光致异构系统中孤子的内部振荡及其传输特性的研究
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摘要
研究光学孤子的内部振荡和相互作用。具体研究了光致异构聚合物中矢量孤子和贝塞尔光子晶格中项链孤子的内部模式(简称内模)和内部振荡,三维空间贝塞尔孤立波以及一维孤子的相互作用。研究内容及所取得的研究成果如下：
1、光致异构矢量孤子的内部模式和内部振荡
研究了光致异构聚合物中的矢量孤子。当矢量孤子受到外部的扰动时,在矢量孤子内部激发出内模。首次得到矢量空间孤子的内模,发现光致异构矢量孤子的微扰本征值是实数,这说明光致异构矢量孤子是稳定的,并且其对应的微扰本征函数就是矢量孤子的内模。我们还发现这种内模只有实部。对矢量孤子的传播进行动态仿真后发现,无外加微扰时光致异构矢量孤子在传播过程中保持波形的宽度和幅度不变：在内模的扰动下,矢量孤子的振幅出现准周期性振荡,但不导致孤子的坍塌；加入小的白噪声后,矢量孤子的振幅在传播过程中发生轻微的抖动,同样不会导致矢量孤子的坍塌,这些数值结果验证了矢量孤子的稳定性。
2、贝塞尔晶格中项链孤子的稳定性、内模和内部振荡
以十珍珠项链孤子为例,研究了贝塞尔晶格中项链孤子的稳定性、内模及其内部振荡。获得项链孤子的数值解后,对解进行了线性稳定性分析。发现项链孤子的微扰本征值为复数,随着传播常数6的增加,本征值实部的最大值出现了多个为零的窗口。零窗口对应于项链孤子稳定传播的区域,非零窗口对应于非稳定传播的区域。在稳定区域内,我们首次得到了项链孤子的内模。数值模拟的结果表示,在内模的作用下,项链孤子在传播过程中出现准周期性的振荡,但不会出现孤子的坍塌。而非稳定区域的项链孤子在传播过程中出现了扩散和坍塌。所研究的贝塞尔晶格项链孤子与贝塞尔晶格多极孤子不同,当传播常数b趋近于零时,多极孤子的能量发散而项链孤子的能量收敛。
3、对非线性介质中三维贝塞尔孤子簇的研究
研究柱坐标系和球坐标系下的非线性薛定谔方程,得到了在圆柱体和球体中半径趋于无穷大时的近似孤子解。发现了在圆柱体中的空间光孤子在横截面上的半径满足m阶贝塞尔函数关系,在方位角上分裂为2m个离散孤子,而在沿轴向方向上分裂为l个离散孤子,在圆柱体中出现了不同状态的贝塞尔孤子簇；而在球体中的空间光孤子的半径满足m阶球贝塞尔函数关系,在方位角上分裂为2m个离散孤子,在俯仰角上满足缔合勒让德多项式关系,在球体中出现了不同状态的球贝塞尔勒让德孤子簇。发现非线性效应使贝塞尔孤子簇在方位角和俯仰角方向上产生了偏转角,也就是出现了相移的现象。
4、一维时间光孤子传播和相互作用分析
首先研究了增益或损耗介质对可积系统中一维亮孤子和暗孤子传播的影响,发现损耗使孤子振幅减小同时使脉宽增大,而增益则使振幅增大同时使脉宽压缩变窄。接着对一维亮亮孤子间和暗暗孤子间的相互碰撞进行演化和分析,得到了亮亮孤子间相互作用与相位差相关,而暗暗孤子间相互作用与相位差无关的结论。最后,通过对在外势场中亮孤子和暗孤子的演化进行数值模拟,发现亮孤子和暗孤子在势阱中受到吸引力的作用,使孤子在势阱中心附近来回振荡,相反,在势垒中则受到排斥力的作用,使孤子远离势垒中心。另外还发现孤子的横向速度越快,其振幅的振荡也就越大。
Internal oscillations and interactions of optical soliton are investigated. Internal modes and internal oscillations of vector solitons in photoisomerization polymer and necklace solitons in Bessel photonic lattices, three-dimensional Bessel solitary wave and interaction of one-dimensional solitons are studied in detail. The contents and achievements are as follows:
1. Internal mode and internal oscillations of vector solitons ossociated with photoisomerization
Vector solitons in photoisomerization polymer are studied. When the vector solitons are perturbed, internal modes will be aroused. Internal modes of vector spatial solitons are obtained for the first time. It is found that the perturbation eigenvalue of photoisomerization vector solitons are real numbers, which indicates that such vector solitons are stable, and the corresponding perturbation eigenfunctions are internal modes of the vector solitons. We also find the internal modes have only real parts. The simulation to the propagation of the vector solitons shows that the vector solitons keep their shapes and amplitude unchanged when no perturbation is presented; when the vector solitons are perturbed by its internal modes, quazi-periodic oscillation of the amplitude of the vector solitons appears, however no collapse of solitons are observed; the amplitude of the vector solitons jitter when a white noise is added, however no collapse occurs either when the white noise is not large enough. The stability of the vector solitons is thus confirmed.
2. Stability, internal model and internal oscillationof necklace solitons in Bessel lattices
Stability, internal model and internal oscillation of necklace soliton in Bessel lattice are studied, taking ten-pearled necklaces for example. Afer the numerical solutions of necklace solitons are obtained, the linear stability analysis on the solutions is provided. It is found that perturbation eigenvalues of the necklace solitons are complex numbers, but with the increase of propagation constant b, several zero windows emerge from the maxima of the real part of eigenvalue. Zero-windows correspond to the stable regions of the necklace solitons; non-zero windows correspond to the non-stable regions. In the stable region, we obtain internal modes of necklace soliton for the first time to our knowledge. Simulation results indicates that when the vector solitons are perturbed by the internal modes, quasi-periodic oscillations of the necklace soliton in the stable area occurs, however on collapse occurs. On the other hand the necklace solitons in the unstable area collapse when propagating. The necklace soliton studied here are different with the previously investigated multi-pole solitons in Bessel lattices. When the propagation constant b tends to zero, the energy of multi-pole solitons divrges while the energy of necklace solitons studied here converges.
3. Three-dimensional Bessel soltion clusters in the nonlinear media
Approximate soliton solutions to nonlinear Schrodinger equation under cylindrical coordinates and spherical coordinates are obtained when the radius of the cylinder and the sphere tend to infinity. We find that the soliton solution in the cylindrical coordinates meets the m-order Bessel function. The solitons split into both 2m discrete solitons in the azimuthal direction and ldiscrete solitons in the axial direction. There are many different quantum states of the Bessel solitons clusters in the cylinder. On the other hand, the spatial optical solitons in the sphere meet m-order spherical Bessel function. The spatial optical soliton split into 2m discrete solitons in the azimuthal direction. There are many different quantum states in the form of the spherical Bessel solitons clusters. The spatial optical solitons experience a small deflection angle because of the nonlinear effects, that is, the phenomenon of phase shift accurs.
