinate r, standing for the interactions are attractive or repulsive, respectively. All physical parameters V (r), g(r), and G(r) are functions of radial coordinate r. [21] The remaining paper is organized as follows. In Sec. 2, we use the similarity transformation to construct the exact Supported by the National Natural Science Foundation of China under Grant No and the Natural Science Foundation of Zhejiang Province of China under Grant No. LY12A04001 Corresponding author, lihuamei@zjnu.cn c 2013 Chinese Physical Society and IOP Publishing Ltd

2 No. 3 Communications in Theoretical Physics 291 vortex solutions (S 1) and the approximate fundamental soliton solutions (S = 0) in harmonic potential. In Sec. 3, the linear stability analysis and numerical simulation are employed to verify the stability of those solutions, and some stable solutions are proposed. Finally, Sec. 4 presents the main consequences of the paper. 2 Vortex Solution and Fundamental Soliton Solution In order to construct explicit stationary solution of Eq. (1), the wave function can be wrote in the form of ψ(r, θ, t) = φ(r)e (isθ iµt), (2) where S is the topological charge, θ stands for the azimuthal angle, µ denotes the chemical potential, and the real function φ(r) describes the amplitude profile of the vortex solution. When topological charge S 0, the substitution of Eq. (2) into Eq. (1) leads to the second-order differential equation µφ+φ rr +r 1 φ r g(r)φ 3 G(r)φ 5 [S 2 r 2 + V (r)]φ = 0. Then defining φ(r) = ρ(r)u[r(r)], one can find that U[R(r)] satisfies U RR + g 0 U 3 + G 0 U 5 = 0, (3) where g 0 and G 0 are constants. Thus, we can get a set of ordinary differential equations. After some algebra, one can find that g(r) = g 0 r 2 ρ 6, G(r) = G 0 r 2 ρ 8, R(r) = r 0 s 1 ρ(s) 2 ds, ρ(r), and V (r) satisfy the equation: ρ rr + r 1 ρ r + (µ V (r) S 2 r 2 )ρ = 0. (4) For constructing exact vortex solutions, Eq. (4) must be solvable. For harmonic potential V (r) = kr 2, ρ(r) is formed in terms of Whittaker s M and W functions: [22] ρ(r) = r 1 [c 1 M(µ/4 k, S/2, kr 2 ) + c 2 W(µ/4 k, S/2, kr 2 )], where ρ(r) must satisfy µ < 2(1 + S) k and c 1 c 2 > 0. Then, combining Eq. (2) with the solutions of Eqs. (3) and (4), the exact vortex solution of Eq. (1) can be obtained [6] ψ = 3ρ(r)sn(ςR, m) 3a0 [3 (m 2 + 1)sn 2 (ςr, m)] e(isθ iµt), (5) equation ςr( ) = 2nK(m) must be satisfied, where K(m) = π/2 [1 m 2 sin 2 (τ)] 1/2 dτ is the first kind elliptic integral and n = 1, 2, 3,... is the radial quantum 0 number. The amplitude profiles of exact vortex solutions and the corresponding profiles of CQ nonlinearities for S = 1, 2, 3 and n = 1 are shown in Fig. 1. Comparing Fig. 1(a) with 1(b), one can see that the amplitudes of vortex solutions with repulsive quintic nonlinearity are larger than those with attractive quintic nonlinearity for the same topological charge. In Figs. 2(a) and 2(b), we demonstrate the amplitude profiles and the density distributions of the vortex solutions with the attractive cubic and repulsive quintic nonlinearities for S = 1 and n = 1, 2, 3, respectively. It is displayed from Fig. 2(a) that the vortex solutions with n bright rings surrounding the vortex core. And although these vortex solutions share the same chemical potential, their energies increase with the increase of number n. The properties of vortex solutions for the attractive CQ nonlinearities are similar as shown in Fig. 2 and we do not show them. g 0 = 2 3 a 0ς 2 (m 4 m 2 + 1), G 0 = 1 9 a2 0ς 2 (m 2 2)(2m 2 1)(m 2 + 1), (6) where a 0, ς are constants, and m (0 m 1) is the modulus of the Jacobi elliptic function. Equation (5) requires a 0 > 0, which obviously gives g 0 < 0 for any m. From the second equation of Eq. (6), one can find that G 0 is negative for 0 m < 2/2 and positive for 2/2 < m 1. Thus, the cubic nonlinearity in CQNLSE (1) must be attractive, while the quintic one can be either attractive or repulsive, effecting by m. When m = 2/2, it is remarkable that G 0 = 0, which means the quintic nonlinearity identically vanishes. To impose the boundary conditions for vortex solution as lim r 0 ψ(r, θ, t) = lim r ψ(r, θ, t) = 0, the Fig. 1 (a) (b) The amplitude profiles of exact vortex solution (5) and the corresponding profiles of the CQ nonlinearities for radial quantum number n = 1 and various topological charges S = 1, 2, 3. Other parameters are c 1 = c 2 = 1, µ = 0.3, k = 0.01, a 0 = 2, (a) m = 0.8, and (b) m = 0.01.

