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Exact spatial soliton solution for nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index
Optics Communications 285 (2012) 2171–2173
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Exact spatial soliton solution for nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index Yan Wang ⁎, Ruiyu Hao Department of Electronic Information and Physics, Changzhi University, Changzhi 046011, China
a r t i c l e
i n f o
Article history: Received 2 October 2011 Received in revised form 18 December 2011 Accepted 24 December 2011 Available online 5 January 2012
a b s t r a c t In this paper, we analyze (2 + 1)-dimensional nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index in the transverse direction, and obtain an exact solution in explicit form using an ansatz method. Finally, the stability of the solution is discussed numerically, and the result reveals that the solution is stable to the finite initial perturbations. © 2011 Elsevier B.V. All rights reserved.
Keywords: Spatial soliton Inhomogeneous nonlinearity Nonlinearity management Nonperiodic modulation
A lot of studies have been focused on the existence and stability of (2 + 1)-dimensional ((2 + 1)D, for brevity) spatial solitons. Strong periodic longitudinal modulation can be used to suppress diffraction of linear beam [1], such as a nonlinear waveguide array built of the properly designed segments supports diffraction-management soliton [2–7]. On the other hand, the way to stabilize the spatial solitons is that nonlinearity coefficient obeys a weak modulation along the propagation direction [8–11]. For example, (2 + 1)D spatial solitons propagate across a nonlinear bulk medium with layered structure in which Kerr nonlinearity alternates between self-focusing and selfdefocusing [9]. Besides, (2 + 1)D spatial solitons can exist in optical lattices which represent a spatially modulation of linear refractive index[12–20], or mixed linear-nonlinear lattices [21,22], i.e., the linear and nonlinear refractive indices are subjected to a spatially modulation in the transverse plane. Recently, a comprehensive survey of results obtained for solitons supported by nonlinear lattices (NLs) and their combinations with linear lattices has been offered in Ref. [23]. It is worth pointing out that the studies about (2 + 1)D spatial solitons are still far from completion. In this paper, we consider (2 + 1)D nonlinear optical system with spatially Kerr-type nonlinearity, and obtain an exact solution for nonlinear Schrödinger equation (NLSE) with nonperiodic transverse modulation of nonlinearity and linear refractive index. Upon a proper change of the notation, the NLSE may be interpreted as the Gross– Pitaevskii equation (GPE) for Bose–Einstein condensates (BECs). Accordingly, the exact solutions found in terms of the NLSE may also find applications to BECs. ⁎ Corresponding author. E-mail address: [email protected] (Y. Wang). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.090
We consider (2 + 1)D NLSE which describes laser beam propagating along the z direction in (2 + 1)D optical media with spatially inhomogeneous nonlinearities.
iqz þ
1 2 qxx þ qyy þ g ðx; yÞjqj q þ pRðx; yÞq ¼ 0; 2
ð1Þ
where q(x, y, z) = (Ldif/Lnl) 1/2A(x, y, z)I0− 1/2, and A(x, y, z) is the complex envelope of the electrical field. I0 is the input intensity, x = X/r0, y = Y/r0, r0 is the input beam width unit, z = Z/Ldif, Ldif = n0wr02/c, Lnl = 2c/wn2I0, w is the frequency, and n2 is of the order of the nonlinear correction to the refractive index because of the Kerr effect.p = Ldif/Lref represents the guiding parameter, Lref = c/(δnw), and δn denotes the refractive index modulation depth, which is small compared with the unperturbed index n0. The dimensionless profile function g(x, y) is positive for nonlinear self-focusing but becomes negative in a selfdefocusing medium. The weak modulation of linear refractive index in the transverse direction is defined by the real function R(x, y). In this case, we consider a Gaussian nonlinearity modulation such as the one may be realized in a linear waveguide with one narrow nonlinear channel embedded into it, and a nonperiodic modulation of linear refractive index in the transverse direction with a quadratic + Gaussian distribution, i.e. h i 2 2 g ðx; yÞ ¼ sg 0 exp −g 1 x þ y ;
ð2Þ
h i 2 2 2 2 ; Rðx; yÞ ¼ − x þ y þ s1 R0 exp −R1 x þ y
ð3Þ
2172
Y. Wang, R. Hao / Optics Communications 285 (2012) 2171–2173
Fig. 1. The 2D trapping potential in the form of a combination of parabolic and Gaussian terms at y = 0, here the parameters are as follows: R0 = 2, R1 = 2.