4. The propagation and interaction of the one-dimensional temporal solitons
The effects of gain or loss on the propagation of a bright solitons and a dark soliton are studied firstly in one-dimensional integrable system. It is found that the soliton amplitude attenuated and the width increased in the loss media however the amplitude enlarged and the width decreased in the gain media. And then by analysing the collision between two bright solitons or two dark solitons, the conclusion is obtained that the interaction between the bright solitons dependeds on the phase difference however the interaction between the dark solitons was independent on the phase difference on the contrary. Finally, using numerical simulation of the bright and dark solitons in the external potential field, it is found that the bright and dark solitons were acted upon by attractive force in the potential well, so solitons oscillate back and forth near the equilibrium position. On the contrary, the bright and dark solitons were affected by the repulsive force in the potential barrier, whch lead to their departure from the potential barrier center. The faster the transverse velocity of solitons is, the greater the amplitude of oscillation is.
1、光致异构矢量孤子的内部模式和内部振荡
研究了光致异构聚合物中的矢量孤子。当矢量孤子受到外部的扰动时,在矢量孤子内部激发出内模。首次得到矢量空间孤子的内模,发现光致异构矢量孤子的微扰本征值是实数,这说明光致异构矢量孤子是稳定的,并且其对应的微扰本征函数就是矢量孤子的内模。我们还发现这种内模只有实部。对矢量孤子的传播进行动态仿真后发现,无外加微扰时光致异构矢量孤子在传播过程中保持波形的宽度和幅度不变：在内模的扰动下,矢量孤子的振幅出现准周期性振荡,但不导致孤子的坍塌；加入小的白噪声后,矢量孤子的振幅在传播过程中发生轻微的抖动,同样不会导致矢量孤子的坍塌,这些数值结果验证了矢量孤子的稳定性。
2、贝塞尔晶格中项链孤子的稳定性、内模和内部振荡
以十珍珠项链孤子为例,研究了贝塞尔晶格中项链孤子的稳定性、内模及其内部振荡。获得项链孤子的数值解后,对解进行了线性稳定性分析。发现项链孤子的微扰本征值为复数,随着传播常数6的增加,本征值实部的最大值出现了多个为零的窗口。零窗口对应于项链孤子稳定传播的区域,非零窗口对应于非稳定传播的区域。在稳定区域内,我们首次得到了项链孤子的内模。数值模拟的结果表示,在内模的作用下,项链孤子在传播过程中出现准周期性的振荡,但不会出现孤子的坍塌。而非稳定区域的项链孤子在传播过程中出现了扩散和坍塌。所研究的贝塞尔晶格项链孤子与贝塞尔晶格多极孤子不同,当传播常数b趋近于零时,多极孤子的能量发散而项链孤子的能量收敛。
3、对非线性介质中三维贝塞尔孤子簇的研究
研究柱坐标系和球坐标系下的非线性薛定谔方程,得到了在圆柱体和球体中半径趋于无穷大时的近似孤子解。发现了在圆柱体中的空间光孤子在横截面上的半径满足m阶贝塞尔函数关系,在方位角上分裂为2m个离散孤子,而在沿轴向方向上分裂为l个离散孤子,在圆柱体中出现了不同状态的贝塞尔孤子簇；而在球体中的空间光孤子的半径满足m阶球贝塞尔函数关系,在方位角上分裂为2m个离散孤子,在俯仰角上满足缔合勒让德多项式关系,在球体中出现了不同状态的球贝塞尔勒让德孤子簇。发现非线性效应使贝塞尔孤子簇在方位角和俯仰角方向上产生了偏转角,也就是出现了相移的现象。
4、一维时间光孤子传播和相互作用分析
首先研究了增益或损耗介质对可积系统中一维亮孤子和暗孤子传播的影响,发现损耗使孤子振幅减小同时使脉宽增大,而增益则使振幅增大同时使脉宽压缩变窄。接着对一维亮亮孤子间和暗暗孤子间的相互碰撞进行演化和分析,得到了亮亮孤子间相互作用与相位差相关,而暗暗孤子间相互作用与相位差无关的结论。最后,通过对在外势场中亮孤子和暗孤子的演化进行数值模拟,发现亮孤子和暗孤子在势阱中受到吸引力的作用,使孤子在势阱中心附近来回振荡,相反,在势垒中则受到排斥力的作用,使孤子远离势垒中心。另外还发现孤子的横向速度越快,其振幅的振荡也就越大。
Internal oscillations and interactions of optical soliton are investigated. Internal modes and internal oscillations of vector solitons in photoisomerization polymer and necklace solitons in Bessel photonic lattices, three-dimensional Bessel solitary wave and interaction of one-dimensional solitons are studied in detail. The contents and achievements are as follows:
1. Internal mode and internal oscillations of vector solitons ossociated with photoisomerization
Vector solitons in photoisomerization polymer are studied. When the vector solitons are perturbed, internal modes will be aroused. Internal modes of vector spatial solitons are obtained for the first time. It is found that the perturbation eigenvalue of photoisomerization vector solitons are real numbers, which indicates that such vector solitons are stable, and the corresponding perturbation eigenfunctions are internal modes of the vector solitons. We also find the internal modes have only real parts. The simulation to the propagation of the vector solitons shows that the vector solitons keep their shapes and amplitude unchanged when no perturbation is presented; when the vector solitons are perturbed by its internal modes, quazi-periodic oscillation of the amplitude of the vector solitons appears, however no collapse of solitons are observed; the amplitude of the vector solitons jitter when a white noise is added, however no collapse occurs either when the white noise is not large enough. The stability of the vector solitons is thus confirmed.
2. Stability, internal model and internal oscillationof necklace solitons in Bessel lattices
Stability, internal model and internal oscillation of necklace soliton in Bessel lattice are studied, taking ten-pearled necklaces for example. Afer the numerical solutions of necklace solitons are obtained, the linear stability analysis on the solutions is provided. It is found that perturbation eigenvalues of the necklace solitons are complex numbers, but with the increase of propagation constant b, several zero windows emerge from the maxima of the real part of eigenvalue. Zero-windows correspond to the stable regions of the necklace solitons; non-zero windows correspond to the non-stable regions. In the stable region, we obtain internal modes of necklace soliton for the first time to our knowledge. Simulation results indicates that when the vector solitons are perturbed by the internal modes, quasi-periodic oscillations of the necklace soliton in the stable area occurs, however on collapse occurs. On the other hand the necklace solitons in the unstable area collapse when propagating. The necklace soliton studied here are different with the previously investigated multi-pole solitons in Bessel lattices. When the propagation constant b tends to zero, the energy of multi-pole solitons divrges while the energy of necklace solitons studied here converges.
3. Three-dimensional Bessel soltion clusters in the nonlinear media
Approximate soliton solutions to nonlinear Schrodinger equation under cylindrical coordinates and spherical coordinates are obtained when the radius of the cylinder and the sphere tend to infinity. We find that the soliton solution in the cylindrical coordinates meets the m-order Bessel function. The solitons split into both 2m discrete solitons in the azimuthal direction and ldiscrete solitons in the axial direction. There are many different quantum states of the Bessel solitons clusters in the cylinder. On the other hand, the spatial optical solitons in the sphere meet m-order spherical Bessel function. The spatial optical soliton split into 2m discrete solitons in the azimuthal direction. There are many different quantum states in the form of the spherical Bessel solitons clusters. The spatial optical solitons experience a small deflection angle because of the nonlinear effects, that is, the phenomenon of phase shift accurs.