3 292 Communications in Theoretical Physics Vol. 59 Actually, in the harmonic trap, we also consider 2D fundamental soliton solutions for topological charge S = 0. [11,23] The problem can be dealt by choosing the CQ nonlinearities as the form, in Fig. 3. g(r) = g s (0 r < r s ), g 0 r 2 ρ 6 (r r s ), G(r) = G s (0 r < r s ), G 0 r 2 ρ 8 (r r s ), where g s and G s are constants. When r r s, we can construct exact solutions as above, except that now R(r) = r r s s 1 ρ(s) 2 ds. For r < r s, if r s is small in comparison with the spatial scale of the external potential (when r s 1/k, the harmonic potential V = kr 2 is satisfied), the real wave function may be approximated as a constant, φ = 2G s ( g s + g 2 + 4G s µ + 4G s V (0))/2G s. Here, φ(r) and φ (r) must be continuous at r = r s, so ρ (r s ) = 0 and du(0)/dr = 0 should be satisfied. And g s = ±G s can be supposed to obtain g s = [µ V (0)][ρ(r s )U(0)] 2 [1 ± (ρ(r s )U(0)) 2 ] 1. Thus, the approximate fundamental soliton solution of Eq. (1) is given by Eq. (5) with ςr = nk(m)r/r( )+K(m), and n = 1, 3, 5... to make lim r ψ(r, θ, t) = 0. The amplitude profiles of fundamental soliton solutions and the corresponding profiles of CQ nonlinearities are presented Fig. 2 (a) The amplitude profiles of exact vortex solution (5) and (b) the corresponding density distributions ψ(r, θ, t) with the attractive cubic and repulsive quintic nonlinearities for topological charge S = 1 and different radial quantum numbers n = 1,2, 3. All domains in (b) are (x, y) [ 15, 15] [ 15, 15]. Other parameters are c 1 = c 2 = 1, µ = 0.3, k = 0.01, a 0 = 2, and m = 0.8. Fig. 3 (a) (b) The amplitude profiles of fundamental soliton solutions and the corresponding profiles of the CQ nonlinearities with different radial quantum numbers n = 1, 3,5. Other parameters are ρ(r s) = 3, r s = c 1 = c 2 = 1, a 0 = 2, µ = 0.18, k = 0.01, (a) m = 0.8, and (b) m = Stability Analysis Obviously, only stable vortices and fundamental solitons can be observed experimentally. Therefore the stability of our solutions against small perturbations is a crucial issue. In this section, the linear stability analysis [15,24] and numerical simulation are employed to analyze the stability of our vortex solutions and fundamental soliton solutions. In the linear stability analysis, small perturbations are given on the solution to Eq. (1) of the form as ψ(r, θ, t) = [φ(r) + u(r)e iλt+ipθ + v (r)e iλ t ipθ ] e isθ iµt, where u and v are the perturbation components, p is an integer azimuthal index of the perturbation, and λ is the corresponding eigenvalue. As usual, the solutions are stable when the imaginary component of λ equals to 0. By substituting the perturbed solution into Eq. (1) and linearizing them around φ(r), we obtain the eigenvalue problem in a matrix form, λ ( u v ) ( C + D = D C ) ( ) u, (7) v

4 No. 3 Communications in Theoretical Physics 293 where C ± = d2 d 2 r 1 d (S ± p)2 r dr r 2 + µ 2g(r)φ(r) 2 3G(r)φ(r) 4 kr 2, p = 1, 2,...,5, D = g(r)φ(r) 2 2G(r)φ(r) 4. First, the linear stability analysis of vortex solutions with two different profiles of CQ nonlinearities (for m = 0.8 and m = 0.01, respectively) are performed through solving Eq. (7) numerically. The curves of the largest instability growth rate are shown in Fig. 4, where the vortex solutions for S = 3 and n = 1 are stable within some region of values of µ. From Figs. 4(a) and 4(b) for n = 1, one can find