RESUMO
A nonlinear Schrödinger equation that includes two terms with power-law nonlinearity and external potential modulated both on time and on the spatial coordinates is considered. The model appears in various branches of contemporary physics, especially in the case of lower values of the nonlinearity power. A significant generalization of the similarity transformations approach to construct explicit localized solutions for the model with arbitrary power-law nonlinearities is introduced. We obtain the exact analytical bright and kink soliton solutions of the governing equation for different nonlinearities and potentials that are of particular interest in applications to Bose-Einstein condensates and nonlinear optics. Necessary conditions on the physical parameters for propagating envelope formation are presented. The obtained results can be straightforwardly applied to a large variety of nonlinear Schrödinger models and hence would be of value to understand nonlinear phenomena in a diversity of nonlinear media.
RESUMO
We observe that the fully nonlinear evolution equations of Rosenau and Hymann, often abbreviated as K(n,m) equations, can be reduced to Hamiltonian form only on a zero-energy hypersurface belonging to some potential function associated with the equations. We treat the resulting Hamiltonian equations by the dynamical systems theory and present a phase-space analysis of their stable points. The results of our study demonstrate that the equations can, in general, support both compacton and soliton solutions. For the K(2,2) and K(3,3) cases one type of solutions can be obtained from the other by continuously varying a parameter of the equations. This is not true for the K(3,2) equation for which the parameter can take only negative values. The K(2,3) equation does not have any stable point and, in the language of mechanics, represents a particle moving with constant acceleration.