4. The propagation and interaction of the one-dimensional temporal solitons
The effects of gain or loss on the propagation of a bright solitons and a dark soliton are studied firstly in one-dimensional integrable system. It is found that the soliton amplitude attenuated and the width increased in the loss media however the amplitude enlarged and the width decreased in the gain media. And then by analysing the collision between two bright solitons or two dark solitons, the conclusion is obtained that the interaction between the bright solitons dependeds on the phase difference however the interaction between the dark solitons was independent on the phase difference on the contrary. Finally, using numerical simulation of the bright and dark solitons in the external potential field, it is found that the bright and dark solitons were acted upon by attractive force in the potential well, so solitons oscillate back and forth near the equilibrium position. On the contrary, the bright and dark solitons were affected by the repulsive force in the potential barrier, whch lead to their departure from the potential barrier center. The faster the transverse velocity of solitons is, the greater the amplitude of oscillation is.
引文
[1]J. S. Russell, In reports of the meetings of the british association for the advancement of science,1844, Liveerpool meeting,1838.
[2]D. V. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new-type of long stationary wave, Phil. Mag.,1895, 39:422-425.
[3]A. C. Newell and J. V. Moloney, Nonlinear Opticas, Addison-Wesley Publishing Company, Redwood,1991.
[4]R. W. Boyd, Nonlinear Optics, Academic Press, Boston,2003.
[5]F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poission problem: existence, uniqueness and approximation, Math.Models Meth.Appl.Sci.1991,14:35.
[6]A. Hasegawa, Optical Solitons in Fibers, Springer-Verlag, Berlin,1989.
[7]R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York,1982.
[8]F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys.,1999,71:463.
[9]R. Fedele, G Miele, L Palumbo, and V. G Vaccaro, Thermal wave model for nonlinear longitudinal dynamics in particle accelerators, Phys. Lett. A 1993,173: 407.
[10]Y. S. Kivshar and B. Luther-Davies, Dark optical solitons:physics and applications, Phys. Rep.,1998,298:81-97.
[11]R. Y. Chiao, E. Garmire, C. H. Townes, Self-Trapping of Optical Beams, Phys. Rev. Lett.,1964,13:479-482.
[12]G. I. Stegeman and M. Segev, Optical Spatial Solitons and Their Interactions: Universality and Diversity, Science 1999,286:1518-1523.
[13]Y. S. Kivshar and G. I. Stegeman, Spatial Optical Solitons:Guiding Light for Future Technologies, Opt. Photon. News,2002,13:59-63.
[14]S. Trillo and W. Torruellas, Spatial Soltions, Springer, New York,2001.
[15]B. Crosignani and G. Salamo, Photorefractive Soltions, Opt. Photon. News,2002,13: 38-41.
[16]T. Hong, Spatial weak-light solitons in an electromagnetically induced nonlinear waveguide, Phys. Rev. Lett.,2003,90:183901(1-4).
[17]N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, Ferroelectrics,1979,22:949.
[18]X. S. Wang, W. L. She, and W. K. Lee, Opt. Lett.,2004,29:277.
[19]S. Bian and M. G. Kuzyk, Appl. Phys. Lett.,2004,85:1104.
[20]X. S. Wang and W. L. She, Phys. Rev. E,2005,71:026601.
[21]X. S. Wang, W. L. She, and W. K. Lee, J. Opt. Soc. Am. B,2006,23:212.
[22]Y. Kartashov, V. Vysloukh, and L. Torner, Tunable Soliton Self-Bending in Optical Lattices with Nonlocal Nonlinearity, Phys. Rev. Lett.,2004,93:153903.
[23]N. Nikolov, D. Neshev, O. Krolikowski, W. Bang, J. Rasmussen, and P. Christiansen, Attraction of nonlocal dark optical solitons, Opt. Lett.,2004,29:286.
[24]K. Yu. S. and L.-D. B, Dark optical solitons:physics and applications, Phys. Rep., 1998,298:81.
[25]D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature,2003,424:817.
[26]T. Pertsch, T. Zentgraf, U. Peschel, et al., Phys. Rev. Lett.,2002,88:093901.
[27]D. N. Christodoulides, and R. I. Joseph, Opt. Lett.,1988,13:794.
[28]M. Shih, Z. Chen, M. Mitchell, et al., J. Opt. Soc. Am. B,1997,14:3091.
[29]楼慈波等,光子晶格中新颖的空间带隙孤子,光学前沿专题,2008,37：239-246.
[30]F. Chen, M. Stepic, C. Ruter, et al., Opt. Express,2005,13:4314.
[31]G. Bartal, O. Cohen, O. Manela, et al., Opt. Lett.,2006,31:483.
[32]J. Yang, I. Makasyuk, A. Bezryadina, et al., Opt. Lett.,2004,29:1662.
[33]J. Yang, I. Makasyuk, A. Bezryadina, et al., Stud. Appl. Math.,2004,113:389.
[34]D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, et al., Phys. Rev. Lett.,2004,92: 123903.
[35]J. W. Fleischer, G. Bartal, O. Cohen, et al., Phys. Rev. Lett.,2004,92:23904.
[36]G. Bartal, O. Manela, O. Cohen, et al., Phys. Rev. Lett.,2005,95:053904.
[37]A. Hasegawa and F. D.Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fiber (I) Anomalous dispersion, Appl. Phys. Lett.,1973,23: 142.
[38]L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett.,1980,45: 1095.
[39]G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego,1995.
[40]L. F. Mollenauer, J. P. Gordon, and S. G Evangelides, Laser Focus World, November 1991,159.
[41]F. Wise and P. Di Trapani, The Hunt for Light Bullets:Spatiotemporal Solitons, Opt. Photon. News,2002,13:28-32.
[42]X. Liu, L. J. Qian, and F. W. Wise, Generation of Optical Spatiotemporal Solitons, Phys. Rev. Lett.,1999,82:4631-4634.
[43]A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, Appl. Phys. Lett.,1973,23:142.
[44]L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett.,1980,45: 1095.
[45]A. Hasegawa and Y. Kodama, Signal transmission by optical solitons in monomode fiber, Proc. IEEE,1981,69:1145.
[46]S. R. Nagel, J. B. MacChesncy, and K. L. Walker, in Optical Fiber Communications, T. Li. Academic, Orlando,1985.
[47]胡国绛,黄超,非线性光纤光学,天津大学出版社,1992.
[48]D. Marcuse, Pulse distortion in single-mode fiber, Appl. Opt.,1981,20:3573.
[49]R. H. Stolen and C. Lin, Self-phase-modulation in silica optical fibers, Phys. Rev., 1978, A17:1448.
