## Effective equations for matter-wave gap solitons in higher-order transversal states    [PDF]

We demonstrate that an important class of nonlinear stationary solutions of the 3D Gross-Pitaevskii equation (GPE) exhibiting nontrivial transversal configurations can be found and characterized in terms of an effective 1D model. Using a variational approach we derive effective equations of lower dimensionality for BECs in $(m,n_{r})$-transversal-states (states featuring a central vortex of charge $m$ as well as $n_{r}$ concentric zero-density rings at every $z$ plane) which provides us with a good approximate solution of the original 3D problem. Since the specifics of the transversal dynamics can be absorbed in the renormalization of a couple of parameters, the functional form of the equations obtained is universal. This permits the study of systems of increasing complexity with the same computational effort. The model proposed finds its principal application in the study of the existence and classification of 3D gap solitons supported by 1D optical lattices, where in addition to providing a good estimate for the 3D wavefunctions it is able to make very good predictions for the $\mu(N)$ curves characterizing the different fundamental families. We have corroborated the validity of our model by comparing its predictions with those from the exact numerical solution of the full 3D GPE.