where s = 1 and s = − 1 represent self-focusing and self-defocusing medium, respectively. s1 = ± 1, g0, g1, R0, and R1 are real constants. Fig. 1 presents the 2D trapping potential at y = 0 with the parameters: R0 = 2, R1 = 2. Taking into account Gaussian optical beam coming from laser devices, we take a solution to the (2 + 1)D NLSE (Eq. (1)) in the form: h i 2 2 expð−ikzÞ; q ¼ η exp −ξ x þ y
ð4Þ
where η, ξ and k are real constants. Substituting the ansatz (Eq. (4)) into Eq. (1), one obtains four independent equations: kη−2ηξ ¼ 0; 2
2ηξ −pη ¼ 0; 3
ð5Þ ð6Þ
sη g 0 þ s1 ηpR0 ¼ 0;
ð7Þ
g 1 þ 3ξ ¼ R1 þ ξ:
ð8Þ
Solving the four real algebraic equations, we can obtain the following solvable conditions for Eq. (1) in which an exact spatial soliton solution can be found, k¼
pffiffiffiffiffiffi 2p;
ð9Þ
pffiffiffiffiffiffiffiffi ξ ¼ p=2; R1 ¼ g 1 þ η¼
pffiffiffiffiffiffi 2p;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pR0 =g 0 ;
s1 ¼ −s:
ð10Þ ð11Þ ð12Þ ð13Þ
Under these relations, the spatial soliton solution for Eq. (1) is of the form: q¼
h pffiffiffiffiffiffiffiffi i pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 exp −i 2pz : pR0 =g 0 exp − p=2 x þ y
Fig. 2. The evolution behavior of the spatial soliton (Eq. (14)) in nonlinear optical system with spatial parameters given by Eqs. (2) and (3) at y = 0, here the parameters adopted arep = 0.5, g0 = 1, (a) s = 1, (b) s = − 1, the other parameters are the same as Fig. 1.
ð14Þ
The exact solution (Eq. (14)) is presented using an ansatz method under potential profiles (Eqs. (2) and (3)). The way to generate exact solutions is, as a matter of fact, an inverse problem, aiming to construct an appropriate potential for a given wave-function ansatz representing an appropriate solution. The GPE with homogeneous nonlinearity have been solved using the inverse-problem technique in Ref. [24].
The propagation of spatial soliton (Eq. (14)) is presented as follow. Fig. 2 shows the evolution behavior of the spatial soliton (Eq. (14)) in nonlinear optical system with spatial parameters given by Eqs. (2) and (3) at y = 0. As shown in Fig. 2, the shape of the spatial soliton in this nonlinear system with inhomogeneous nonlinearity and transverse refractive-index modulation does not visibly vary during the propagation except for the intensity. The shapes shown in Fig. 2a and b are similar to each other. The result shown in Fig. 2b demonstrates that the soliton could exist in the system, where the diffraction term is positive and the nonlinearity is defocusing. In fact, any mode will be localized in the presence of the harmonic-oscillator trapping potential. To demonstrate stability with respect to finite perturbations, we perform numerical experiments with white noise. Fig. 3 shows the beam shape of the output beam when the initial beam is perturbed by white noise in this system at y = 0. As shown in Fig. 3, although the spatial soliton is affected weakly by the noise, the spatial soliton is stable during the propagating. Under the condition: R0 b b 1, we find that the Eq. (3) can be replaced by the following equation: 2 2 Rðx; yÞ ¼ − x þ y :
ð15Þ
In other words, the 2D trapping potential is simplified to the purely parabolic potential .We consider the input beam in the following form: h pffiffiffiffiffiffiffiffi i 2 2 q ¼ η exp − p=2 x þ y :
ð16Þ
When s = − 1, g0 b b 1, the evolution behavior of the spatial soliton (Eq. (16)) in nonlinear optical system with spatial parameters given by Eq. (15) is depicted in Fig. 4. From it, we can see that the
Y. Wang, R. Hao / Optics Communications 285 (2012) 2171–2173
2173
Fig. 5. The comparison plot of the evolution behavior of the spatial soliton (Eq. (16)) in three cases, here the parameters are as follows: p = 0.5, η = 1.