[50]K. J. Blow and N. J. Doran, Multiple Dark Soliton Solutions of the Nonlinear Schrodinger Equation, Phys. Lett.,1985, A107:55.
[51]J. Satsuma and N. Yajima, Initial value problems of one dimensional self-modulation of nonlinear waves in dispersive media, Prog. Theor. Phys. Suppl., 1974,55:284.
[52]L. Dong, F. Ye, J. Wang, T. Cai, and Y. Li, Internal Modes of Localized Vortex Soliton in a Cubic-Quintic Nonlinear Medium, Physica D,2004,194:219.
[53]D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afannasjev, Internal modes of envelope solitons, Physica D,1997,116:121.
[54]Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, and M. Peyrard, Internal Modes of Solitary Waves, Phys. Rev. Lett.,1998,80:5032.
[55]P. Anninos, S. Oliveira, and R. A. Matzner, Fractal structure in the scalar λ(φ2-1)2 theory, Phys. Rev. D,1991,44:1147.
[56]J. Yang and Y. Tan, Fractal Structure in the Collision of Vector Solitons, Phys. Rev. Lett.,2000,85:3624.
[57]J. A. Gonzalez, A. Bellorin, and L. E. Guerrero, Internal modes of sine-Gordon solitons in the presence of spatiotemporal perturbations, Phys. Rev. E,2002,65: 065601.
[58]J. K. Yang, Internal oscillations and instability characteristics of (2+1)-dimensional solitons in a saturable nonlinear medium, Phys. Rev. E,2002,66:026601.
[59]L. W. Dong, F. W. Ye, J. D. Wang, T. Cai, and Y. P. Li, Internal modes of localized optical vortex soliton in a cubic-quintic nonlinear medium, Physica D,2004,194: 219.
[60]I. Shadrivov and A. A. Zharov, Opt. Commun.,2003,216:47.
[61]J. Yang and D. E. Pelinovsky, Phys. Rev. E,2003,67:016608.
[62]G D. Montesinos, V. M. Perez-Garcia, H. Michinel, Phys. Rev. Lett.,2004,92: 133901.
[63]M. Shen, X. Chen, J. Shi, Q. Wang, and W. Krolikowski, Opt. Commun.,2009,282: 4805.
[64]M. I. Carvalho, S. R. Singh, D. N. Christodoulides, and R. I. Joseph, Phys. Rev. E, 1996,53:R53.
[65]J. Yang, Phys. Rev. E,2001,64:026607.
[66]Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, and L. Tomer, Phys. Rev. E,2004, 70:066623.
[67]Z. Y. Sun, Y T. Gao, X. Yu, W. J. Liu, and Y. Liu, Phys. Rev. E,2009,80:066608.
[68]B. A. Malomed and R. S. Tasgal, Phys. Rev. E,1998,58:2564.
[69]J. Liang and X. Zhou, Application of continuous-wave laser Z-scan technique to photoisomerization, J. Opt. Soc. Am. B,2005,22:2468.
[70]J. C. Liang, H. Zhao, X. Zhou, and H. Wang, Polarization-dependent effects of refractive index change associated with photoisomerization investigated with Z-scan technique, J. Appl. Phys.,2007,101:013106.
[71]X. S. Wang, W. L. She, and W. K. Lee, Optical spatial solitons supported by photoisomerization nonlinearity in a polymer, Opt. Lett.,2004,29:277.
[72]X. S. Wang and W. L. She, Spontaneous one-and two-dimensional optical spatial solitons supported by photoisomerization nonlinearity in a bulk polymer, Phys. Rev. E,2005,71:026601.
[73]H. Wang, B. Zhang, L. Yan, Y. Li, G Zheng, and W. She, Gray and dark spatial solitons supported by two-photon isomerization, Opt. Commun.,2007,275:234.
[74]B. Z. Zhang, H. C. Wang, and W. L. She, Influence of molecular reorientation on photoisomerization optical spatial solitons, J. Opt. A:Pure Appl. Opt.,2007,9:395.
[75]B. Z. Zhang, H. Cui, and W. L. She, Modulational instability of incoherently coupled beams in azobenzene-containing polymer with photoisomerization nonlinearity, Chin. Phys. B,2009,18:209.
[76]X. S. Wang, W. L. She, S. Z. Wu, and F. Zeng, Circularly polarized optical spatial solitons, Opt. Lett.,2005,30:863.
[77]S. Bian and M. G Kuzyk, Dark spatial solitons in bulk azo-dye-doped polymer using photoinduced molecular reorientation, Appl. Phys. Lett.,2004,85:1104.
[78]X. S. Wang, W. L. She, and W. K. Lee, Optical spatial soliton supported by photoisomerization nonlinearity in a polymer with a background beam, J. Opt. Soc. Am. B,2006,23:2127.
[79]J. C. Liang, Z. B. Cai, Y. Z. Sun, S. L. Xu, and L. Yi, Various kinds of spatial solitons associated with photoisomerization, J. Opt. Soc. Am. B,2009,26:36.
[80]J. C. Liang, Z. B. Cai, Y. Z. Sun, L. Yi, and H. C. Wang, "What types of spatial soliton can be formed based on two-photon-isomerisation?" Eur. Phys. J. D,2009, 55:685.
[81]A. Xie, L. Meer, W. Hoff, and R. H. Austin, Long-lived amide I vibrational modes in myoglobin, Phys. Rev. Lett.,2000,84:5435.
[82]D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides, Opt. Lett.,1998,13:794.
[83]J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature(London),2003,422:147.
[84]Z. G Chen, H. Martin, E. D. Eugenieva, J. J. Xu, and A. Bezryadina, Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains, Phys. Rev. Lett.,2004,92:143902.
[85]X. S. Wang, A. Bezryadina, Z. G Chen, K. G. Makris, D. N. Christodoulides, and G I. Stegeman, Observation of two-dimensional surface solitons, Phys. Rev. Lett., 2007,98:123903.
[86]B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, Bright Bose-Einstein gap solitons of atoms with repulsive interaction, Phys. Rev. Lett.,2004,92:230401.
[87]A. Trombettoni and A. Smerzi, Discrete solitons and breathers with dilute Bose-Einstein condensates, Phys. Rev. Lett.,2001,86:2353-2356.
[88]M. Sato, B. E. Hubbard, A. J. Sievers, B. Llic, D. A. Czaplewski, and H. C. Craighead, Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array, Phys. Rev. Lett.,2003,90:044102.
[89]H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Discrete spatial optical solitons in waveguide arrays, Phys. Rev. Lett.,1998,81: 3383-3386.
[90]Z. G Chen, H. Martin, E. Eugenieva, J. J. Xu, and J. K. Yang, Formation of discrete solitons in light-induced photonic lattices, Opt. Express,2005,13:1816.
[91]R. Iwanow, R. Schiek, G I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Observation of discrete quadratic solitons, Phys. Rev. Lett.,2004,93:113902.
[92]A. Fratalocchi, G Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Discrete propagation and spatial solitons in nematic liquid crystals, Opt. Lett.,2004,29:1530.
