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1、arXiv:nlin/0006028v1 nlin.PS 19 Jun 2000 Solitons in nonlocal nonlinear media: exact results Wies law Kr olikowski Australian Photonics Cooperative Research Centre, Laser Physics Centre, Research School of Physical Science and Engineering, The Australian National University, Canberra ACT 0200, Austr
2、alia Ole Bang Department of Mathematical Modelling, Technical University Denmark, Building 305/321, DK-2800 Kgs. Lyngby, Denmark We investigate the propagation of one-dimensional bright and dark spatial solitons in a nonlocal Kerr-like media, in which the nonlocality is of general form. We fi nd an
3、exact analytical solution to the nonlinear propagation equation in the case of weak nonlocality. We study the properties of these solitons and show their stability. I. INTRODUCTION Let us consider a phenomenological model of nonlocal nonlinear Kerr type media, in which the refractive index change n
4、induced by a beam with intensity I(x,z) can be represented in general form as n(I) = Z R(x x)I(x,z)dx,(1) where the positive (negative) sign corresponds to a focus- ing (defocusing) nonlinearity and x and z denote trans- verse and propagation coordinates, respectively.The real, localized, and symmet
5、ric function R(x) is the re- sponse function of the nonlocal medium, whose width determines the degree of nonlocality. For a singular re- sponse, R(x) = (x), the refractive index change becomes a local function of the light intensity, n(I) = I(x,z), i.e. the refractive index change at a given point
6、is solely determined by the light intensity at that very point. With increasing width of R(x) the light intensity in the vicinity of the point x also contributes to the index change at that point.In the limit of a highly nonlo- cal response Snyder and Mitchell showed that the beam evolution was desc
7、ribed by the simple equation for a lin- ear harmonic oscillator 1. The infl uence of nonlocality of the nonlinear response on the dynamics of beams was illustrated for the special logarithmic nonlinearity, which allows exact analytical treatment 2. While Eq. (1) is a phenomenological model, it never
8、- theless describes several real physical situations. Possible physical mechanisms responsible for this type of nonlin- ear response includes various transport eff ects, such as heat conduction in materials with thermal nonlinearity 35, diff usion of molecules or atoms accompanying non- linear light
9、 propagation in atomic vapours 6, and drift and/or diff usion of photoexcited charges in photorefrac- tive materials 7,8. A highly nonlocal nonlinearity of the form (1) has also been identifi ed in plasmas 912 and it appears as a result of many body interaction processes in the description of Bose-E
10、instein condensates 13. Even though it is quite apparent in some physical situ- ations that the nonlinear response in general is nonlocal (as in the case of thermal lensing), the nonlocal contri- bution to the refractive index change was often neglected 14,15. This is justifi ed if the spatial scale
11、 of the beam is large compared to the characteristic response length of the medium (given by the width of the response func- tion).However, for very narrow beams the nonlocal- ity can be of crucial importance and has to be taken into account. For instance, it has been shown theoreti- cally that a we
12、ak nonlocal contribution arrests collapse (catastrophic self-focusing) of high power optical beams in a self-focusing medium and leads to the formation of stable 2D (diff racting in two transverse dimensions) soli- tons 11,12,16,17. On the other hand, a purely nonlocal nonlinearity leads to formatio
13、n of so-called cusp solitons, which, however, are unstable 9. Some of the consequences of nonlocality in the nonlin- ear response have been observed experimentally. Suter and Blasberg reported stabilization of 2D solitary beams in atomic vapors due to atomic diff usion, which carries excitation away
14、 from the interaction region 6. Also, the discrepancy between the theoretical model of dark soli- tons and that observed experimentally in a medium with thermal nonlinearity has been associated with nonlocal- ity of the nonlinearity 14,15. Here, we investigate the propagation of 1D beams in nonlinea
15、r media having a weakly nonlocal response of the general form (1). Our goal is to fi nd exact analyti- cal solutions for bright and dark spatial solitons and use them to determine soliton properties, such as existence and stability. We start with the paraxial wave equation describing propagation of
16、a 1D beam with envelope func- tion = (x,z) and intensity I = I(x,z) = |(x,z)|2, iz + 1 2 2 x + n(I) = 0. (2) When the nonlocality is weak, i.e. when the response 1 function R(x) is narrow compared to the extent of the beam, we can expand I(x,z) around the point x= x to obtain n(I) = (I + 2 xI), (3)
17、where the nonlocality parameter 0 is given by = 1 2 Z R(x)x2dx,(4) and where we have assumed that the response function is normalized, R R(x)dx = 1. Note that for R(x) = (x), = 0 and Eq. (3) describes the local Kerr nonlinearity. For weakly nonlocal meda 1 is a small parameter. 0 xx R(x-x) I(x) FIG.
