Solitons riding on solitons and the quantum Newton's cradle

The reduced dynamics for dark and bright soliton chains in the one-dimensional nonlinear Schrödinger equation is used to study the behavior of collective compression waves corresponding to Toda lattice solitons. We coin the term hypersoliton to describe such solitary waves riding on a chain of solitons. It is observed that in the case of dark soliton chains, the formulated reduction dynamics provides an accurate an robust evolution of traveling hypersolitons. As an application to Bose-Einstein condensates trapped in a standard harmonic potential, we study the case of a finite dark soliton chain confined at the center of the trap. When the central chain is hit by a dark soliton, the energy is transferred through the chain as a hypersoliton that, in turn, ejects a dark soliton on the other end of the chain that, as it returns from its excursion up the trap, hits the central chain repeating the process. This periodic evolution is an analog of the classical Newton's cradle.

Figure 1

Dynamics of a slightly perturbed equidistant chain of 12 BSs of amplitude $a_i=1$ in the periodic domain $x\in [-54,54]$, namely with a separation of $r_0=9$. The top-left and top-right panels depict, respectively, the spatiotemporal evolution of the (square root of the) density $\left|u\right(x,t\left)\right|$ for the IP and OOP initial chains. The initial conditions are set as the numerically exact steady states found by Newton iterations and then the BSs positions are perturbed with randomly distributed displacements between $-r_0/18$ and $r_0/18$. The dashed lines represents the orbits obtained from the corresponding reduced Toda lattice model (9) (for the IP case this model can only be integrated until the first collision time for $t\lesssim 50$). The bottom two panels depict the phase difference between consecutive density maxima. These correspond to the relative phases $\mathrm{\Delta }\varphi$ between consecutive BSs (top and bottom subpanels corresponding, respectively, to the IP and OOP cases).

Figure 2

BdG stability spectrum for the steady state for a chain of 10 equidistant IP (top left) and OOP (top right) BSs in the periodic domain $x\in [-35,35]$. The bottom set of panels depict, from top to bottom, the steady state and the first nine most unstable eigenfunctions where the real part is depicted by the blue (dark) solid line and the imaginary part by the red (gray) dashed line.

Figure 3

Interaction dynamics for 12 DSs initialized at random positions with random velocities in a periodic domain. The top panel depicts the spatiotemporal evolution of the (square root of the) density. The dashed lines represent the orbits obtained from the corresponding reduced Toda lattice model (12). The middle panel depicts the DS orbits for the original GP model (lines) and the orbits produced by the reduced Toda lattice model (small dots). The DS orbits for the GP model were extracted using a local (piecewise) fitting procedure to the DS ansatz of Eq. (3). The bottom panel depicts the difference between the GP and the Toda lattice orbits.

Figure 4

Toda lattice soliton riding on a chain of $N=10$ BSs in the periodic domain $x\in [-43,43]$. The top panels depict the initial conditions in terms of the relative displacements from equilibrium $r_n(t=0)$ (top left) and their corresponding velocities ${\dot{r}}_n(t=0)$ (top right). The bottom-left and bottom-right panels depict the evolution of the (square root of the) density of the original GP model initialized with the Toda lattice soliton for, respectively, a total time equivalent to two and ten cycles of the Toda lattice soliton around the periodic domain. The dashed lines correspond to the reduced ODE model of the Toda lattice (9).

Figure 5

Toda lattice soliton riding in a chain of $N=12$ DSs on a periodic domain. The top two rows of panels depict the initial conditions in terms of the relative displacement from equilibrium of the mutual distances $r_n\left(0\right)$ (top panels) and their corresponding velocities (bottom panels). The middle and bottom rows of panels depict, respectively, the evolution of the (square root of the) density $\left|u\right(x,t\left)\right|$ and its time derivative for the original GP model initialized with the Toda lattice soliton. The left, middle, and right columns correspond to increasingly large Toda lattice velocities for a background density $n_0=1$ and, from left to right, $r_0=6, r_0=5$, and $r_0=4$, which correspond, respectively, to $c=0.0180, c=0.0489$, and $c=0.1329$. The dashed lines correspond to the reduced ODE model of the Toda lattice (12).

Figure 6

Stability of the hypersoliton. The layout of the different panels is the same as in Fig. 5. The initial condition corresponds to the case of the middle column of panels of Fig. 5 ($r_0=5$) with random perturbations added to the initial condition. The initial condition is perturbed by adding a random perturbation of size, from left to right, 30%, 50%, and 80% on the initial separations and velocities of the DSs in the chain.

Figure 7

Hypersoliton collisions. The left, middle, and right columns depict, respectively, the collision for (a) a head-on collision with equal but opposite velocities, (b) a head-on collision with different speeds, and (c) a chasing collision where a faster hypersoliton (seeded on the left) chases a slower hypersoliton (seeded on the right) until they collide in a nonlinear fashion. All panels have the same meaning and layout as previous figures.

Figure 8

Hypersoliton Newton's cradle. A DS soliton (leftmost one) is released at various distances away from a stationary lattice of 12 DSs placed at the center of the parabolic trapping potential (4) with $\mathrm{\Omega }=0.01$. The DS is released at a distance equivalent to (a) $3r_0$ (left column of panels), (b) $12r_0$ (middle column of panels), and (c) $20r_0$ (right column of panels), where $r_0$ is the separation between the central DSs. All panels have the same meaning and layout as previous figures.

Figure 9

Schematic of the mechanism for the loss of the hypersoliton Newton's cradle through the synchronization of the outer DS [see red (gray) circle] with the central DS chain [see black circles]. (a) Release of a DS from the left with a stationary DS chain in the central region. (b) After the hypersoliton travels through the chain, the rightmost DS is ejected to the right. The remaining central DS chain is now asymmetric with respect to the center (see vertical dashed line) and starts moving to the right. (c) Both outer DS and central DS chains perform half an oscillation after their respective excursion up the trap. (d) After the DS collides and transfers its energy (through a propagating hypersoliton through the central DS chain) the leftmost DS is ejected. The central DS chain still has the extra energy of moving towards the left. (e) Both the DS and the central chain now oscillate in the same direction (although slightly OOP). This process continues until the energy of the outer DS is completely transferred to the central chain and all that remains is a central chain undergoing left-to-right oscillations in the IP normal mode.

Figure 10

Long term evolution of the hypersoliton Newton's cradle corresponding to the case depicted in the middle column of Fig. 8. Notice the beating of energy exchange between the outer DS and the central chain. All panels have the same meaning and layout as previous figures.