[93]X. S. Wang, Z. G. Chen, and P. G. Kevrekidis, Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices, Phys. Rev. Lett.,2006,96: 083904.
[94]R. Fischer, D. N. Neshev, S. L. Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, Observation of light localization in modulated Bessel optical lattices, Opt. Express,2006,14:2825.
[95]X. S. Wang, Z. G Chen, and J. K. Yang, Guiding light in optically induced ring lattices with a low-refractive-index core, Opt. Lett.,2006,31:1887.
[96]J. Arlt and K. Dholakia, Generation of high-order Bessel beams by use of an axicon, Opt. Commun.,2000,177:297.
[97]A. Vasara, J. Turunen, and A. T. Friberg, Realization of general nondiffracting beams with computer-generated holograms, J. Opt. Soc. Am. A,1989,6:1748.
[98]Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Stable ring-profile vortex solitons in bessel optical lattices, Phys. Rev. Lett.,2005,94:043902.
[99]J. C. Liang, Z. B. Cai, L. Yi, and H. C. Wang, Bessel solitary waves in strongly nonlocal nonlinear media with distributed parameters, Opt. Commun.,2010,283: 386.
[100]Y. V. Kartashov, R. Carretero-Gonzalez, B. A. Malomed, V. A. Vysloukh, and L. Torner, Opt. Express,2005,13:10703.
[101]Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Rotary dipole-mode solitons in Bessel optical lattices, J. Opt. B:Quantum Semiclass. Opt.,2004,6: 444-447.
[102]Y. V. Kartashov, A. A. Egorov, V. A. Vysoukh, and L. Torner, Phys. Rev. E,2004,70: 065602(R).
[103]Y. S. Kivshar and B. A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys.,1989,61:763.
[104]D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, Internal modes of envelope solitons, Physica D,1997,116:121-142.
[105]Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, and M. Peyrard, Internal modes of solitary waves, Phys. Rev. Lett.,1998,80:5032-5035.
[106]D. K. Campell, J. F. Schonfeld, and L. E. Guerrero, Resonance structure in kink-antikink interactions in φ4 theory, Physica D,1983,9:1.
[107]M. Peyrard and D. K. Campbell, Kink-antikink interactions in a modified sine-Gordon model, Physica D,19839:33.
[108]D. K. Campbell, M. Peyrard, and P. Sodano, Kink-antikink interactions in the double sine-Gordon equation, Physica D,1986,19:165.
[109]J. A. Gonzalez, A. Bellorin, and L. E. Guerrero, Internal modes of sine-Gordon solitons in the presence of spatiotemporal perturbations, Phys. Rev. E,2002,65: 065601(R).
[110]V. Skarka, N. Aleksic, and V. Berzhiani, Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media, Phys. Lett. A,2001,291:124.
[111]L. W. Dong, F. W. Ye, J. D. Wang, T. Cai, and Y. P. Li, Internal modes of localized optical vortex soliton in a cubic-quintic nonlinear medium, Physica D,2004,194: 219.
[112]M. Soljacic, S. Sears and M. Segev, Phys. Rev. Lett.,1998,81:4851.
[113]M. Soljacic and M. Segev, Phys. Rev. E,2000,62:2810.
[114]J. Yang, I. Makasyuk, P. G Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, Phys. Rev. Lett.,2005,94:113902.
[115]A. W. Snyder and D. J. Mitchell, Accessible solitons, Science,1997,276:1538.
[116]Y. R. Shen, Solitons made simple, Science,1997,276:1520.
[117]W. Krolikowski, O. Bang, J. Rasmussen, et al., Modulational instability in nonlocal nonlinear Kerr media, Phys. Rev. E,2001,64:016612-1.
[118]O. Bang, W. Krolikowski, J. Wyller, et al., Collapse arrest and soliton stabilization in nonlocal nonlinear media, Phys. Rev. E,2002,66:046619-1.
[119]W. Krolikowski and O. Bang, Solitons in nonlocal nonlinear media:exact solutions, Phys. Rev. E,2000,63:016610-1.
[120]Ming Shen, Qi Wang, Jielong Shi, et al., Partially coherent accessible solitons in strongly nonlocal media, Phys. Rev. E,2006,73:056602-1.
[121]Zhiyong Xu, Yaroslav V. Kartashov, and Lluis Tomer, Stabilization of vector soliton complexes in nonlocal nonlinear media, Phys. Rev. E,2006,73:055601-1.
[122]Mousumi Ballav and A. Roy Chowdhury, A generalized nonlinear Schrodinger equation and optical soliton in a gradient index cylindrical media, Chaos, Solitons and Fractals,2007,31:794-803.
[123]Qi Guo, Boren Luo, Fhuai Yi, et al., Large phase shift of nonlocal optical spatial solitons, Phys. Rev. E,2004,69:016602-1.
[124]Yiqun Xie and Qi Guo, Phase modulations due to collisions of beam pairs in nonlocal nonlinear media, Opt. and Quant. Electron,2004,36:1335-1351.
[125]Q. Guo and X. Jiang, Induced focusing from co-propagation of a pair of bright-dark optical beams in self-defocusing Kerr media, Opt. Commun.,2005,254:19-29.
[126]Qi Guo, Boren Luo, and Sien Chi, Optical beams in sub-strongly non-local nonlinear media:a variational solution, Opt. Commun.,2006,259:336-341.
[127]Yi Huang, Qi Guo, and Jueneng Cao, Optical beams in lossy non-local Kerr media, Opt. Commun.,2006,161:175-180.
[128]Qi Guo, Xiaping Zhang, Wei Hu, et al., Photonic switching and logic gating with strongly nonlocal spatial optical solitons, Acta Physica Sinica,2006,55:1832-1839.
[129]M. Peccianti, C. Conti, G. Assanto, et al., All optical switching and logic gating with spatial solitons in liquid crystals, Appl. Phys. Lett.,2002,81:3335-3337.
[130]M. Peccianti, C. conti, and G Assanto, Optical modulational instability in a nonlocal medium, Phys. Rev. E,2003,68:025602-1.
[131]M. Peccianti, K. A. Brzdakiewicz, and G Assanto, Nonlocal spatial soliton interactions in nematic liquid crystals, Opt. Lett.,2002,27:1460-1463.
[132]G Assanto, M. Peccianti, and C. Conti, Spatial optical solitons in bulk nematic liquid crystals, Acta Physica Polonica (A),2003,103:161-167.
[133]M. Peccianti, C. Conti, and G Assanto, Optical multisoliton generation in bulk nematic liquid crystals, Opt. Lett.,2003,28:2231-2233.
[134]C. Rotschild, O. Cohen, O. Manela, et al., Solitons in nonlinear media with an infinite range of nonlocality:first observation of coherent elliptic solitions and of vortex-ring solitions, Phys. Rev. Lett.,2005,95:213904-1.
[2]D. V. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular channel, and on a new-type of long stationary wave, Phil. Mag.,1895, 39:422-425.
[3]A. C. Newell and J. V. Moloney, Nonlinear Opticas, Addison-Wesley Publishing Company, Redwood,1991.
