Fermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap

We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schrödinger (NLS) equation with an isotropic harmonic oscillator potential. The dynamics in this regime is dominated by resonant interactions between quartets of linear normal modes, accurately captured by the corresponding resonant approximation. Within this approximation, we identify Fermi-Pasta-Ulam–like recurrence phenomena, whereby the normal-mode spectrum passes in close proximity of the initial configuration, and two-mode states with time-independent mode amplitude spectra that translate into long-lived breathers of the original NLS equation. We comment on possible implications of these findings for nonlinear optics and matter-wave dynamics in Bose-Einstein condensates.

Figure 1

Energies Eq. (4) corresponding to normal modes Eq. (6) labeled by quantum numbers $n$ and $m$. A few possible restrictions to smaller sets of modes, addressed within the resonant approximation in Sec. 3, are highlighted by colors. The rightmost highlighted diagonal line is the lowest Landau-level truncation which was used in Refs. [30, 31]. The vertical highlighted line represents the fixed-angular-momentum truncations employed in Ref. [32]. The remaining highlighted line represents a generic restriction to a “one-dimensional” subset of modes that play a central role here, in Secs. 4 and 5.

Figure 2

The upper plot: the evolution of $|{\alpha }_l|$. The second plot: the contribution to $N$ from modes with $l>1$, defined as per Eq. (27). The third plot: the position-space representation of ${\left|\psi \right|}^2$ at times highlighted by red dashed lines in the middle plot. The initial configuration consists of the two-mode initial data (26). The two modes in the initial state in these plots are chosen as $(n_0,m_0)=(4,-2)$ and $(n_1,m_1)=(9,-3)$ with initial values ${\alpha }_0=0.6$ and ${\alpha }_1=0.8$, such that Eq. (24) gives $N=1$. The lower plot: the maximum and minimum values of $\mathrm{\Delta }\left(\tau \right)$ over a succession of 50 direct-reverse cascades (with the first direct cascade excluded) for the restriction given in the upper plots but different distributions of the initial energy between modes 0 and 1 (with $N=1$).

Figure 3

The spatial configuration of ${\left|\psi \right|}^2$ in the stationary solutions given by Eq. (36) [rows from 1 to 3, (a)] and Eq. (37) [the lower row, (b)]. Each plot specifies two excited modes $(n_0,m_0)$ and $(n_1,m_1)$, as indicated by white numbers.

Figure 4

Upper row: The evolution within different mode restrictions (specified in each plot) that exhibit accurate energy returns for some two-mode initial data; from left to right $({\alpha }_0,{\alpha }_1)=(0.6,0.8)$, $({\alpha }_0,{\alpha }_1)=(0.7,0.7)$, and $({\alpha }_0,{\alpha }_1)=(0.8,0.6)$. Middle row: The evolution under a specific mode restriction for different initial distributions of energy, from left to right $({\alpha }_0,{\alpha }_1)=(0.4,0.92)$, $({\alpha }_0,{\alpha }_1)=(0.7,0.7)$, and $({\alpha }_0,{\alpha }_1)=(0.8,0.6)$. Even when some initial configurations of the two lowest modes display accurate energy returns, for others such behavior may be much less pronounced. Lower row: This mode restriction demonstrates a wider range of behaviors than what we have discussed in the main presentation. In the first plot the two-mode initial data $({\alpha }_0,{\alpha }_1)=(0.8,0.6)$ results in accurate FPU returns. The second plot shows that the single mode 1 is unstable because small perturbations like $({\alpha }_0,{\alpha }_1)=(\delta ,1)$ with $\delta \ll 1$ drive the system away from this configuration; however, even if the energy does not accurately return close to the initial distribution we observe that there are accurate returns around a new reference configuration. Finally, the third plot shows that three-mode initial data, specifically $({\alpha }_0,{\alpha }_1,{\alpha }_2)=(0.15,0.7,0.15)$, can exhibit accurate FPU returns. Mode labeling in the plots is identical to that for the first plot shown.