Fig. 3. The evolution of an initial beam given by Eq. (14) in nonlinear optical system with spatial parameters given by Eqs. (2) and (3) under the perturbation of white noise at y = 0. (a) s = 1, (b) s = − 1, the other parameters are the same as Fig. 2.
propagation of the input beam is stable. When s = 1, g0 b b 1, we can observe the propagation behavior is similar to Fig. 4. In order to explain the phenomenon, we take g0 = 0, and find exact solution of Eq. (1) with Eq. (15) as follows: h pffiffiffiffiffiffiffiffi i pffiffiffiffiffiffi 2 2 exp −i 2pz : q ¼ η exp − p=2 x þ y
ð17Þ
When z = 0, solution (Eq. (17)) denotes input beam (Eq. (16)). When g0 = 0, numerical simulations show the propagation behavior of input beam (Eq. (16)) is also similar to Fig. 4. The comparison of the evolution behavior of the spatial soliton (Eq. (16)) in three cases is shown in Fig. 5. These reveal that the existence of spatial soliton is the result of the parabolic modulation of linear refractive index. Obviously, a spatially Gaussian modulation of nonlinearity does not influence the existence of spatial soliton.
In conclusion, we have considered (2+ 1)D NLSE with spatially Kerrtype nonlinearity, which describes the propagation of the beam in nonlinear optical medium with the Gaussian nonlinearity modulation and the parabolic + Gaussian modulation of the linear refractive index in the transverse direction. The exact soliton solution has been presented using an ansatz method. Numerical simulations show that the Gaussian optical beam can stably propagate in the nonlinear optical medium. Based on the solution, we analyze the propagation of Gaussian optical beam in nonlinear optical system with the quadratic modulation of the linear refractive index. Numerical simulations reveal that the existence of spatial soliton is the result of the parabolic modulation of the linear refractive index, and further confirm that any mode will be localized in the presence of the harmonic-oscillator trapping potential. Acknowledgments This research is supported by the Research and Development Program of Science and Technology of Higher Education of Shanxi Province (Grant No. 20111027). References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Fig. 4. The evolution behavior of the spatial soliton (Eq. (16)) in nonlinear optical system with spatial parameters given by Eqs. (2) and (15), here the parameters adopted are p = 0.5, s = − 1, g0 = 0.01, g1 = 1, η = 1.
[23] [24]
K. Staliunas, R. Herrero, Physical Review E 73 (2006) 016601. M.J. Ablowitz, Z.H. Musslimani, Physical Review Letters 87 (2001) 254102. M.J. Ablowitz, Z.H. Musslimani, Physical Review E 65 (2002) 056618. H.S. Eisenberg, Y. Silberberg, R. Morandotti, J.S. Aitchison, Physical Review Letters 85 (2000) 1863. K. Staliunas, R. Herrero, G.J. de Valcárcel, Physical Review E 73 (2006) 065603(R). I.L. Garanovich, A.A. Sukhorukov, Y.S. Kivshar, Physical Review E 74 (2006) 066609. A. Szameit, I.L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A.A. Sukhorukov, T. Pertsch, D.N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, Y.S. Kivshar, Physical Review A 78 (2008) 031801. L. BergÃ, V.K. Mezentsev, J. Juul Rasmussen, P. Leth Christiansen, Yu.B. Gaididei, Optics Letters 25 (2000) 1037. I. Towers, B.A. Malomed, Journal of the Optical Society of America B 19 (2002) 537. F.Kh. Abdullaev, J.G. Caputo, R.A. Kraenkel, B.A. Malomed, Physical Review A 67 (2003) 013605. H. Saito, M. Ueda, Physical Review Letters 90 (2003) 040403. J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, Physical Review Letters 92 (2004) 123904. D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Physical Review Letters 92 (2004) 123903. Z. Chen, A. Bezryadina, I. Makasyuk, J. Yang, Optics Letters 29 (2004) 1656. J. Yang, I. Makasyuk, A. Bezryadina, Z. Chen, Optics Letters 29 (2004) 1662. O. Manela, O. Cohen, G. Bartal, J.W. Fleischer, M. Segev, Optics Letters 29 (2004) 