# Unique Features of Nonlocally Nonlinear Systems with Oscillatory Responses
## 1 Introduction  
The optical Kerr effect (OKE), a phenomenon that refers to the dependence of the refractive index on the optical intensity, is one of the most important effects in nonlinear optics [1; 2]. The physical mechanism of OKE includes molecular reorientation, thermal nonlinearity, photorefractive effect, electronic contribution and electrostriction, etc. No matter what the physical mechanism is, the caused refractive index can always be phenomenologically represented as \(n(\mathbf{r},z,t)=n_{0}(\mathbf{r},z)+n_{nl}(\mathbf{r},z,t)\), with \(n_{0}\) and \(n_{nl}\) being, respectively, the linear refractive index and nonlinear refractive index (NRI). If NRI at a certain point in space is not determined solely by the optical intensity at that point but also depends on its vicinity, nonlinearity is spatially nonlocal [3; 4; 5]. Mathematically, NRI can be expressed as an integral of a kernel function (also called the response function) and an optical intensity [3]. The media of nonlocal nonlinearity are referred to as nonlocally nonlinear media [6; 7; 8; 9], which ranges from nematic liquid crystal (NLC) with positive dielectric anisotropy [10; 11; 12; 13; 14], lead glass [15], nonlinear ion gas [16], photorefractive crystal [17], dipolar Bose-Einstein condensate [18] and so on. For the physical systems mentioned above, response functions are positive definite, for example, the exponential-decay response function for the \((1+1)\)-dimensional case [19] and the zeroth-order modified Bessel function for the \((1+2)\)-dimensional infinite case [10; 20].  
In fact, there is the other kind of response function without positive definiteness: the sine-oscillation function, brought out in the study of quadratic solitons by the formal equivalence in mathematics between quadratic and nonlocal solitons [21; 22]. This kind of sine-oscillation function was also obtained in a system of coupled Gross-Pitaevskii-Poisson equations [23], which govern the evolutions for matter-wave components and the microwave magnetic field in atomic Bose-Einstein condensates. Recently, we investigated the nonlocally nonlinear system with oscillatory responses [24; 25; 26; 27; 28; 29; 30; 31; 32] and found various new features such as the nonlocality-controllable transitions between focusing and defocusingnonlinearities, unique modulational instabilities (MI) and new kinds of solitons. The nonlocally nonlinear system with oscillatory responses can model the propagation of optical beams in NLC with negative dielectric anisotropy.  
In this review article, we provide a brief overview of unique features of nonlocally nonlinear system with oscillatory responses, including its mathematical model, the treatment of the model and the nonlocality-controllable transition between focusing and defocusing nonlinearities in Section 2, the short-term and long-term MI properties in Section 3, different kind of soliton solutions in Section 4 and a special algorithm for finding soliton solutions in Section 5. In Section 6, we discuss the nonlinearity characteristic of NLC with negative dielectric anisotropy and show that it can be modeled by the nonlinear system with oscillatory responses.