[4]R. W. Boyd, Nonlinear Optics, Academic Press, Boston,2003.
[5]F. Brezzi and P. A. Markowich, The three-dimensional Wigner-Poission problem: existence, uniqueness and approximation, Math.Models Meth.Appl.Sci.1991,14:35.
[6]A. Hasegawa, Optical Solitons in Fibers, Springer-Verlag, Berlin,1989.
[7]R. K. Dodd, J. C. Eilbeck, J. D. Gibbon, and H. C. Morris, Solitons and Nonlinear Wave Equations, Academic Press, New York,1982.
[8]F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys.,1999,71:463.
[9]R. Fedele, G Miele, L Palumbo, and V. G Vaccaro, Thermal wave model for nonlinear longitudinal dynamics in particle accelerators, Phys. Lett. A 1993,173: 407.
[10]Y. S. Kivshar and B. Luther-Davies, Dark optical solitons:physics and applications, Phys. Rep.,1998,298:81-97.
[11]R. Y. Chiao, E. Garmire, C. H. Townes, Self-Trapping of Optical Beams, Phys. Rev. Lett.,1964,13:479-482.
[12]G. I. Stegeman and M. Segev, Optical Spatial Solitons and Their Interactions: Universality and Diversity, Science 1999,286:1518-1523.
[13]Y. S. Kivshar and G. I. Stegeman, Spatial Optical Solitons:Guiding Light for Future Technologies, Opt. Photon. News,2002,13:59-63.
[14]S. Trillo and W. Torruellas, Spatial Soltions, Springer, New York,2001.
[15]B. Crosignani and G. Salamo, Photorefractive Soltions, Opt. Photon. News,2002,13: 38-41.
[16]T. Hong, Spatial weak-light solitons in an electromagnetically induced nonlinear waveguide, Phys. Rev. Lett.,2003,90:183901(1-4).
[17]N. V. Kukhtarev, V. B. Markov, and S. G. Odulov, Ferroelectrics,1979,22:949.
[18]X. S. Wang, W. L. She, and W. K. Lee, Opt. Lett.,2004,29:277.
[19]S. Bian and M. G. Kuzyk, Appl. Phys. Lett.,2004,85:1104.
[20]X. S. Wang and W. L. She, Phys. Rev. E,2005,71:026601.
[21]X. S. Wang, W. L. She, and W. K. Lee, J. Opt. Soc. Am. B,2006,23:212.
[22]Y. Kartashov, V. Vysloukh, and L. Torner, Tunable Soliton Self-Bending in Optical Lattices with Nonlocal Nonlinearity, Phys. Rev. Lett.,2004,93:153903.
[23]N. Nikolov, D. Neshev, O. Krolikowski, W. Bang, J. Rasmussen, and P. Christiansen, Attraction of nonlocal dark optical solitons, Opt. Lett.,2004,29:286.
[24]K. Yu. S. and L.-D. B, Dark optical solitons:physics and applications, Phys. Rep., 1998,298:81.
[25]D. N. Christodoulides, F. Lederer, and Y. Silberberg, Nature,2003,424:817.
[26]T. Pertsch, T. Zentgraf, U. Peschel, et al., Phys. Rev. Lett.,2002,88:093901.
[27]D. N. Christodoulides, and R. I. Joseph, Opt. Lett.,1988,13:794.
[28]M. Shih, Z. Chen, M. Mitchell, et al., J. Opt. Soc. Am. B,1997,14:3091.
[29]楼慈波等,光子晶格中新颖的空间带隙孤子,光学前沿专题,2008,37：239-246.
[30]F. Chen, M. Stepic, C. Ruter, et al., Opt. Express,2005,13:4314.
[31]G. Bartal, O. Cohen, O. Manela, et al., Opt. Lett.,2006,31:483.
[32]J. Yang, I. Makasyuk, A. Bezryadina, et al., Opt. Lett.,2004,29:1662.
[33]J. Yang, I. Makasyuk, A. Bezryadina, et al., Stud. Appl. Math.,2004,113:389.
[34]D. N. Neshev, T. J. Alexander, E. A. Ostrovskaya, et al., Phys. Rev. Lett.,2004,92: 123903.
[35]J. W. Fleischer, G. Bartal, O. Cohen, et al., Phys. Rev. Lett.,2004,92:23904.
[36]G. Bartal, O. Manela, O. Cohen, et al., Phys. Rev. Lett.,2005,95:053904.
[37]A. Hasegawa and F. D.Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fiber (I) Anomalous dispersion, Appl. Phys. Lett.,1973,23: 142.
[38]L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett.,1980,45: 1095.
[39]G. P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego,1995.
[40]L. F. Mollenauer, J. P. Gordon, and S. G Evangelides, Laser Focus World, November 1991,159.
[41]F. Wise and P. Di Trapani, The Hunt for Light Bullets:Spatiotemporal Solitons, Opt. Photon. News,2002,13:28-32.
[42]X. Liu, L. J. Qian, and F. W. Wise, Generation of Optical Spatiotemporal Solitons, Phys. Rev. Lett.,1999,82:4631-4634.
[43]A. Hasegawa and F. Tappert, Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers, Appl. Phys. Lett.,1973,23:142.
[44]L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Experimental observation of picosecond pulse narrowing and solitons in optical fibers, Phys. Rev. Lett.,1980,45: 1095.
[45]A. Hasegawa and Y. Kodama, Signal transmission by optical solitons in monomode fiber, Proc. IEEE,1981,69:1145.
[46]S. R. Nagel, J. B. MacChesncy, and K. L. Walker, in Optical Fiber Communications, T. Li. Academic, Orlando,1985.
[47]胡国绛,黄超,非线性光纤光学,天津大学出版社,1992.
[48]D. Marcuse, Pulse distortion in single-mode fiber, Appl. Opt.,1981,20:3573.
[49]R. H. Stolen and C. Lin, Self-phase-modulation in silica optical fibers, Phys. Rev., 1978, A17:1448.
[50]K. J. Blow and N. J. Doran, Multiple Dark Soliton Solutions of the Nonlinear Schrodinger Equation, Phys. Lett.,1985, A107:55.
[51]J. Satsuma and N. Yajima, Initial value problems of one dimensional self-modulation of nonlinear waves in dispersive media, Prog. Theor. Phys. Suppl., 1974,55:284.
[52]L. Dong, F. Ye, J. Wang, T. Cai, and Y. Li, Internal Modes of Localized Vortex Soliton in a Cubic-Quintic Nonlinear Medium, Physica D,2004,194:219.
[53]D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afannasjev, Internal modes of envelope solitons, Physica D,1997,116:121.
[54]Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, and M. Peyrard, Internal Modes of Solitary Waves, Phys. Rev. Lett.,1998,80:5032.
[55]P. Anninos, S. Oliveira, and R. A. Matzner, Fractal structure in the scalar λ(φ2-1)2 theory, Phys. Rev. D,1991,44:1147.
[56]J. Yang and Y. Tan, Fractal Structure in the Collision of Vector Solitons, Phys. Rev. Lett.,2000,85:3624.