2049. A. Szameit, Y.V. Kartashov, F. Dreisow, M. Heinrich, V.A. Vysloukh, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, L. Torner, Optics Letters 33 (2008) 663. Y.V. Kartashov, B.A. Malomed, V.A. Vysloukh, L. Torner, Optics Letters 33 (2008) 2949. M. Heinrich, Y.V. Kartashov, L.P.R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V.A. Vysloukh, L. Torner, Optics Letters 34 (2009) 3701. A. Szameit, Y.V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V.A. Vysloukh, L. Torner, Optics Letters 34 (2009) 797. Y.V. Kartashov, V.A. Vysloukh, L. Torner, Optics Letters 33 (2008) 2173. Y.V. Kartashov, V.A. Vysloukh, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, T. Pertsch, L. Torner, Optics Letters 33 (2008) 1120. Y.V. Kartashov, B.A. Malomed, L. Torner, Reviews of Modern Physics 83 (2011) 247. B.A. Malomed, Y.A. Stepanyants, Chaos 20 (2010) 013130.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Exact spatial soliton solution for nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index Yan Wang ⁎, Ruiyu Hao Department of Electronic Information and Physics, Changzhi University, Changzhi 046011, China
a r t i c l e
i n f o
Article history: Received 2 October 2011 Received in revised form 18 December 2011 Accepted 24 December 2011 Available online 5 January 2012
a b s t r a c t In this paper, we analyze (2 + 1)-dimensional nonlinear Schrödinger equation with nonperiodic modulation of nonlinearity and linear refractive index in the transverse direction, and obtain an exact solution in explicit form using an ansatz method. Finally, the stability of the solution is discussed numerically, and the result reveals that the solution is stable to the finite initial perturbations. © 2011 Elsevier B.V. All rights reserved.
Keywords: Spatial soliton Inhomogeneous nonlinearity Nonlinearity management Nonperiodic modulation
A lot of studies have been focused on the existence and stability of (2 + 1)-dimensional ((2 + 1)D, for brevity) spatial solitons. Strong periodic longitudinal modulation can be used to suppress diffraction of linear beam [1], such as a nonlinear waveguide array built of the properly designed segments supports diffraction-management soliton [2–7]. On the other hand, the way to stabilize the spatial solitons is that nonlinearity coefficient obeys a weak modulation along the propagation direction [8–11]. For example, (2 + 1)D spatial solitons propagate across a nonlinear bulk medium with layered structure in which Kerr nonlinearity alternates between self-focusing and selfdefocusing [9]. Besides, (2 + 1)D spatial solitons can exist in optical lattices which represent a spatially modulation of linear refractive index[12–20], or mixed linear-nonlinear lattices [21,22], i.e., the linear and nonlinear refractive indices are subjected to a spatially modulation in the transverse plane. Recently, a comprehensive survey of results obtained for solitons supported by nonlinear lattices (NLs) and their combinations with linear lattices has been offered in Ref. [23]. It is worth pointing out that the studies about (2 + 1)D spatial solitons are still far from completion. In this paper, we consider (2 + 1)D nonlinear optical system with spatially Kerr-type nonlinearity, and obtain an exact solution for nonlinear Schrödinger equation (NLSE) with nonperiodic transverse modulation of nonlinearity and linear refractive index. Upon a proper change of the notation, the NLSE may be interpreted as the Gross– Pitaevskii equation (GPE) for Bose–Einstein condensates (BECs). Accordingly, the exact solutions found in terms of the NLSE may also find applications to BECs. ⁎ Corresponding author. E-mail address: [email protected] (Y. Wang). 0030-4018/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2011.12.090
We consider (2 + 1)D NLSE which describes laser beam propagating along the z direction in (2 + 1)D optical media with spatially inhomogeneous nonlinearities.
