Solitons Riding on Solitons: Hypersolitons and the Quantum Newton's Cradle

Abstract

The reduced dynamics for dark and bright soliton chains in the one-dimensional nonlinear Schr\"odinger equation is used to study the behavior of collective compression waves. For appropriate conditions, the reduced dynamics derived from perturbation and variational techniques allows to describe a chain of dark or bright solitons as a chain of effective masses connected by nonlinear springs taking the form of a Toda lattice model on the soliton's positions. In turn, the Toda lattice possesses exact solitary travelling compression wave solutions corresponding to travelling compression waves in the original soliton chain. We coin the term hypersoliton to describe such solitary waves riding on a chain of solitons. We corroborate our analytical results with direct numerical simulations of the nonlinear Schr\"odinger equation. It is observed that in the case of dark soliton chains, the formulated reduction dynamics provides an accurate description for the robust evolution of travelling compression waves. In contrast, bright soliton chains do not have such stable propagating solutions due to the desynchronization of the mutual phases between consecutive solitons during evolution. Finally, as an application to Bose-Einstein condensates trapped in a standard external harmonic trapping potential, we study the case of finite dark soliton chain confined at the center of the trap. We find that when the central chain is hit by a dark soliton initiated at the edge of the external trap, the energy is transferred through the chain as a hypersoliton. When the hypersoliton reaches the end of the central chain, a dark soliton is ejected away from the center of the trap and, as it returns from its excursion up the trap, hits the central chain producing again a hypersoliton. This periodic evolution is the equivalent of the classical Newton's cradle.