that three-body interaction plays an important role to affect the stable region. The stable region of vortex solution (5) with the repulsive three-body interaction is larger than those with the attractive one. Further, we find that the vortex solutions with the larger radial quantum numbers n 2 are all unstable. Finally, we select the solutions which share the same chemical potential and use the split-step Fourier method to verify their stability. Figure 5 shows the evolutions and phase diagrams of vortex solutions for S = 3 and n = 1, 2. The vortex solutions for n = 1 are stable against initially perturbation with white noise of level 3%. We can see that the vortex solutions with the repulsive quintic nonlinearity are more stable than those with the attractive quintic nonlinearity in Figs. 5(a) and 5(b) for n = 2, where the vortex solutions begin to split into several humps at t = 45 and 4.5, respectively. Although we have displayed here the results of stability study only for the vortex solutions with S = 3, similar conclusions hold for the vortex solutions with other topological charges and the fundamental soliton solutions as well. Fig. 4 (a) The largest instability growth rate for vortex solutions with the attractive cubic and repulsive quintic nonlinearities for topological charge S = 3 and different radial quantum numbers n = 1, 2. (b) The same as (a) when the CQ nonlinearities are both attractive. Other parameters are c 1 = c 2 = 1, k = 0.01, a 0 = 2, (a) m = 0.8, and (b) m = Fig. 5 (a) The evolutions and phase diagrams of vortex solutions ψ(r, θ, t) with the attractive cubic and repulsive quintic nonlinearities for topological charge S = 3 and different radial quantum numbers n = 1, 2. (b) The same as (a) when the CQ nonlinearities are both attractive. In (a) and (b) for n = 1, the vortex solutions with an initial white noise of level 3%. For all cases, the domains are (x,y) [ 18, 18] [ 18, 18]. Other parameters are c 1 = c 2 = 1, µ = 0.74, k = 0.01, a 0 = 2, (a) m = 0.8, and (b) m = 0.01.

5 294 Communications in Theoretical Physics Vol Conclusions In conclusion, we have studied the exact vortex solutions and approximate fundamental soliton solutions of the two-dimensional CQNLSE with spatially modulated nonlinearities and harmonic potential by applying the similarity transformation. Significantly, within some region of values of µ, the vortex solutions for every topological charge S 1 and the lowest radial quantum number n = 1 are stable under the specific profiles of CQ nonlinearities (see Fig. 4). Whereas for n 2, all vortex solutions are unstable. And the fundamental soliton solutions share the similar properties as well. What is more, for the attractive two-body interaction, the repulsive three-body interaction can be selected to amplify the stable region and enhance the stability of our solutions. The profiles of CQ nonlinearities can be managed by the Feshbach-resonance technique of scattering length of interatomic interactions in BECs. We hope these results can motivate future studies on the vortex solutions and help to guide possible experimental work in nonlinear physical systems. References [1] Y.S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals, Academic, New York (2003). [2] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gase, Cambridge University Press, Cambridge (2002); V.A. Brazhnyi and V.V. Konotop, Mod. Phys. Lett. B 18 (2004) 