Black and gray spatial optical solitons with Kerr type nonlocal nonlinearity

We develop one numerical method to compute black and gray solitons with Kerr-type nonlocal nonlinearity. As two examples of nonlocal cases, the gray soliton with exponentially decaying nonlocal response or with Gaussian nonlocal response are discussed. For such two nonlocal cases, the analytical form of the tails of nonlocal gray soliton is presented and the analytical relationship for the maximal transverse velocity of nonlocal gray soliton to the characteristic nonlocal length is obtained.


I. INTRODUCTION
In present years spatial solitons with Kerr-type nonlocal nonlinearity have attracted a great amount of studies. It is indicated the nonlocality of the nonlinearity ensures the existence of stable multidimensional solitons [1,2,3,4]. The nonlocal bright spatial solitons have been experimentally observed in nematic liquid crystal [5,6,7] and in the lead glass [8]. The propagation and interaction properties of nonlocal bright/black solitons are greatly different from that of local solitons. For example, the dependent functions of the beam power and phase constant on the beam width for nonlocal bright solitons are very different from those for local bright solitons [9,10,11]; There exists higher order Hermite-Gaussian like (1+1) dimensional bright nonlocal solitons [9,10,12] and Laguerre-Gaussian like (1+2) dimensional bright nonlocal solitons [9,11,12,13,14,15]; Bright nonlocal solitons with π phase difference attract rather repel each other [12,16,17,18]; Nonlocal black solitons can attract each other and two black solitons can form a bound state [19,20,21]. The nonlocal gray solitons also can form a bound state [24].
In this paper we present a numerical method to compute black and gray soliton solutions with Kerr-type nonlocal nonlinearity. This numerical method is a generalized version of the so called spectral renormalization method presented by M. J. Ablowitz and Z. H. Musslimani [22] with which they compute the bright soliton solutions. Our numerical method can be easily generalized to compute a large family of gray soliton solutions with other type of nonlinearity. It is worth to note that Yaroslav V. Kartashov and Lluis Torner [24] has rather successfully studied the nonlocal gray soliton with an exponential decaying nonlocal response in a different manner. They has revealed that the gray soliton velocity depends on the nonlocality degree and pointed out the maximal velocity of gray soliton monotonically decreases with the degree of nonlocality [24]. In our paper, besides of the nonlocal case of an exponential de- * Electronic address: guoq@scnu.edu.cn caying nonlocal response, the nonlocal case of Gaussian nonlocal response is also discussed. The analytical form of the tails of nonlocal gray soliton is presented. It is indicated that nonlocal gray solitons can have exponentially decaying tails or have exponentially decaying oscillatory tails. The analytical relationship for the maximal transverse velocity of nonlocal gray soliton to the characteristic nonlocal length is presented. It is indicated when the characteristic nonlocal length is less than some critical value, such maximal transverse velocity is a constant and equal to that of local gray soliton, otherwise such maximal velocity will decrease with the increasing of the characteristic nonlocal length. That mostly agrees with the results of reference [24] with slight but not trivial exception. It was said in reference [24] that such maximal velocity monotonically decreases with the characteristic nonlocal length. But as will be shown in our paper the maximal velocity does not vary with the characteristic nonlocal length when the characteristic nonlocal length is less than some critical value. Our paper poses a problem for such inconsistence.
In this paper we numerically compute the black and gray soliton solutions of NNLSE (1) that take these following form Substitution of Eq. (4a) into (1), we have an equation for ψ(x) and φ(x). Since both the real and imaginary part of such resulting equation must vanish, we have From Eq. (6),we have d dx (ψ 2 φ ′ ) = 0, which results in ψ 2 φ ′ = const. We set this constant as µη 2 , where µ * = µ is another constant. So we have Since The phase jump through the soliton is defined as 2c. Substituting Eq. (7) into (5), we get and So the gray soliton takes a form of u( (10) and (7). It is easy to prove that the Galilean transformed solution where , the gray soliton moves in velocity −µ with respect to the coordinate system. On the other hand, since , which implies the background of gray soliton is at rest with respect to the coordinate system. So the soliton u 1 (x, z) moves in transverse velocity −µ with respect to the background intensity of gray soliton. In this paper we simply refer −µ as the transverse velocity of the soliton.
