Effective equations for matter-wave gap solitons in higher-order transversal states

TL;DR

This paper derives effective 1D equations to describe complex 3D matter-wave gap solitons in Bose-Einstein condensates with higher-order transversal states, simplifying analysis and predicting properties accurately.

Contribution

It introduces a universal effective 1D model for 3D gap solitons with higher-order transversal states, validated against full 3D GPE solutions.

Findings

Effective 1D equations accurately approximate 3D solutions.

Model successfully predicts $(N)$ curves for different soliton families.

Validation shows strong agreement with numerical 3D GPE results.

Abstract

We demonstrate that an important class of nonlinear stationary solutions of the three-dimensional (3D) Gross-Pitaevskii equation (GPE) exhibiting nontrivial transversal configurations can be found and characterized in terms of an effective one-dimensional (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. 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 wave functions 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.