iqz þ
1 2 qxx þ qyy þ g ðx; yÞjqj q þ pRðx; yÞq ¼ 0; 2
ð1Þ
where q(x, y, z) = (Ldif/Lnl) 1/2A(x, y, z)I0− 1/2, and A(x, y, z) is the complex envelope of the electrical field. I0 is the input intensity, x = X/r0, y = Y/r0, r0 is the input beam width unit, z = Z/Ldif, Ldif = n0wr02/c, Lnl = 2c/wn2I0, w is the frequency, and n2 is of the order of the nonlinear correction to the refractive index because of the Kerr effect.p = Ldif/Lref represents the guiding parameter, Lref = c/(δnw), and δn denotes the refractive index modulation depth, which is small compared with the unperturbed index n0. The dimensionless profile function g(x, y) is positive for nonlinear self-focusing but becomes negative in a selfdefocusing medium. The weak modulation of linear refractive index in the transverse direction is defined by the real function R(x, y). In this case, we consider a Gaussian nonlinearity modulation such as the one may be realized in a linear waveguide with one narrow nonlinear channel embedded into it, and a nonperiodic modulation of linear refractive index in the transverse direction with a quadratic + Gaussian distribution, i.e. h i 2 2 g ðx; yÞ ¼ sg 0 exp −g 1 x þ y ;
ð2Þ
h i 2 2 2 2 ; Rðx; yÞ ¼ − x þ y þ s1 R0 exp −R1 x þ y
ð3Þ
2172
Y. Wang, R. Hao / Optics Communications 285 (2012) 2171–2173
Fig. 1. The 2D trapping potential in the form of a combination of parabolic and Gaussian terms at y = 0, here the parameters are as follows: R0 = 2, R1 = 2.
where s = 1 and s = − 1 represent self-focusing and self-defocusing medium, respectively. s1 = ± 1, g0, g1, R0, and R1 are real constants. Fig. 1 presents the 2D trapping potential at y = 0 with the parameters: R0 = 2, R1 = 2. Taking into account Gaussian optical beam coming from laser devices, we take a solution to the (2 + 1)D NLSE (Eq. (1)) in the form: h i 2 2 expð−ikzÞ; q ¼ η exp −ξ x þ y
ð4Þ
where η, ξ and k are real constants. Substituting the ansatz (Eq. (4)) into Eq. (1), one obtains four independent equations: kη−2ηξ ¼ 0; 2
2ηξ −pη ¼ 0; 3
ð5Þ ð6Þ
sη g 0 þ s1 ηpR0 ¼ 0;
ð7Þ
g 1 þ 3ξ ¼ R1 þ ξ:
ð8Þ
Solving the four real algebraic equations, we can obtain the following solvable conditions for Eq. (1) in which an exact spatial soliton solution can be found, k¼
pffiffiffiffiffiffi 2p;
ð9Þ
pffiffiffiffiffiffiffiffi ξ ¼ p=2; R1 ¼ g 1 þ η¼
pffiffiffiffiffiffi 2p;
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pR0 =g 0 ;
s1 ¼ −s:
ð10Þ ð11Þ ð12Þ ð13Þ
Under these relations, the spatial soliton solution for Eq. (1) is of the form: q¼
h pffiffiffiffiffiffiffiffi i pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 exp −i 2pz : pR0 =g 0 exp − p=2 x þ y
Fig. 2. The evolution behavior of the spatial soliton (Eq. (14)) in nonlinear optical system with spatial parameters given by Eqs. (2) and (3) at y = 0, here the parameters adopted arep = 0.5, g0 = 1, (a) s = 1, (b) s = − 1, the other parameters are the same as Fig. 1.
