Three-dimensional Gross–Pitaevskii solitary waves in optical lattices: Stabilization using the artificial quartic kinetic energy induced by lattice shaking

•We propose a way to create mobile, self-supporting, 3D solitary matter waves.•The stability is achieved by changing the dispersion law to quadratic.•The change in the dispersion law is achieved by shaking a lattice.•We outline an experimental scheme for realizing such objects.•Such objects could be very useful for building atomic Sagnac interferometers.

In this Letter, we show that a three-dimensional Bose–Einstein solitary wave can become stable if the dispersion law is changed from quadratic to quartic. We suggest a way to realize the quartic dispersion, using shaken optical lattices. Estimates show that the resulting solitary waves can occupy as little as ∼1/20∼1/20-th of the Brillouin zone in each of the three directions and contain as many as N=103N=103 atoms, thus representing a fully mobile macroscopic three-dimensional object.