No abstract

Recent studies have demonstrated that defocusing cubic nonlinearity with local strength growing from the center to the periphery faster than r D , in space of dimension D with radial coordinate r, supports a vast variety of robust bright solitons. In the framework of the same model, but with a weaker spatial-growth rate, ∼ r α with α ≤ D, we here test the possibility to create stable localized continuous waves (LCWs) in one-and two-dimensional (1D and 2D) geometries, localized dark solitons (LDSs) in 1D, and localized dark vortices (LDVs) in 2D, which are all realized as loosely confined states with a divergent norm. Asymptotic tails of the solutions, which determine the divergence of the norm, are constructed in a universal analytical form by means of the Thomas-Fermi approximation (TFA). Global approximations for the LCWs, LDSs, and LDVs are constructed on the basis of interpolations between analytical approximations available far from (TFA) and close to the center. In particular, the interpolations for the 1D LDS, as well as for the 2D LDVs, are based on a "deformed-tanh" expression, which is suggested by the usual 1D dark-soliton solution. The analytical interpolations produce very accurate results, in comparison with numerical findings, for the 1D and 2D LCWs, 1D LDSs, and 2D LDVs with vorticity S = 1. In addition to the 1D fundamental LDSs with the single notch, and 2D vortices with S = 1, higher-order LDSs with multiple notches are found too, as well as double LDVs, with S = 2. Stability regions for the modes under the consideration are identified by means of systematic simulations, the LCWs being completely stable in 1D and 2D, as they are ground states in the corresponding settings. Basic evolution scenarios are identified for those vortices which are unstable. The settings considered in this work may be implemented in nonlinear optics and in Bose-Einstein condensates.

We present a systematic analysis of the self-similar propagation of optical pulses within the framework of the generalized cubic-quintic nonlinear Schrödinger equation with distributed coefficients. By appropriately choosing the relations between the distributed coefficients, we not only retrieve the exact self-similar solitonic solutions, but also find both the approximate self-similar Gaussian-Hermite solutions and compact solutions. Our analytical and numerical considerations reveal that proper choices of the distributed coefficients could make the unstable solitons stable and could restrict the nonlinear interaction between the neighboring solitons.

By using the lens-type transformation, exact soliton and quasi-soliton similaritons are found in (1+1), (2+1) and (3+1)-dimensional nonlinear Schrödinger equations in the context of nonlinear optical fiber amplifiers and graded-index waveguide amplifiers. The novel analytical and numerical results show that, in addition to the exact solitonic optical waves, quasi-solitonic optical waves with Gaussian, parabolic, vortex and ring soliton profiles can evolve exact self-similarly without any radiation.