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تسجيل مستخدم جديدFermi-Pasta-Ulam phenomena and persistent breathers in the harmonic trap
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We consider the long-term weakly nonlinear evolution governed by the two-dimensional nonlinear Schr{o}dinger (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 Hamiltonian system. In the framework of this system, 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.
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The effect of discrete breathers (DBs) on macroscopic properties of the Fermi-Pasta-Ulam chain with symmetric and asymmetric potentials is investigated. The total to kinetic energy ratio (related to specific heat), stress (related to thermal expansio
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rameter $Sequiv Ebeta(N+1)$. Our numerics also reveal that for small $left|Sright|$, $T_{r}$ is linear in $S$ with positive slope for both positive and negative $beta$. For large $left|Sright|$, $T_{r}$ is proportional to $left|Sright|^{-1/2}$ for both positive and negative $beta$ but with different multiplicative constants. In the continuum limit, the $beta$-FPUT chain approaches the modified Korteweg-de Vries (mKdV) equation, which we investigate numerically to better understand the FPUT recurrences on the lattice. In the continuum, the recurrence time closely follows the $|S|^{-1/2}$ scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the $alpha$ chain. The difference in the multiplicative factors between positive and negative $beta$ arises from soliton-kink interactions which exist only in the negative $beta$ case. We complement our numerical results with analytical considerations in the nearly linear regime (small $left|Sright|$) and in the highly nonlinear regime (large $left|Sright|$). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for $T_{r}$ which depends only on $S$. In the latter regime, we show that $T_{r}proptoleft| Sright|^{-1/2}$ is predicted by the soliton theory in the continuum limit. We end by discussing the striking differences in the amount of energy mixing as well as the existence of the FPUT recurrences between positive and negative $beta$ and offer some remarks on the thermodynamic limit.
Instabilities are common phenomena frequently observed in nature, sometimes leading to unexpected catastrophes and disasters in seemingly normal conditions. The simplest form of instability in a distributed system is its response to a harmonic modula
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