Two-dimensional Whittaker solitons in nonlocal nonlinear media Wei-Ping Zhong, 1,6 Milivoj Belić, 2 Rui-Hua Xie, 3,4,5, * Goong Chen, 5 and Lin Yi 6 1 Department of Electronic Engineering, Shunde College, Guangdong Province, Shunde 528300, China 2 Texas A&M University at Qatar, P.O. Box 23874, Doha, Qatar 3 Department of Physics, Hubei University, Wuhan, Hubei 430062, China 4 Department of Applied Physics , Xi’an Jiaotong University, Xi’an 710049, China 5 Department of Mathematics, Texas A&M University, College Station, Texas 77843 USA 6 Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China Received 25 January 2008; revised manuscript received 23 April 2008; published 18 July 2008; corrected 23 September 2008 1 Two-dimensional Whittaker solitons WSs are introduced and investigated numerically in nonlocal non- linear media. Different classes of stable and unstable higher-order spatial optical solitons are discovered among the solutions of the generalized nonlocal nonlinear Schrödinger equation, in analogy with the linear Whittaker eigenmodes of the quantum harmonic oscillator. Specific values of the modulation depth parameter for differ- ent values of the topological charge are discussed. Our results reveal that in media with a Gaussian response function higher-order spatial WSs can exist in various families, such as two-dimensional Gaussian solitons, vortex-ring solitons, half-moon solitons, and symmetric and asymmetric single-layer and multilayer necklace solitons. The stability of WSs is addressed numerically. We establish that two-dimensional Gaussian solitons and vortex-ring solitons are stable, while other families of spatial WSs are unstable, although their stability can be improved by moving into the strongly nonlocal regime. DOI: 10.1103/PhysRevA.78.013826 PACS numbers: 42.65.Tg, 42.65.Sf I. INTRODUCTION The recent surge of interest in the study of nonlocal opti- cal solitons was initiated by a number of experimental obser- vations and theoretical treatments of self-trapping effects and spatial solitary waves in different types of nonlocal nonlinear NN media 1. Nonlocality is ubiquitous in nature and has already been demonstrated in a diversity of nonlinear NL media, such as atomic vapors 2, nematic liquid crystals 3, photorefractive media 4, Bose-Einstein condensates 5, and so on. Nonlocality has recently become important in NL optics 6–14. Studies of spatially nonlocal nonlinearities have revealed a number of interesting effects 7–13. Perhaps most importantly, nonlocality tends to suppress modulational instability of plane waves propagating in self-focusing me- dia. It is well known that localized multidimensional waves in media with a focusing nonlinearity may exhibit cata- strophic collapse over finite propagation distances. Nonlocal- ity can prevent beam collapse and stabilize multidimensional solitons 10–14. In this paper, starting from a set of linear Whittaker modes, we construct higher-order spatial solitons in NN me- dia, in the form of Whittaker solitons WSs. We display different possible WS families: Gaussian solitons, vortex- ring solitons, half-moon solitons, and symmetric and asym- metric single-layer and multilayer necklace solitons. We find that some classes of WSs display well-defined symmetry and give rise to stable solitons, while others display unstable be- havior typical of multidimensional soliton clusters 9, al- though their stability may be improved. Our conclusions on enhanced stability are based on numerical study. More de- finitive answers must await more thorough analysis, which is beyond the scope of this contribution. The paper is organized as follows. In Sec. II we introduce the two-dimensional 2D WS model in NN media. Different families of WSs, investigated numerically, are presented in Sec. III. Section IV gives the conclusions. II. THE NONLOCAL NONLINEAR WS MODEL We consider the propagation of optical beams in a NN medium in the paraxial approximation, described by the gen- eralized nonlocal nonlinear Schrödinger equation NNSE 8,11–14 for the scalar electric field envelope ur  , z: i  u  z + 1 2   2 u + NIr , zu =0, 1 NIr , z =  - +  - + Rr  - r '|ur ', z| 2 dr ' , 2 where z and r  = x , y are the dimensionless propagation dis- tance and the transverse position vector, respectively. In po- lar coordinates the transverse Laplacian is   2 = 1 r   r  r   r  + 1 r 2  2   2 , where  is the azimuthal angle and r 2 = x 2 + y 2 . The nonlin- earity NIr  , z is represented in a general nonlocal form, with Rr  - r  ' being the medium response function and I = |ur  ' , z| 2 the beam intensity. Some general statements about the NNSE follow from this general form of the re- sponse function. In the limit when the response function is the delta func- tion Rr  - r  ' = r  - r  ', the nonlinearity becomes propor- tional to the intensity distribution, NIr  , z = |ur  , z| 2 , and we recover the local limit of the NNSE, i.e., the simple non- * Corresponding author. Email: rhxie@physics.tamu.edu PHYSICAL REVIEW A 78, 013826 2008 1050-2947/2008/781/0138267 ©2008 The American Physical Society 013826-1