# Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses
## 4 Characteristics of Solitons
### Fundamental Solitons  
Integrating Equation (10) once yields \(\frac{1}{2}\Big{(}\frac{dw}{dz}\Big{)}^{2}+V_{D}(w)=0\), where \(V_{D}(w)\) is the equivalent potential in the form of [28]  
\[V_{D}(w)=-\int_{1}^{w}\Biggl{[}\frac{1}{w_{1}^{3}}-\frac{2 sP_{0}}{\pi^{D}}N_{D} (w_{1},\sigma_{0})\Biggr{]}dw_{1}. \tag{22}\]  
The soliton solutions correspond to the extremum points of the equivalent potential [28; 33]; therefore, by allowing \(\partial_{w}V_{D}=0\) the critical power of the solitons can be obtained as  
\[P_{c}=\frac{\pi^{D}}{2 sN_{D}(1,\sigma_{0})}. \tag{23}\]  
It should be noted that, in Equation (23), the critical power \(P_{c}\) must be positive, which requires \(N_{D}>0\) for \(s=1\) and \(N_{D}<0\) for \(s=-1\). Furthermore, the solitons are stable at the minimum points of \(V_{D}(w)\), then we have:  
\[\frac{d^{2}V_{D}}{dw^{2}}\Big{|}_{w=1,P_{0}=P_{c}}>0. \tag{24}\]  
From Equation (24), the existence ranges of the degree of nonlocality \(\sigma_{0}\) for the solitons can be determined, which depend on the values of \(D\) and the sign of \(s\). When \(D=1\), the solitons exist if \(\sigma_{0}\in[0,0.76]\) for \(s=1\) and \([1.05,+\infty)\) for \(s=-1\). When \(D=2\), the existence ranges of \(\sigma_{0}\) are \([0.33,0.61]\) for \(s=1\) and \([0.90,+\infty)\) for \(s=-1\). These variational results agree well with the numerical ones, which are given in the following. In order to numerically iterate the soliton solution, we assume \(u(x,z)=u(x)\exp(i\lambda z)\). Substitution of the solution into Equations (1) and (2) yields the following:  
\[\frac{1}{2}\frac{d^{2}u}{dx^{2}}-\lambda u+\Delta nu = 0, \tag{25}\] \[w_{m}^{2}\frac{d^{2}\Delta n}{dx^{2}}+\Delta n-su^{2} = 0. \tag{26}\]  
By the imaginary-time method, the solitons can be numerically found in higher ranges of the degree of nonlocality, \(\sigma_{0}\in[1.05,+\infty]\) for \(D=1\) and \(\sigma_{0}\in[1.11,+\infty]\) for \(D=2\) in the case of \(s=-1\). These ranges of \(\sigma_{0}\) are all within the self-focusing sides of such a nonlinear system and agree well with the variational results. The bright solitons have two abnormal properties. The first one is the negative propagation constants, while it is the opposite of the cases in local nonlinear media [39] and nonlocally nonlinear media with the positively defining attenuating response functions [3]. The other is the negative slope of the \(P_{c}(\lambda)\) (\(P_{c}\) and \(\lambda\) being the critical power and the propagation constant), as shown in Figure 8. By means of linear stability analysis, all solitons are shown to be stable [27]. This means that the stability criterion obeys an anti-Vakhitov-Kolokolov stability criterion [40], which is also obeyed by the in-phase and out-of-phase bound-state solitons and the multi-peak solitons reviewed in the next two sections. However, in nonlocally nonlinear media with the positively defining attenuating response functions, stable solitons should comply with the Vakhitov-Kolokolov criterion [9]. We have noticed that the evolutions for matter-wave components and the microwave magnetic field in the atomic Bose-Einstein condensates can also be governed by the mathematical model of Equation (1) plus (2) with \(s=-1\) in some specific conditions [23; 41; 42], and in such a system, bright solitons are obtained in strongly nonlocal cases. Although what Equations (25) and (26) determine are the solitons of the nonlocally nonlinear system (7), the coupled equations with \(s=1\) are formally the same as the model governing parametric spatial solitary waves in quadratic nonlinear materials [21; 22; 43]. The fundamental and the second harmonic waves are mathematically equivalent to the paraxial beam and its induced NRI [21]. [PERSON] et al. employed the formal equivalence between two distinct physical systems to investigate the stabilities of solitons and found that the solitons are all unstable.