# Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses
## 2 Nonlocality-Controllable Kerr Nonlinearities
The Lagrangian density of the system described by Equation (7) is the following: ([PERSON] and [PERSON], 2016; [PERSON] and [PERSON], 2016)  
\[\mathcal{L}=\frac{i}{2}\bigg{(}u^{*}\frac{\partial u}{\partial z}-u\frac{ \partial u^{*}}{\partial z}\bigg{)}-\frac{1}{2}|\
abla_{D}u|^{2}+\frac{s}{2}| u|^{2}\Big{(}R_{D}\otimes|u|^{2}\Big{)}. \tag{8}\]  
We introduce a trial Gaussian beam:  
\[u(\mathbf{r},z)=A(z)\exp[i\alpha(z)]\exp\biggl{[}-\frac{r^{2}}{2w(z)^{2}}+ic(z)r^{ 2}\biggr{]}, \tag{9}\]  
where all meanings of parameters \(A,\alpha\) and \(c\) can be found in Ref. ([PERSON], 2016). According to the standard procedure of variational approach ([PERSON] and [PERSON], 2016; [PERSON] and [PERSON], 2016), we obtain the evolution of the beam width \(w\)([PERSON], 2016):  
\[\frac{d^{2}w}{dz^{2}}=\frac{1}{w^{3}}-\frac{2 sP_{0}}{\pi^{D}}N_{D}(w,w_{m}), \tag{10}\]  
where \(P_{0}=\int_{-\infty}^{+\infty}|u(\mathbf{r},z)|^{2}d^{D}\mathbf{r}=\pi^{D/2}A^{2}w^{D}\) is the input power, and the mathematical expression of \(N_{D}\) can be found in Ref. ([PERSON], 2016). By assuming \(w|_{z=0}=w_{0}=1\) and \(dw/dz|_{z=0}=0\), Equation (10) can be solved:  
\[w_{m}^{2}\
abla_{D}^{2}\Delta n+\Delta n-s|u|^{2}=0. \tag{2}\]0  
where \(\sigma_{0}=w_{m}/w_{0}|_{w_{0}=1}=w_{m}\). For linear propagations of the input Gaussian beam, the beam width \(w_{0}=1\) is widened to be \(w(z)=\sqrt{1+z^{2}}\)([PERSON] and [PERSON], 2016). If the beam is in a focusing state, its width is \(w(z)<\sqrt{1+z^{2}}\), while in a defocusing state it is \(w(z)>\sqrt{1+z^{2}}\). Therefore, when nonlinearities transit between focusing and defocusing as the degree of nonlocality changes, by replacing \(w\) with \(\sqrt{1+z^{2}}\) in Equation (11) transition point \(\sigma_{c}\) can be obtained as  
\[w_{m}^{2}\
abla_{D}^{2}\Delta n+\Delta n-s|u|^{2}=0. \tag{2}\]1  
Equation (12) provides an implicit function of the critical degree of nonlocality \(\sigma_{c}\) on \(z\), \(P_{0}\), \(s\) and \(D\). For a given \(s\), \(D\) and \(P_{0}\), the function \(\sigma_{c}(z)\) can be obtained by numerically solving the integral Equation (12) at different propagation distances \(z\), which is shown in \[w_{m}^{2}\
abla_{D}^{2}\Delta n+\Delta n-s|u|^{2}=0. \tag{2}\]2  
At the specific distance \(z=1\), the critical degree of nonlocality for the \((1+1)\)- and \((1+2)\)-dimensional cases can be obtained from Equation (12) in that \(\sigma_{c}|_{z=1}=0.82\) and \(\sigma_{c}|_{z=1}=0.63\), respectively. The dependence of beam width on the degree of nonlocality is given in \[w_{m}^{2}\
abla_{D}^{2}\Delta n+\Delta n-s|u|^{2}=0. \tag{2}\]3 For the case of \(s=-1\), optical beams experience self-defocusing nonlinearity when \(\sigma_{0}<\sigma_{c}\), and self-focusing nonlinearity when \(\sigma_{0}>\sigma_{c}\). The case of \(s=1\) is on the contrary. Figure 3 shows the evolutions of the optical beam for different degree of nonlocality in two cases of \(s=1\) and \(s=-1\), where the degree-of-nonlocality-dependence self-focusing and self-defocusing effects can be obviously observed. \[w_{m}^{2}\
abla_{D}^{2}\Delta n+\Delta n-s|u|^{2}=0. \tag{2}\]4 Beam widths at \(z=1\) as the function of \(\sigma_{0}\) for different input powers. (**a**) \(D=1\), (**b**) \(D=2\). Solid color curves represent the numerical simulation results, agreeing well with the variational results denoted by dashed color curves. For comparison, the output beam width at \(z=1\) in the linear case is also plotted by the horizontal solid straight line (After Ref. [28]).