18、 1. Intensity profi le I(x) = I(x,z) and response func- tion R(x x) in the weakly nonlocal limit. Substituting n(I), given by Eq. (3), into Eq. (2) gives the modifi ed nonlinear Schr odinger equation iz + 1 2 2 x (| 2 + 2 x| 2) = 0. (5) The weak nonlocality appears thus as a perturbation to the loca
19、l nonlinear refractive index change. For a single peak beam in a self-focusing medium this perturbation is of negative sign in the central part of the beam, where it serves to decrease the refractive index change. Hence, even for very narrow and sharp intensity distributions, the resulting self-indu
20、ced waveguide will be wide and a smooth function of the transverse coordinates. In some sense, this is similar to saturation of the nonlinearity. One may therefore expect that certain features of soli- tons of Eq. (5) will be reminiscent of those exhibited by solitons in saturable nonlinear media. I
21、t transpires, how- ever, that nonlocality also results in new eff ects, espe- cially for self-defocusing nonlinearities. In the following we consider separately the case of self-focusing and self- defocusing nonlinearities. II. BRIGHT SOLITONS For a self-focusing nonlinearity the sign of the refrac-
22、 tive index change is positive. We search for a stationary bright soliton solution to Eq. (5) of the form (x,z) = u(x)exp(iz),(6) where the profi le u = u(x) is real, symmetric, and expo- nentially localized and the propagation constant 0 is positive. For this solution Eq. (5) reduces to 2 xu + 2(u
23、2 )u + 2u2 xu 2 = 0,(7) which can be integrated once to give the equation (1 + 4u2)(xu)2+ (u2 2)u2= C,(8) where C is an integration constant. For exponentially localized solutions C = 0. Using that u(x) has its max- imum u0at the center x = 0 we futher obtain the well- known relation between the pro
24、pagation constant and the amplitude u0 u2 0= 2. (9) Equation (8) can therefore be simplifi ed to (xu)2= u2(u2 0 u 2)/(1 + 4u2). (10) Interestingly, had the local and nonlocal contributions been of opposite signs, i.e. 0 and unstable otherwise 18. For the model considered here P() can be found analyt
25、ically to P = 0+ 1 + 40 4tan1(p40),(13) by using Eq. (10). For weakly nonlocal media with 1 the power is approximately given by P = P0(1 + 4 30 16 15 22 0+ .), (14) where P0= 20is the soliton power in the limit of a local response with =0. The derivative dP/d can easily be found from Eq. (13) and it
26、 transpires that the power is a monotonically increasing function of the prop- agation constant (for 0), indicating that the solitons are stable. Interestingly again, had been negative, then the bright solitons would exist for suffi ciently low peak in- tensities, 0 0.7/|4|, for which dP/d is negati
27、ve (see again 10). To demonstrate the stability of the bright solitons for 0 we numerically integrated Eq. (5) using the split- step fourier method and the exact soliton solution as ini- tial condition.In all simulations (for both bright and dark solitons) we used a steplength of dz = 0.001 and a tr
28、ansverse resolution of dx = 0.05. Results of the numeri- cal simulations are shown in Fig. 4 where we demonstrate propagation and collision of two bright solitons with unit amplitude 0= 1. It is evident that the solitons propa- gate in a stable manner. The collision, on the other hand, is not comple
29、tely elastic and causes the soliton amplitude and width to oscillate slightly. FIG. 4. Collision of unit amplitude bright solitons in a weakly nonlocal Kerr-like medium with = 0.1 Thebrightsolitonsofthenonlocalnonlinear Schr odinger Eq. (5) were considered by Davydova and Fischuk in the context of f
30、ocusing of upper hybrid waves in plasma 12. In particular, the existence of bright soli- ton solutions and their stability was reported by these authors. However, the explicit expression (11) was not given and no numerical confi rmation was presented. III. DARK SOLITONS We now consider the impact of