[57]J. A. Gonzalez, A. Bellorin, and L. E. Guerrero, Internal modes of sine-Gordon solitons in the presence of spatiotemporal perturbations, Phys. Rev. E,2002,65: 065601.
[58]J. K. Yang, Internal oscillations and instability characteristics of (2+1)-dimensional solitons in a saturable nonlinear medium, Phys. Rev. E,2002,66:026601.
[59]L. W. Dong, F. W. Ye, J. D. Wang, T. Cai, and Y. P. Li, Internal modes of localized optical vortex soliton in a cubic-quintic nonlinear medium, Physica D,2004,194: 219.
[60]I. Shadrivov and A. A. Zharov, Opt. Commun.,2003,216:47.
[61]J. Yang and D. E. Pelinovsky, Phys. Rev. E,2003,67:016608.
[62]G D. Montesinos, V. M. Perez-Garcia, H. Michinel, Phys. Rev. Lett.,2004,92: 133901.
[63]M. Shen, X. Chen, J. Shi, Q. Wang, and W. Krolikowski, Opt. Commun.,2009,282: 4805.
[64]M. I. Carvalho, S. R. Singh, D. N. Christodoulides, and R. I. Joseph, Phys. Rev. E, 1996,53:R53.
[65]J. Yang, Phys. Rev. E,2001,64:026607.
[66]Y. V. Kartashov, A. S. Zelenina, V. A. Vysloukh, and L. Tomer, Phys. Rev. E,2004, 70:066623.
[67]Z. Y. Sun, Y T. Gao, X. Yu, W. J. Liu, and Y. Liu, Phys. Rev. E,2009,80:066608.
[68]B. A. Malomed and R. S. Tasgal, Phys. Rev. E,1998,58:2564.
[69]J. Liang and X. Zhou, Application of continuous-wave laser Z-scan technique to photoisomerization, J. Opt. Soc. Am. B,2005,22:2468.
[70]J. C. Liang, H. Zhao, X. Zhou, and H. Wang, Polarization-dependent effects of refractive index change associated with photoisomerization investigated with Z-scan technique, J. Appl. Phys.,2007,101:013106.
[71]X. S. Wang, W. L. She, and W. K. Lee, Optical spatial solitons supported by photoisomerization nonlinearity in a polymer, Opt. Lett.,2004,29:277.
[72]X. S. Wang and W. L. She, Spontaneous one-and two-dimensional optical spatial solitons supported by photoisomerization nonlinearity in a bulk polymer, Phys. Rev. E,2005,71:026601.
[73]H. Wang, B. Zhang, L. Yan, Y. Li, G Zheng, and W. She, Gray and dark spatial solitons supported by two-photon isomerization, Opt. Commun.,2007,275:234.
[74]B. Z. Zhang, H. C. Wang, and W. L. She, Influence of molecular reorientation on photoisomerization optical spatial solitons, J. Opt. A:Pure Appl. Opt.,2007,9:395.
[75]B. Z. Zhang, H. Cui, and W. L. She, Modulational instability of incoherently coupled beams in azobenzene-containing polymer with photoisomerization nonlinearity, Chin. Phys. B,2009,18:209.
[76]X. S. Wang, W. L. She, S. Z. Wu, and F. Zeng, Circularly polarized optical spatial solitons, Opt. Lett.,2005,30:863.
[77]S. Bian and M. G Kuzyk, Dark spatial solitons in bulk azo-dye-doped polymer using photoinduced molecular reorientation, Appl. Phys. Lett.,2004,85:1104.
[78]X. S. Wang, W. L. She, and W. K. Lee, Optical spatial soliton supported by photoisomerization nonlinearity in a polymer with a background beam, J. Opt. Soc. Am. B,2006,23:2127.
[79]J. C. Liang, Z. B. Cai, Y. Z. Sun, S. L. Xu, and L. Yi, Various kinds of spatial solitons associated with photoisomerization, J. Opt. Soc. Am. B,2009,26:36.
[80]J. C. Liang, Z. B. Cai, Y. Z. Sun, L. Yi, and H. C. Wang, "What types of spatial soliton can be formed based on two-photon-isomerisation?" Eur. Phys. J. D,2009, 55:685.
[81]A. Xie, L. Meer, W. Hoff, and R. H. Austin, Long-lived amide I vibrational modes in myoglobin, Phys. Rev. Lett.,2000,84:5435.
[82]D. N. Christodoulides and R. I. Joseph, Discrete self-focusing in nonlinear arrays of coupled waveguides, Opt. Lett.,1998,13:794.
[83]J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices, Nature(London),2003,422:147.
[84]Z. G Chen, H. Martin, E. D. Eugenieva, J. J. Xu, and A. Bezryadina, Anisotropic enhancement of discrete diffraction and formation of two-dimensional discrete-soliton trains, Phys. Rev. Lett.,2004,92:143902.
[85]X. S. Wang, A. Bezryadina, Z. G Chen, K. G. Makris, D. N. Christodoulides, and G I. Stegeman, Observation of two-dimensional surface solitons, Phys. Rev. Lett., 2007,98:123903.
[86]B. Eiermann, Th. Anker, M. Albiez, M. Taglieber, P. Treutlein, K.-P. Marzlin, and M. K. Oberthaler, Bright Bose-Einstein gap solitons of atoms with repulsive interaction, Phys. Rev. Lett.,2004,92:230401.
[87]A. Trombettoni and A. Smerzi, Discrete solitons and breathers with dilute Bose-Einstein condensates, Phys. Rev. Lett.,2001,86:2353-2356.
[88]M. Sato, B. E. Hubbard, A. J. Sievers, B. Llic, D. A. Czaplewski, and H. C. Craighead, Observation of locked intrinsic localized vibrational modes in a micromechanical oscillator array, Phys. Rev. Lett.,2003,90:044102.
[89]H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, Discrete spatial optical solitons in waveguide arrays, Phys. Rev. Lett.,1998,81: 3383-3386.
[90]Z. G Chen, H. Martin, E. Eugenieva, J. J. Xu, and J. K. Yang, Formation of discrete solitons in light-induced photonic lattices, Opt. Express,2005,13:1816.
[91]R. Iwanow, R. Schiek, G I. Stegeman, T. Pertsch, F. Lederer, Y. Min, and W. Sohler, Observation of discrete quadratic solitons, Phys. Rev. Lett.,2004,93:113902.
[92]A. Fratalocchi, G Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, Discrete propagation and spatial solitons in nematic liquid crystals, Opt. Lett.,2004,29:1530.
[93]X. S. Wang, Z. G. Chen, and P. G. Kevrekidis, Observation of discrete solitons and soliton rotation in optically induced periodic ring lattices, Phys. Rev. Lett.,2006,96: 083904.
[94]R. Fischer, D. N. Neshev, S. L. Aguayo, A. S. Desyatnikov, A. A. Sukhorukov, W. Krolikowski, and Y. S. Kivshar, Observation of light localization in modulated Bessel optical lattices, Opt. Express,2006,14:2825.
