Soliton localization in Bose–Einstein condensates with time-dependent harmonic potential and scattering length

Abstract

We derive exact solitonic solutions of a class of Gross-Pitaevskii equations with time-dependent harmonic trapping potential and interatomic interaction. We find families of exact single-solitonic, multi-solitonic, and solitary wave solutions. We show that, with the special case of an oscillating trapping potential and interatomic interaction, a soliton can be localized indefinitely at an arbitrary position. The localization is shown to be experimentally possible for sufficiently long time even with only an oscillating trapping potential and a constant interatomic interaction.

Binding energy of soliton molecules in time-dependent harmonic potential and nonlinear interaction

“…This corresponds, for instance, to a matter-wave soliton in a vibrating harmonic trapping potential, which we have studied previously [15]. Since γ (t) is bounded and g(t) is not, the energy of the soliton will be bounded only when the coefficient of g(t) vanishes, namely, v 1 + x 1γ (0) = 0.…”

“…The single-and two-soliton solutions, which are well known in the literature, may be derived using the inverse-scattering method [13]. For completeness and to establish the notation, we present them here in the suggestive form that we have published recently [14,15]. The single-soliton solution is given by…”

Soliton dynamics and interaction in the Bose–Einstein condensates with harmonic trapping potential and time-varying interatomic interaction

“…Besides the interatomic interaction, external trapping potential V ext is another factor affecting the macroscopic behaviors of the BECs [38][39][40][41][42]. It has been proposed that the external trapping potential has such forms as the optical lattice one, elliptic function one, harmonic one, and double well one [38][39][40][41].…”

“…It has been proposed that the external trapping potential has such forms as the optical lattice one, elliptic function one, harmonic one, and double well one [38][39][40][41]. Among those, the harmonic trapping potential in three dimensions, i.e., r = (x, y, z) in expression (1), has been given as [42] …”

“…Such harmonic trapping potential has been realized, for instant, the dipole trap formed by a strong off-resonant laser field [42]. When ω ⊥ ω x in expression (2), i.e., the confinement of the BECs in the y and z directions is much larger than that in the x direction, (1) could be considered to be effective only along the x direction [42]. In such case, the external harmonic trapping potential can be reduced to [42] …”