Nonlinear Trapping Potentials and Nonlinearity Management

Abstract

The trapping potentials considered in Chap. 5 provide a traditional method for maintaining and stabilizing localized states, which, by itself, applies independently of the presence of nonlinearity in the system. Nonlinear potentials, induced by spatial modulation of the local strength of the cubic or other nonlinearity, offer a completely different method for the creation of self-trapped states (quasi-solitons). A highly efficient implementation of the latter method was proposed by Borovkova et al. [Opt. Lett. 36, 3088–3090 (2011a)] and Borovkova et al. [Phys. Rev. E 84, 035602(R) (2011b)], in the form of the self-repulsive cubic term with the coefficient growing fast enough from the center to periphery, as per Eqs. (2.27) and (2.30) or (2.31) (see Chap. 2). This scheme offers options for the creation of various localized states that would not exist or would be unstable without the use of nonlinear potentials. These are 2D and 3D vortex states with high values of the winding number, vortex gyroscopes, hopfions (vortex tori with intrinsic twist of the toroidal core), and hybrid modes in the form of vortex–antivortex pairs supported by an effective nonlinear potential with a peanut-like shape. Additionally considered are 2D solutions in the form of localized dark modes, which feature a confined spatial profile with a divergent integral norm. Although experimental realization of the scheme has not yet been reported, many possibilities of its use have been explored theoretically, as summarized in this chapter. In particular, an essential asset of the theoretical work in this direction is that, while it is chiefly based on numerical methods, many important results may be obtained in an analytical form, approximately or exactly. The chapter also reports a summary of results for the nonlnearity management, with the cubic nonlinearity periodically switching between self-attraction and repulsion.