[95]X. S. Wang, Z. G Chen, and J. K. Yang, Guiding light in optically induced ring lattices with a low-refractive-index core, Opt. Lett.,2006,31:1887.
[96]J. Arlt and K. Dholakia, Generation of high-order Bessel beams by use of an axicon, Opt. Commun.,2000,177:297.
[97]A. Vasara, J. Turunen, and A. T. Friberg, Realization of general nondiffracting beams with computer-generated holograms, J. Opt. Soc. Am. A,1989,6:1748.
[98]Y. V. Kartashov, V. A. Vysloukh, and L. Torner, Stable ring-profile vortex solitons in bessel optical lattices, Phys. Rev. Lett.,2005,94:043902.
[99]J. C. Liang, Z. B. Cai, L. Yi, and H. C. Wang, Bessel solitary waves in strongly nonlocal nonlinear media with distributed parameters, Opt. Commun.,2010,283: 386.
[100]Y. V. Kartashov, R. Carretero-Gonzalez, B. A. Malomed, V. A. Vysloukh, and L. Torner, Opt. Express,2005,13:10703.
[101]Y. V. Kartashov, A. A. Egorov, V. A. Vysloukh, and L. Torner, Rotary dipole-mode solitons in Bessel optical lattices, J. Opt. B:Quantum Semiclass. Opt.,2004,6: 444-447.
[102]Y. V. Kartashov, A. A. Egorov, V. A. Vysoukh, and L. Torner, Phys. Rev. E,2004,70: 065602(R).
[103]Y. S. Kivshar and B. A. Malomed, Dynamics of solitons in nearly integrable systems, Rev. Mod. Phys.,1989,61:763.
[104]D. E. Pelinovsky, Y. S. Kivshar, and V. V. Afanasjev, Internal modes of envelope solitons, Physica D,1997,116:121-142.
[105]Y. S. Kivshar, D. E. Pelinovsky, T. Cretegny, and M. Peyrard, Internal modes of solitary waves, Phys. Rev. Lett.,1998,80:5032-5035.
[106]D. K. Campell, J. F. Schonfeld, and L. E. Guerrero, Resonance structure in kink-antikink interactions in φ4 theory, Physica D,1983,9:1.
[107]M. Peyrard and D. K. Campbell, Kink-antikink interactions in a modified sine-Gordon model, Physica D,19839:33.
[108]D. K. Campbell, M. Peyrard, and P. Sodano, Kink-antikink interactions in the double sine-Gordon equation, Physica D,1986,19:165.
[109]J. A. Gonzalez, A. Bellorin, and L. E. Guerrero, Internal modes of sine-Gordon solitons in the presence of spatiotemporal perturbations, Phys. Rev. E,2002,65: 065601(R).
[110]V. Skarka, N. Aleksic, and V. Berzhiani, Dynamics of electromagnetic beam with phase dislocation in saturable nonlinear media, Phys. Lett. A,2001,291:124.
[111]L. W. Dong, F. W. Ye, J. D. Wang, T. Cai, and Y. P. Li, Internal modes of localized optical vortex soliton in a cubic-quintic nonlinear medium, Physica D,2004,194: 219.
[112]M. Soljacic, S. Sears and M. Segev, Phys. Rev. Lett.,1998,81:4851.
[113]M. Soljacic and M. Segev, Phys. Rev. E,2000,62:2810.
[114]J. Yang, I. Makasyuk, P. G Kevrekidis, H. Martin, B. A. Malomed, D. J. Frantzeskakis, and Z. Chen, Phys. Rev. Lett.,2005,94:113902.
[115]A. W. Snyder and D. J. Mitchell, Accessible solitons, Science,1997,276:1538.
[116]Y. R. Shen, Solitons made simple, Science,1997,276:1520.
[117]W. Krolikowski, O. Bang, J. Rasmussen, et al., Modulational instability in nonlocal nonlinear Kerr media, Phys. Rev. E,2001,64:016612-1.
[118]O. Bang, W. Krolikowski, J. Wyller, et al., Collapse arrest and soliton stabilization in nonlocal nonlinear media, Phys. Rev. E,2002,66:046619-1.
[119]W. Krolikowski and O. Bang, Solitons in nonlocal nonlinear media:exact solutions, Phys. Rev. E,2000,63:016610-1.
[120]Ming Shen, Qi Wang, Jielong Shi, et al., Partially coherent accessible solitons in strongly nonlocal media, Phys. Rev. E,2006,73:056602-1.
[121]Zhiyong Xu, Yaroslav V. Kartashov, and Lluis Tomer, Stabilization of vector soliton complexes in nonlocal nonlinear media, Phys. Rev. E,2006,73:055601-1.
[122]Mousumi Ballav and A. Roy Chowdhury, A generalized nonlinear Schrodinger equation and optical soliton in a gradient index cylindrical media, Chaos, Solitons and Fractals,2007,31:794-803.
[123]Qi Guo, Boren Luo, Fhuai Yi, et al., Large phase shift of nonlocal optical spatial solitons, Phys. Rev. E,2004,69:016602-1.
[124]Yiqun Xie and Qi Guo, Phase modulations due to collisions of beam pairs in nonlocal nonlinear media, Opt. and Quant. Electron,2004,36:1335-1351.
[125]Q. Guo and X. Jiang, Induced focusing from co-propagation of a pair of bright-dark optical beams in self-defocusing Kerr media, Opt. Commun.,2005,254:19-29.
[126]Qi Guo, Boren Luo, and Sien Chi, Optical beams in sub-strongly non-local nonlinear media:a variational solution, Opt. Commun.,2006,259:336-341.
[127]Yi Huang, Qi Guo, and Jueneng Cao, Optical beams in lossy non-local Kerr media, Opt. Commun.,2006,161:175-180.
[128]Qi Guo, Xiaping Zhang, Wei Hu, et al., Photonic switching and logic gating with strongly nonlocal spatial optical solitons, Acta Physica Sinica,2006,55:1832-1839.
[129]M. Peccianti, C. Conti, G. Assanto, et al., All optical switching and logic gating with spatial solitons in liquid crystals, Appl. Phys. Lett.,2002,81:3335-3337.
[130]M. Peccianti, C. conti, and G Assanto, Optical modulational instability in a nonlocal medium, Phys. Rev. E,2003,68:025602-1.
[131]M. Peccianti, K. A. Brzdakiewicz, and G Assanto, Nonlocal spatial soliton interactions in nematic liquid crystals, Opt. Lett.,2002,27:1460-1463.
[132]G Assanto, M. Peccianti, and C. Conti, Spatial optical solitons in bulk nematic liquid crystals, Acta Physica Polonica (A),2003,103:161-167.
[133]M. Peccianti, C. Conti, and G Assanto, Optical multisoliton generation in bulk nematic liquid crystals, Opt. Lett.,2003,28:2231-2233.
[134]C. Rotschild, O. Cohen, O. Manela, et al., Solitons in nonlinear media with an infinite range of nonlocality:first observation of coherent elliptic solitions and of vortex-ring solitions, Phys. Rev. Lett.,2005,95:213904-1.