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

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 μ(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.

Three-dimensional hybrid vortex solitons

“…Dipole modes have been previously studied in diverse 1D and 2D settings [38], including vortex dipoles created in a common plane [31][32][33]. In 3D, such dipole structures can be approximately described by assuming that a quasi-1D dark soliton is embedded into an originally symmetric 3D mode around its midplane (z = 0), as suggested in a different context in [6]. In particular, for the fundamental states (S = 0) approximated by the TFA expression (6), the respective antisymmetric solution can be easily found from equation (2), assuming that the width of the dark soliton in the z direction is much smaller than the intrinsic scale of the TFA mode, i.e., μ is large enough: For the vortex states, a similar approximation is available too, but its applicability condition does not hold around the inner hole of the vortex.…”

“…Self-trapping of three-dimensional (3D) confined modes (solitons or, more properly, solitary waves) in optics [1][2][3], Bose-Einstein condensates (BECs) [4][5][6], ferromagnetic media [7], superconductors [8], semiconductors [9], baryonic matter [10], and general field theory [11,12] is a fundamental problem of nonlinear physics. An apparent condition is that an attractive, or self-focusing, nonlinearity is necessary for the creation of localized states; however, the attractive cubic nonlinearity simultaneously gives rise to collapse [13] of localized modes in higher-dimensional settings and, additionally, to strong azimuthal modulational instability of states with intrinsic vorticity [14], thus making the search for stable 3D fundamental and topological solitons in materials with the cubic (Kerr) nonlinearity a challenging issue.…”

Accurate one-dimensional effective description of realistic matter-wave gap solitons

“…Gap solitons with (m, n r ) = (0, 0) have been rarely considered in the literature [57,58]. They feature a nontrivial radial topology and require a more elaborate treatment [58]. The inset in Fig.…”

Vortex lines attached to dark solitons in Bose-Einstein condensates and boson-vortex duality in 3+1 dimensions

“…In spite of the fact that there have been an intensive research about vortices and solitons in scalar BECs, as far as we know, the study of topological structures showing junctions between them has not been performed. Only a particular configuration of this type has been addressed in the context of an effective low-dimensional model in trapped condensates [17]. Note that in a trapped BEC without dark solitons stable vortex lines which do not form loops have to end at the condensate boundary.…”