RESUMO
It was recently found [Fujioka et al., Phys. Lett. A 374, 1126 (2010)] that the propagation of solitary waves can be described by a fractional extension of the nonlinear Schrödinger (NLS) equation which involves a temporal fractional derivative (TFD) of order α > 2. In the present paper, we show that there is also another fractional extension of the NLS equation which contains a TFD with α < 2, and in this case, the new equation describes the propagation of radiating solitons. We show that the emission of the radiation (when α < 2) is explained by resonances at various frequencies between the pulses and the linear modes of the system. It is found that the new fractional NLS equation can be derived from a suitable Lagrangian density, and a fractional Noether's theorem can be applied to it, thus predicting the conservation of the Hamiltonian, momentum and energy.
RESUMO
Extending the recent work on models with spatially nonuniform nonlinearities, we study bright solitons generated by the nonpolynomial self-defocusing (SDF) nonlinearity in the framework of the one-dimensional (1D) Muñoz-Mateo-Delgado (MM-D) equation (the 1D reduction of the Gross-Pitaevskii equation with the SDF nonlinearity), with the local strength of the nonlinearity growing at |x|â∞ faster than |x|. We produce numerical solutions and analytical ones, obtained by means of the Thomas-Fermi approximation, for nodeless ground states and for excited modes with one, two, three and four nodes, in two versions of the model, with steep (exponential) and mild (algebraic) nonlinear-modulation profiles. In both cases, the ground states and the single-node ones are completely stable, while the stability of the higher-order modes depends on their norm (in the case of the algebraic modulation, they are fully unstable). Unstable states spontaneously evolve into their stable lower-order counterparts.
RESUMO
We consider solitons in a system of linearly coupled Korteweg-de Vries (KdV) equations, which model two-layer settings in various physical media. We demonstrate that traveling symmetric solitons with identical components are stable at velocities lower than a certain threshold value. Above the threshold, which is found exactly, the symmetric modes are unstable against spontaneous symmetry breaking, which gives rise to stable asymmetric solitons. The shape of the asymmetric solitons is found by means of a variational approximation and in the numerical form. Simulations of the evolution of an unstable symmetric soliton sometimes produce its breakup into two different asymmetric modes. Collisions between moving stable solitons, symmetric and asymmetric ones, are studied numerically, featuring noteworthy features. In particular, collisions between asymmetric solitons with identical polarities are always elastic, while in the case of opposite polarities the collision leads to a switch of the polarities of both solitons. Three-soliton collisions are studied too, featuring quite complex interaction scenarios.
RESUMO
A model for a non-Kerr cylindrical nematic fiber is presented. We use the multiple scales method to show the possibility of constructing different kinds of wave packets of transverse magnetic modes propagating through the fiber. This procedure allows us to generate different hierarchies of nonlinear partial differential equations which describe the propagation of optical pulses along the fiber. We go beyond the usual weakly nonlinear limit of a Kerr medium and derive a complex modified Korteweg-de Vries equation (CM KdV) which governs the dynamics for the amplitude of the wave packet. In this derivation the dispersion, self-focussing, and diffraction in the nematic fiber are taken into account. It is shown that this CM KdV equation has two-parameter families of bright and dark complex solitons. We show analytically that under certain conditions, the bright solitons are actually double-embedded solitons. We explain why these solitons do not radiate at all, even though their wave numbers are contained in the linear spectrum of the system. We study (numerically and analytically) the stability of these solitons. Our results show that these embedded solitons are stable solutions, which is an interesting property since in most systems the embedded solitons are weakly unstable solutions. Finally, we close the paper by making comments on the advantages as well as the limitations of our approach, and on further generalizations of the model and method presented.