627. [3] C.S. Gardner, J.M. Greene, M.D. Kruskal, and R.M. Miura, Phys. Rev. Lett. 19 (1967) 1095; V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz. 61 (1971) 118; V.E. Zakharov and A.B. Shabat, Zh. Eksp. Teor. Fiz. 64 (1973) 1627; V.N. Serkin and A. Hasegawa, Phys. Rev. Lett. 85 (2000) 4502; V.N. Serkin, A. Hasegawa, and T.L Belyaeva, Phys. Rev. Lett. 98 (2007) [4] J. Belmonte-Beitia, M. Pérez-García Víctor, V. Vekslerchil, and V.V. Konotop, Phys. Rev. Lett. 100 (2008) [5] J. Belmonte-Beitia, M. Pérez-García Víctor, and V. Vekslerchil, Phys. Rev. Lett. 98 (2007) [6] X.Y. Tang and P.K. Shukla, Phys. Rev. A 76 (2007) [7] D.S. Wang and X.G. Li, J. Phys. B 45 (2012) [8] I. Towers and B.A. Malomed, J. Opt. Soc. Am. B 19 (2002) 537; S.K. Adhikari, Phys. Rev. A 69 (2004) ; Phys. Rev. E 70 (2004) ; M. Beli`c, N. Petrovi`c, W.P. Zhong, R.H. Xie, and G. Chen, Phys. Rev. Lett. 101 (2008) [9] B.A. Malomed, D. Mihalache, F. Wise, and L. Torner, J. Optics B: Quant. Semicl. Opt. 7 (2005 ) R53; R.M. Bradley, J.E. Bernard, and L.D. Carr, Phys. Rev. A 77 (2008) ; X.S. Zhao, L. Li, and Z.Y. Xu, Phys. Rev. A 79 (2009) ; Y.X. Chen and X.H Lu, Commun. Theor. Phys. 55 (2011) 871; C.Y. Liu and C.Q. Dai, Commun. Theor. Phys. 57 (2012) 568. [10] M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77 (1996) 490; F. Dalfovo, S. Giorgini, L. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71 (1999) 463; J.J. Garcia-Ripoll and V.M. Perez-Garcia, Phys. Rev. A 64 (2001) [11] L. Wu, L. Li, J.F. Zhang, D. Mihalache, B.A. Malomed, and W.M. Liu, Phys. Rev. A 81 (2010) (R). [12] D.S. Wang, S.W. Song, B. Xiong, and W.M. Liu, Phys. Rev. A 84 (2011) [13] Q. Tian, L. Wu, Y.H. Zhang, and J.F. Zhang, Phys. Rev. E 85 (2012) ; J.R. He, H.M. Li, and L. Li, Phys. Lett. A 85 (2012) [14] D. Mihalache, D. Mazilu, F. Lederer, B.A. Malomed, Y.V. Kartashov, L.C. Crasovan, and L. Torner, Phys. Rev. Lett. 95 (2005) [15] L.W. Dong, J.D. Wang, H. Wang, and G.Y. Yin, Phys. Rev. A 79 (2009) ; M. Quiroga-Teixeiro and H. Michinel, J. Opt. Soc. Am. B 14 (1997) 8. [16] L.W. Dong, F.W. Ye, J.D. Wang, T. Cai, and Y.P. Li, Physica D 194 (2004) 219. [17] V.I. Berezhiani, V. Skarka, and N.B. Aleksić, Phys. Rev. E 64 (2001) ; D. Mihalache, D. Mazilu, I. Towers, B.A. Malomed, and F. Lederer, Phys. Rev. E 67 (2003) ; R.M. Caplan, R. Carretero-González, P.G. Kevrekidis, and B.A. Malomed, Mathematics and Computers in Simulation 82 (2012) [18] S.L. Cornish, N.R. Claussen, J.L. Roberts, E.A. Cornell, and C.E. Wieman, Phys. Rev. Lett. 85 (2000) [19] M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, Phys. Rev. Lett. 93 (2004) [20] C.Q. Dai, D.S. Wang, L.L. Wang, J.F. Zhang, and W.M. Liu, Annals of Physics 326 (2011) 2356; J.C. Chen, X.F. Zhang, B. Li, and Y. Chen, Chin. Phys. Lett. 29 (2012) ; D.S. Wang, X. Zeng, and Y.Q. Ma, Phys. Lett. A 376 (2012) [21] Y. Sivan, G. Fibich, and M.I. Weinstein, Phys. Rev. Lett. 97 (2006) ; Y. Sivan, G. Fibich, B. Ilan, and M.I. Weinstein, Phys. Rev. E 78 (2008) ; W.P. Zhong, M.R. Belić, and Y.Z. Xia, Phys. Rev. E 83 (2011) [22] E.T. Whittaker and G.N. Watson, A Course in Modern Analysis, 4th ed., Cambridge University Press, Cambridge (1990). [23] D. Mihalache, D. Mazilu, V. Skarka, B.A. Malomed, N.B. Leblond, H. Aleksić, and F. Lederer, Phys. Rev. A 82 (2010) [24] J.M. Soto-Crespo, E.M. Wright, and N.N. Akhmediev, Phys. Rev. A 45 (1992) 3168; A. Vinçotte and L. Bergé, Phys. Rev. Lett. 95 (2005)