We present an overview of recent advances in the understanding of optical beams in nonlinear media with a spatially nonlocal nonlinear response. We discuss the impact of nonlocality on the modulational instability of plane waves, the collapse of finite-size beams, and the formation and interaction of spatial solitons.

We investigate the properties of localized waves in cubic nonlinear materials with a symmetric nonlocal nonlinear response of arbitrary shape and degree of nonlocality, described by a general nonlocal nonlinear Schrödinger type equation. We prove rigorously by bounding the Hamiltonian that nonlocality of the nonlinearity prevents collapse in, e.g., Bose-Einstein condensates and optical Kerr media in all physical dimensions. The nonlocal nonlinear response must be symmetric and have a positive definite Fourier spectrum, but can otherwise be of completely arbitrary shape and degree of nonlocality. We use variational techniques to find the soliton solutions and illustrate the stabilizing effect of nonlocality.

We study modulational instability ͑MI͒ of plane waves in nonlocal nonlinear Kerr media. For a focusing nonlinearity we show that, although the nonlocality tends to suppress MI, it can never remove it completely, irrespective of the particular profile of the nonlocal response function. For a defocusing nonlinearity the stability properties depend sensitively on the response function profile: for a smooth profile ͑e.g., a Gaussian͒ plane waves are always stable, but MI may occur for a rectangular response. We also find that the reduced model for a weak nonlocality predicts MI in defocusing media for arbitrary response profiles, as long as the intensity exceeds a certain critical value. However, it appears that this regime of MI is beyond the validity of the reduced model, if it is to represent the weakly nonlocal limit of a general nonlocal nonlinearity, as in optics and the theory of Bose-Einstein condensates.

No abstract

The modulational instability ͑MI͒ of plane waves in nonlocal Kerr media is studied for a general response function. Several generic properties are proven mathematically, with emphasis on how new gain bands are formed through a bifurcation process when the degree of nonlocality, , passes certain bifurcation values and how the bandwidth and maximum of each individual gain band depends on . The generic properties of the MI gain spectrum, including the bifurcation phenomena, are then demonstrated for the exponential and rectangular response functions. For a focusing nonlinearity the nonlocality tends to suppress MI, but can never remove it completely, irrespectively of the shape of the response function. For a defocusing nonlinearity the stability properties depend sensitively on the profile of the response function. For response functions with a positivedefinite spectrum, such as Gaussians and exponentials, plane waves are always stable, whereas response functions with spectra that are not positive definite ͑such as the rectangular͒ will lead to MI if exceeds a certain threshold. For the square response function, in both the focusing and defocusing case, we show analytically and numerically how new gain bands that form at higher wave numbers when increases will eventually dominate the existing gain bands at lower wave numbers and abruptly change the length scale of the periodic pattern that may be observed in experiments.

The theory of MHD soliton formation parallel to the magnetic field, as described by the DNLS equation and related models, is reviewed. The known results on the physical relevance, solutions and mathematical properties of the DNLS equation, are briefly described. New results on the extension of the theory to finite beta and three space dimensions are presented. A hybrid fluid and kinetic treatment of the case of finite temperature based on the guiding center plasma model, leads to an additional nonlocal term due to resonant particles. The combination of modulational instability and resonant particle damping leads to the conclusion that parallel weakly dispersive Alfvén waves are nonlinearly damped in a finite beta plasma.

The DNLS equation for parallel nonlinear and weakly dispersive MHD waves is extended to finite beta values as well as to three spatial dimensions, by means of the reductive perturbation method. Kinetic effects are included by means of the hybrid fluid and kinetic guiding-centre model of Grad (1961). The resulting equation contains a nonlinear and non-local term representing the effect of resonant particles.

No abstract