On the Properties of a Nonlocal Nonlinear Schrödinger Model and Its Soliton Solutions

Nonlinear waves are normally described by means of certain nonlinear evolution equations. However, finding physically relevant exact solutions of these equations is, in general, particularly difficult. One of the most known nonlinear evolution equation is the nonlinear Schrödinger (NLS), a universal equation appearing in optics, Bose-Einstein condensates, water waves, plasmas, and many other disciplines. In optics, the NLS system is used to model a unique balance between the critical effects that govern propagation in dispersive nonlinear media, namely dispersion/diffraction and nonlinearity. This balance leads to the formation of solitons, namely robust localized waveforms that maintain their shape even when they interact. However, for several physically relevant contexts the standard NLS equation turns out to be an oversimplified description. This occurs in the case of nonlocal media, such as nematic liquid crystals, plasmas, and optical media exhibiting thermal nonlinearities. Here, we study the properties and soliton solutions of such a nonlocal NLS system, composed by a paraxial wave equation for the electric field envelope and a diffusion-type equation for the medium’s refractive index. The study of this problem is particularly interesting since remarkable properties of the traditional NLS—associated with complete integrability—are lost in the nonlocal case. Nevertheless, we identify cases where derivation of exact solutions is possible while, in other cases, we resort to multiscale expansions methods. The latter, allows us to reduce this systems to a known integrable equation with known solutions, which in turn, can be used to approximate the solutions of the initial system. By doing so, a plethora of solutions can be found; solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are predicted to occur.