ð14Þ
The exact solution (Eq. (14)) is presented using an ansatz method under potential profiles (Eqs. (2) and (3)). The way to generate exact solutions is, as a matter of fact, an inverse problem, aiming to construct an appropriate potential for a given wave-function ansatz representing an appropriate solution. The GPE with homogeneous nonlinearity have been solved using the inverse-problem technique in Ref. [24].
The propagation of spatial soliton (Eq. (14)) is presented as follow. Fig. 2 shows the evolution behavior of the spatial soliton (Eq. (14)) in nonlinear optical system with spatial parameters given by Eqs. (2) and (3) at y = 0. As shown in Fig. 2, the shape of the spatial soliton in this nonlinear system with inhomogeneous nonlinearity and transverse refractive-index modulation does not visibly vary during the propagation except for the intensity. The shapes shown in Fig. 2a and b are similar to each other. The result shown in Fig. 2b demonstrates that the soliton could exist in the system, where the diffraction term is positive and the nonlinearity is defocusing. In fact, any mode will be localized in the presence of the harmonic-oscillator trapping potential. To demonstrate stability with respect to finite perturbations, we perform numerical experiments with white noise. Fig. 3 shows the beam shape of the output beam when the initial beam is perturbed by white noise in this system at y = 0. As shown in Fig. 3, although the spatial soliton is affected weakly by the noise, the spatial soliton is stable during the propagating. Under the condition: R0 b b 1, we find that the Eq. (3) can be replaced by the following equation: 2 2 Rðx; yÞ ¼ − x þ y :
ð15Þ
In other words, the 2D trapping potential is simplified to the purely parabolic potential .We consider the input beam in the following form: h pffiffiffiffiffiffiffiffi i 2 2 q ¼ η exp − p=2 x þ y :
ð16Þ
When s = − 1, g0 b b 1, the evolution behavior of the spatial soliton (Eq. (16)) in nonlinear optical system with spatial parameters given by Eq. (15) is depicted in Fig. 4. From it, we can see that the
Y. Wang, R. Hao / Optics Communications 285 (2012) 2171–2173
2173
Fig. 5. The comparison plot of the evolution behavior of the spatial soliton (Eq. (16)) in three cases, here the parameters are as follows: p = 0.5, η = 1.
Fig. 3. The evolution of an initial beam given by Eq. (14) in nonlinear optical system with spatial parameters given by Eqs. (2) and (3) under the perturbation of white noise at y = 0. (a) s = 1, (b) s = − 1, the other parameters are the same as Fig. 2.
propagation of the input beam is stable. When s = 1, g0 b b 1, we can observe the propagation behavior is similar to Fig. 4. In order to explain the phenomenon, we take g0 = 0, and find exact solution of Eq. (1) with Eq. (15) as follows: h pffiffiffiffiffiffiffiffi i pffiffiffiffiffiffi 2 2 exp −i 2pz : q ¼ η exp − p=2 x þ y
ð17Þ
When z = 0, solution (Eq. (17)) denotes input beam (Eq. (16)). When g0 = 0, numerical simulations show the propagation behavior of input beam (Eq. (16)) is also similar to Fig. 4. The comparison of the evolution behavior of the spatial soliton (Eq. (16)) in three cases is shown in Fig. 5. These reveal that the existence of spatial soliton is the result of the parabolic modulation of linear refractive index. Obviously, a spatially Gaussian modulation of nonlinearity does not influence the existence of spatial soliton.
In conclusion, we have considered (2+ 1)D NLSE with spatially Kerrtype nonlinearity, which describes the propagation of the beam in nonlinear optical medium with the Gaussian nonlinearity modulation and the parabolic + Gaussian modulation of the linear refractive index in the transverse direction. The exact soliton solution has been presented using an ansatz method. Numerical simulations show that the Gaussian optical beam can stably propagate in the nonlinear optical medium. Based on the solution, we analyze the propagation of Gaussian optical beam in nonlinear optical system with the quadratic modulation of the linear refractive index. Numerical simulations reveal that the existence of spatial soliton is the result of the parabolic modulation of the linear refractive index, and further confirm that any mode will be localized in the presence of the harmonic-oscillator trapping potential. Acknowledgments This research is supported by the Research and Development Program of Science and Technology of Higher Education of Shanxi Province (Grant No. 20111027). References [1] [2] [3] [4] [5] [6] [7]
[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]
Fig. 4. The evolution behavior of the spatial soliton (Eq. (16)) in nonlinear optical system with spatial parameters given by Eqs. (2) and (15), here the parameters adopted are p = 0.5, s = − 1, g0 = 0.01, g1 = 1, η = 1.
