Starting from the 3D Gross-Pitaevskii equation and using a variational approach, we derive an effective 1D wave-equation that describes the axial dynamics of a Bose condensate confined in an external potential with cylindrical symmetry. The trapping potential is harmonic in the transverse direction and generic in the axial one. Our equation, that is a time-dependent non-polynomial nonlinear Schr\"odinger equation (1D NPSE), can be used to model cigar-shaped condensates, whose dynamics is essentially 1D. We show that 1D NPSE gives much more accurate results than all other effective equations recently proposed. By using 1D NPSE we find analytical solutions for bright and dark solitons, which generalize the ones known in the literature. We deduce also an effective 2D non-polynomial Schr\"odinger equation (2D NPSE) that models disc-shaped Bose condensates confined in an external trap that is harmonic along the axial direction and generic in the transverse direction. In the limiting cases of weak and strong interaction, our approach gives rise to Schr\"odinger-like equations with different polynomial nonlinearities

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The formation of clusters in condition of thermodynamic equilibrium can be easily observed both in two and three dimensions. In two dimensions relevant cases include pattern formation in Langmuir monolayers and ferrofluids, while in three dimensions cluster phases have been observed in colloids and in protein solutions. We have analyzed the problem within the scenario of competing interactions: typically, a short-range attractive interaction against a long-range repulsive one. This simplified approach is suggested by the fact that the forces, governing self-organization, act on a length scale which is larger than the molecular size; as a consequence many specific details of the molecules of interest are not necessary for studying the general features of microphases. We have tackled the microphase formation by simulations in bidimensional fluids, exploiting the parallel tempering scheme. In particular, we have analyzed the density range in which the particles arrange in circular domains (droplets), and the temperature range in which the system goes from microphases to the homogeneous fluid phase. As the density increases, the droplet size increases as well, but above a certain density the morphology changes and stripes are formed. Moreover at low density, we observe the formation of a liquidlike phase of disordered droplets; at higher densities, instead, the droplets tend to arrange onto a triangular superlattice. Such a change affects the features of the static structure factor, which gives well defined signatures of the microphase morphology. In each case, the specific heat exhibits a peak close to the transition from microphases to the homogeneous fluid phase, which is due to the breaking up of the clusters. The saturation of the height of the specific heat peak, with the increasing of the system size, suggests the possibility of a Kosterlitz-Thouless transition.