As an example, we consider this following nonlocal case [1,2,3,4,5,6,7,10,11,16,18,21,24] n − w 2 d 2 n    which results in n( is the exponential decaying nonlocal response function. In another example, we consider the Gaussian nonlocal response [2,3,9,10,11,14,15,17,18] For such two nonlocal cases, w is referred as the characteristic nonlocal length. In Fig. (1) some nonlocal gray soliton solutions are shown. The numerical simulation result in Fig. (2) indicates the numerical nonlocal gray soliton solution obtained can describe the soliton state very well. The phase jump through the nonlocal gray soliton is shown in Fig. (3). As shown by Fig. (4) the nonlocal gray solitons can have exponentially decaying oscillatory tails for both the exponential nonlocal response case and the Gaussian nonlocal response case. Now we investigate the form of the decaying tails of gray solitons. In case |χ(x)| ≪ η, to the leading order Eq. (13) can be linearized to Since Eq. (21) is linear for χ(x), the superposition theorem applies. For the exponential nonlocal response R(x) = 1 2w exp(− |x| w ), Eq. (21) reduces to two coupled equations The solution of (22) can be assumed as Substituting (23) into (22), we have The eigenvalue problem of the above equation provides an equation for λ which results in When w → 0, from Eq. (26) to the leading order we have The roots of Eq. (26) are where For w = 5, η = 1, µ = 0.43, we have λ 1 = 0.0753, λ 2 = 0.596 which are used by the fitting function in Fig. (4).
To obtain exponentially decaying tails, λ must be a real number and we have which in turn result into On the other hand when λ is a complex number and exponentially decaying oscillatory tails are obtained. From inequalities (33), we have In conclusion, when w ≤ 1 2η the maximal transverse velocity does not vary with the characteristic nonlocal length w and is equal to η the maximal velocity of local gray soliton; When w > 1 2η the maximal transverse velocity is 4wη−1 4w 2 . The maximal transverse velocity will decrease when w > 1 2η with the increasing of the nonlocal length w. Such results are shown in Fig. (5). It is worth to note that there is slight but not trivial inconsistence between our results and the results of reference [24]. As shown in figure 4(a) of reference [24] and pointed out in reference [24], the maximal velocity monotonically decreases with the nonlocality degree (the characteristic nonlocal length in this paper). But as has been pointed out by our paper and shown in figure (5) in this paper, the maximal velocity does not vary with the characteristic nonlocal length w when w ≤ 1 2η . On the form of the tail of the nonlocal gray soliton we arrive at, for w ≤ 1 4η , the gray solitons always have exponentially decaying tails; for w > 1 2η , the gray solitons always have exponentially decaying oscillatory tails; for 1 4η < w ≤ 1 2η , the gray solitons can have exponentially decaying oscillatory tails when 0 ≤ µ 2 < 4wη−1 4w 2 or have exponentially decaying tails when 4wη−1 4w 2 ≤ µ 2 ≤ η 2 . Now we consider the nonlocal case with a Gaussian nonlocal response Here we only discuss the exponentially decaying oscillatory tails. However the exponentially decaying tails can be discussed in a similar way. For the exponentially decaying oscillatory tails we have where So Eq. (21) turns into The eigenvalue problem of the above equation provides two coupled equations for λ and κ For example, when w = 5, η = 1, µ = 0.4, from (40) we get λ = 0.0733, κ = 0.713 which are used by the fitting function in Fig. (4).
In a similar way of deducing the parameter space in the exponentially decaying nonlocal case, we also get the parameter space in the Gaussian nonlocal case. When w < 1/η, the maximal transverse velocity is a constant equal to η; when w > 1/η, the maximal velocity is equal to 1+ln(η 2 w 2 ) w 2 . On the other hand when w > 1/η, the gray soliton always has exponentially decaying oscillatory tail; When w ≤ 1/ √ eη, the gray soliton always has exponentially decaying tail; when 1/ √ eη < w < 1/η, the gray soliton can have exponentially decaying tail for or exponentially decaying oscillatory tail for µ < 1+ln(η 2 w 2 ) w 2 . These results are shown in Fig. (6).

IV. CONCLUSION
One numerical method to compute black and gray soliton solutions with Kerr-type nonlocal nonlinearity is developed. As two examples of nonlocal cases, the gray soliton with exponentially decaying nonlocal response or with Gaussian nonlocal response are discussed. The analytical form of the tails of nonlocal gray soliton is presented. It is indicated that nonlocal gray solitons can have exponentially decaying tails or have exponentially decaying oscillatory tails. The analytical relationship for the maximal transverse velocity of nonlocal gray soliton to the characteristic nonlocal length is presented. It is indicated when the characteristic nonlocal length is less than some critical value, such maximal transverse velocity is a constant and equal to that of local gray soliton, otherwise such maximal velocity will decrease with the increasing of the characteristic nonlocal length.