[23] [24]
K. Staliunas, R. Herrero, Physical Review E 73 (2006) 016601. M.J. Ablowitz, Z.H. Musslimani, Physical Review Letters 87 (2001) 254102. M.J. Ablowitz, Z.H. Musslimani, Physical Review E 65 (2002) 056618. H.S. Eisenberg, Y. Silberberg, R. Morandotti, J.S. Aitchison, Physical Review Letters 85 (2000) 1863. K. Staliunas, R. Herrero, G.J. de Valcárcel, Physical Review E 73 (2006) 065603(R). I.L. Garanovich, A.A. Sukhorukov, Y.S. Kivshar, Physical Review E 74 (2006) 066609. A. Szameit, I.L. Garanovich, M. Heinrich, A. Minovich, F. Dreisow, A.A. Sukhorukov, T. Pertsch, D.N. Neshev, S. Nolte, W. Krolikowski, A. Tünnermann, A. Mitchell, Y.S. Kivshar, Physical Review A 78 (2008) 031801. L. BergÃ, V.K. Mezentsev, J. Juul Rasmussen, P. Leth Christiansen, Yu.B. Gaididei, Optics Letters 25 (2000) 1037. I. Towers, B.A. Malomed, Journal of the Optical Society of America B 19 (2002) 537. F.Kh. Abdullaev, J.G. Caputo, R.A. Kraenkel, B.A. Malomed, Physical Review A 67 (2003) 013605. H. Saito, M. Ueda, Physical Review Letters 90 (2003) 040403. J.W. Fleischer, G. Bartal, O. Cohen, O. Manela, M. Segev, J. Hudock, D.N. Christodoulides, Physical Review Letters 92 (2004) 123904. D.N. Neshev, T.J. Alexander, E.A. Ostrovskaya, Y.S. Kivshar, H. Martin, I. Makasyuk, Z. Chen, Physical Review Letters 92 (2004) 123903. Z. Chen, A. Bezryadina, I. Makasyuk, J. Yang, Optics Letters 29 (2004) 1656. J. Yang, I. Makasyuk, A. Bezryadina, Z. Chen, Optics Letters 29 (2004) 1662. O. Manela, O. Cohen, G. Bartal, J.W. Fleischer, M. Segev, Optics Letters 29 (2004) 2049. A. Szameit, Y.V. Kartashov, F. Dreisow, M. Heinrich, V.A. Vysloukh, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, L. Torner, Optics Letters 33 (2008) 663. Y.V. Kartashov, B.A. Malomed, V.A. Vysloukh, L. Torner, Optics Letters 33 (2008) 2949. M. Heinrich, Y.V. Kartashov, L.P.R. Ramirez, A. Szameit, F. Dreisow, R. Keil, S. Nolte, A. Tünnermann, V.A. Vysloukh, L. Torner, Optics Letters 34 (2009) 3701. A. Szameit, Y.V. Kartashov, M. Heinrich, F. Dreisow, T. Pertsch, S. Nolte, A. Tünnermann, F. Lederer, V.A. Vysloukh, L. Torner, Optics Letters 34 (2009) 797. Y.V. Kartashov, V.A. Vysloukh, L. Torner, Optics Letters 33 (2008) 2173. Y.V. Kartashov, V.A. Vysloukh, A. Szameit, F. Dreisow, M. Heinrich, S. Nolte, A. Tünnermann, T. Pertsch, L. Torner, Optics Letters 33 (2008) 1120. Y.V. Kartashov, B.A. Malomed, L. Torner, Reviews of Modern Physics 83 (2011) 247. B.A. Malomed, Y.A. Stepanyants, Chaos 20 (2010) 013130.