# Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses
## 2 Nonlocality-Controllable Kerr Nonlinearities  
The model we considered is the dimensionless system in the form of [27,28]  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\
abla_{D}^{2}u+\Delta nu=0, \tag{1}\]  
\[w_{m}^{2}\
abla_{D}^{2}\Delta n+\Delta n-s|u|^{2}=0. \tag{2}\]  
The equations are the coupled equations for a dimensionless optical beam \(u(\mathbf{r},z)\) and its (dimensionless) induced NRI \(\Delta n(\mathbf{r},z)\), where \(\mathbf{r}\) is \(D\)-dimensional transverse coordinate vectors (\(D=1\) or \(2\)), \(\
abla_{D}^{2}\) is the \(D\)-dimensional transverse Laplacian operator, \(s(=\pm 1)\) is the sign of Kerr coefficient and \(w_{m}\) is the nonlinear characteristic length. We will show in Section 6 that the model, Equation (1) plus (2), can describe the evolutions of optical beams in the NLC with negative dielectric anisotropy. However, the model discussed extensively before is the following [3,5]:  
\[w_{m}^{2}\
abla_{D}^{2}\Delta n-\Delta n+s|u|^{2}=0, \tag{3}\]  
of which the second term has different signs from that of Equation (2). It is such a difference in the signs that makes the two nonlinear systems described by Equations (2) and (3) exhibit utterly different features. Although the NRI described by Equation (2) in an infinite space can also be expressed by the following convolution:  
\[\Delta n(\mathbf{r},z)=sR_{D}\otimes|u|^{2}, \tag{4}\]  
similarly to the case of Equation (3) [3,5], the \(D\)-dimensional response function \(R_{D}\) is completely different. When \(D=1\), the response takes the sin-oscillatory function as [27]  
\[R_{1}=\frac{1}{2w_{m}}\sin\biggl{(}\frac{|\mathbf{x}|}{w_{m}}\biggr{)}. \tag{5}\]  
While, when \(D=2\)[28], the response function is expressed by the following:  
\[R_{2}=\frac{1}{4w_{m}^{2}}Y_{0}\biggl{(}\frac{r}{w_{m}}\biggr{)} \tag{6}\]  
with \(Y_{0}\) being the zeroth order Bessel function of the second kind. Substituting Equation (4) into Equation (1) yields a nonlocally nonlinear Schrodinger equation in the form of  
\[i\frac{\partial u}{\partial z}+\frac{1}{2}\
abla_{D}^{2}u+suR_{D}\otimes|u|^{ 2}=0. \tag{7}\]  
We can glimpse at the self-focusing and self-defocusing property of the nonlinear system described by Equation (7) from two limits where \(w_{m}=0\) and \(w_{m}\rightarrow\infty\). When \(w_{m}=0\), Equation (2) is reduced to \(\Delta n=s|u|^{2}\). Clearly, the focusing nonlinearity occursfor \(s=1\) and the defocusing nonlinearity does for \(s=-1\). When \(w_{m}\to\infty\), Equation (2) changes to \(w_{m}^{2}\
abla_{D}^{2}\Delta n=s|u|^{2}\), which models the focusing nonlinearity in lead glass for \(s=-1\)([PERSON] and [PERSON], 2016) and the defocusing nonlinearity for \(s=1\). Therefore, the nonlinearities reverse if \(w_{m}\) proceeds from \(0\) to \(\infty\) for both cases of \(s=\pm 1\). This indicates that the self-focusing and self-defocusing property of the nonlinear system described by Equation (7) depends on the degree of nonlocality, which is defined by \(\sigma=w_{m}/w\) with \(w\) being the beam width ([PERSON], 2016). In contrast, for the nonlinear system given by Equation (3), the self-focusing and self-defocusing property is different. In the case where \(s=1\), Equation (3) is reduced to \(\Delta n=|u|^{2}\) when \(w_{m}=0\), and to \(w_{m}^{2}\
abla_{D}^{2}\Delta n=-|u|^{2}\) when \(w_{m}\to\infty\). Under the two limits, the system (3) exhibits both self-focusing nonlinearity: that is, the local self-focusing nonlinearity when \(w_{m}=0\) and the (thermally induced) nonlocal self-focusing nonlinearity in lead glass \(w_{m}\to\infty\)([PERSON] and [PERSON], 2016). Conversely, in the case where \(s=-1\), the self-defocusing nonlinearity exists for the two limits of \(w_{m}\). Therefore, no nonlinearities transition can take place in the nonlinear system given by Equation (3). The dependence of the self-focusing and self-defocusing property on the degree of nonlocality can be obtained by both the variational approach and by numerical simulations.