31、 weak nonlocality in the case of a self-defocusing nonlinearity, which corresponds to a negative sign in Eq.(3). We introduce a new spatial variable = xV z, with V being the soliton transverse velocity, and look for stationary solutions of the form 3 (x,z) = p ()exp(iz + i(),(15) where () is the sol
32、iton intensity and () its phase. Substitution of this form into Eq. (5) yields ( V ) = 0,(16) 2(1 4)2 () 2 42()2+ 8V 2 82( + ) = 0.(17) We are interested in dark solitons, i.e. solutions with an intensity dip on a constant background. Integrating the system (16-17) once we obtain that the soliton ba
33、ck- ground intensity 0= u2 0 determines the soliton prop- agation constant and that the center intensity 1= u2 1 determines the transverse velocity V , through the rela- tions V 2 = 1, = 0,(18) which are exactly the same as those obtained for a purely local response. We further fi nd that the solito
34、n intensity () is governed by the equation ()2= 4( 1)(0 )2/(1 4).(19) Obviously, a solution to Eq. (19) exists only if the back- ground intensity does not exceed a certain critical value, 0 cr= 1/|4|,(20) which coinsides with the critical peak intensity above which bright solitons does not exist in
35、a focusing non- local medium when 0.(26) In the case of a nonlocal nonlinearity the expression for the Momentum can be evaluated analytically to Q = 2otan1 ? 0 1 ? + (0 1) ? 1 0 ? +1 + 4(20 1) r 1 4 tan1(p40)(27) The dependence of the momentum on the velocity of dark solitons with background intensi
36、ty 0= 1 is shown in Fig. 8 for several degrees of nonlocality . The mono- tonic increase of this function indicates that the dark solitons are stable. =0.15 =0=0.24 FIG. 8. Momentum Q of dark solitons with 0= 1 versus velocity V for diff erent degrees of nonlocality, . This conclusion is confi rmed
37、by direct numerical sim- ulations of Eq. (5) with initial conditions given by the exact solution (21-22). Fig. 9 illustrates propagation and collision of two identical dark solitons with 0= 101= 1 in a weakly nonlocal medium with = 0.1. While the solitons propagate in a stable fashion, their collisi
38、on is clearly inelastic with radiation being emitted from the impact area. 5 FIG. 9. Propagation and collision of identical dark solitons with 0= 101= 1 in a weakly nonlocal medium with = 0.1 In conclusion, we have studied the properties of bright and dark one-dimensional spatial solitons in a gener
39、al weakly nonlocal Kerr-like medium, in which the change in refractive index due to local and nonlocal contributions are of the same sign. We have found an exact analytical solution for both bright and dark solitons and used it to fi nd the existence and stability regions of both solutions. Stable b
40、right soliton solutions were found to exist in focusing nonlocal media for all parameter values.Al- though they were previously predicted the solution was never explicitly written down. The eff ect of nonlocal- ity is in some sense equivalent to that of saturation, - to smooth out the index profi le
41、 and thereby increase the soliton width. Previously unknown dark soliton solutions were found to exist in defocusing nonlocal media for background in- tensities below a certain critical value, which correspond to the intensity at which the plane wave solutions become unstable. We found that nonlocal
42、ity leads to narrowing of the dark solitons, with all dark solitons acquiring the same width in the strongly nonlocal limit, independent of the soliton contrast. The total phase change across the dark soliton was found to be less that in the purely local case, with the phase profi le resembling that
43、 of a threshold nonlinearity. Both nonlocal bright and dark spatial solitons appear to be stable under propagation. Preliminary studies of soliton collision revealed their inelastic character in anal- ogy to collisions of solitons of nonintegrable models. O. Bang acknowledges support from the Danish
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