关于非局域孤子解存在条件

1、关于非局域孤子解存在条件的一个猜想洪伟毅 郭旗华南师范大学信息光电子科技学院微纳光子功能材料与器件广东省重点实验室OutlineBackgraound Necessary condition for soliton-like solutions Our conjectureBackground模型A(z,t) 未知函数(optical envelope), R(t)已知函数（响应函数，R(0) 不可导）, 变量 z 0,), t (-, )此方程称为非局域非线性薛定谔方程 (nonlocally nonlinear Schrodinger equation, NNLSE)。数学上是非线性微分-

2、积分方程，在物理上有广泛的应用。非局域(nonlocality) 模型对空间问题, A(z,t)为光束函数，t为空间坐标，代表垂直与传输方向z的空间坐标，R(t)为对称函数，比如对时间问题, A(z,t)为脉冲函数，t为时间坐标，由于因果关系，R(t)为非对称函数，比如非局域性及其分类r-r非局域性及其分类局域的(locality)：弱非局域(weak nonlocality)一般性非局域(general nonlocality)强非局域(strong nonlocality) 问题的提出 EditorsSuggestionsPapersthatarejudgedtobeparticularl

3、yimportant,interesting,andwellwrittenarechosenasEditorsSuggestions. 问题的提出Necessary condition for soliton-like solutions mass centerThe “mass center of A is defined asNNLSE（ ） mass center mass center孤子解The soliton-like solution of NNLSE is defined as the form:where can be any arbitrary function.tradi

4、tional solitons that are static or uniformly moving possible accelerated solitons. Invariance Necessary condition for solitons=0By comparison, we can obtain the necessary condition for soliton-like solututions Necessary condition for solitons=0Example 1: local mediaThere are sech-form solitons in su

5、ch media. Necessary condition for solitons=0Example 2: Spatially nonlocally nonlinear bulk mediaThere are symmetric or antisymmetric solitons in such media. (11)-dimensional HermiteGauss breathers and solitonsD. M. Deng, X. Zhao，Q. Guo*, and S. Lan, JOSAB, 24 (2007) 2537-2544HermiteGauss solution Ne

6、cessary condition for solitonsD. M. Deng, X. Zhao，Q. Guo*, and S. Lan, JOSAB, 24 (2007) 2537-2544(1+1)-D Hermite-Gaussian solitons Necessary condition for solitons(12)-dimensional HermiteGauss solutionsD. M. Deng, X. Zhao，Q. Guo*, and S. Lan, JOSAB, 24 (2007) 2537-2544m=1, n=0m=2, n=0m=3, n=0m=1, n=

7、1HG10HG20HG30HG11 Necessary condition for solitonsHermite-Gauss solitonsD. M. Deng, X. Zhao，Q. Guo*, and S. Lan, JOSAB, 24 (2007) 2537-2544HG00HG01HG10HG11 Necessary condition for solitons柱坐标系下, Snyder-Mitchell模型的Laguerre-Gauss解:Laguerre-Gauss solutionsD. Deng and Q. Guo*, J. Opt. A: Pure Appl. Opt.

8、, Vol.10 (2008), 035101 当P0=Pc，得到Laguerre-Gaussian孤子解： Necessary condition for solitonsLaguerre-Gaussian solitonsD. Deng and Q. Guo*, J. Opt. A: Pure Appl. Opt., Vol.10 (2008), 035101 Necessary condition for solitons椭圆柱坐标系中可得到S-M model的Ince-Gauss孤子解为： 其中， 是归一化常数， 是 (p, m) 阶的奇模和偶模Ince多项式， 椭圆参量。Ince-G

9、auss solutions通过以下变换引入椭圆柱坐标系:其中， 表示径向和角向的椭圆坐标变量，f 是半焦距离 (semifocal separation)。D. Deng，and Q. Guo*, Opt. Lett, 32 (2007), 3206; J Phy. B, 41(2008), 155401 Necessary condition for solitons (3,3)阶偶模Ince高斯孤子(P0=Pc) ，(a)-(d)解析解，(e)-(h)数值解(非局域程度为0.1), (i)-(l)数值解(非局域程度为0.3), 椭圆参数=2. (3,3)阶奇模Ince高斯孤子(P0=Pc

10、) ，(a)-(d)解析解，(e)-(h)数值解(非局域程度为0.1), (i)-(l)数值解(非局域程度为0.3). 椭圆参数=2V Solitons in strong temporal nonlocalOur conjecture 0A(z,t) 未知函数(optical envelope), R(t)已知不对称函数（响应函数，R(0) 不可导）, 变量 z 0,), t (-, )We have conjectured that the noninstantaneous Kerr system with generally nonlocality has no soliton-like solutions. This is an open question.ACKNOWLEDGMENTSWe thank Prof. Senyue Lou for helpful discussions.CONCLUSIONWe have conjectured that the noninstant