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Register a new userAsymptotic two-soliton solutions in the Fermi-Pasta-Ulam model
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We prove the existence of asymptotic two-soliton states in the Fermi-Pasta-Ulam model with general interaction potential. That is, we exhibit solutions whose difference in $ell^2$ from the linear superposition of two solitary waves goes to zero as time goes to infinity.
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We perform a thorough investigation of the first FPUT recurrence in the $beta$-FPUT chain for both positive and negative $beta$. We show numerically that the rescaled FPUT recurrence time $T_{r}=t_{r}/(N+1)^{3}$ depends, for large $N$, only on the parameter $Sequiv Ebeta(N+1)$. 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.
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.
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 modulation. Such instability has special names in various branches of physics and is generally known as modulation instability (MI). The MI is tightly related to Fermi-Pasta-Ulam (FPU) recurrence since breather solutions of the nonlinear Schrodinger equation (NLSE) are known to accurately describe growth and decay of modulationally unstable waves in conservative systems. Here, we report theoretical, numerical and experimental evidence of the effect of dissipation on FPU cycles in a super wave tank, namely their shift in a determined order. In showing that ideal NLSE breather solutions can describe such dissipative nonlinear dynamics, our results may impact the interpretation of a wide range of new physics scenarios.
The lifetimes of localized nonlinear modes in both the $beta$-Fermi-Pasta-Ulam-Tsingou ($beta$-FPUT) chain and a cubic $beta$-FPUT lattice are studied as functions of perturbation amplitude, and by extension, the relative strength of the nonlinear interactions compared to the linear part. We first recover the well known result that localized nonlinear excitations (LNEs) produced by a bond squeeze can be reduced to an approximate two-frequency solution and then show that the nonlinear term in the potential can lead to the production of secondary frequencies within the phonon band. This can affect the stability and lifetime of the LNE by facilitating interactions between the LNE and a low energy acoustic background which can be regarded as noise in the system. In the one dimensional FPUT chain, the LNE is stabilized by low energy acoustic emissions at early times; in some cases allowing for lifetimes several orders of magnitude larger than the oscillation period. The longest lived LNEs are found to satisfy the parameter dependence $mathcal{A}sqrt{beta}approx1.1$ where $beta$ is the relative nonlinear strength and $mathcal{A}$ is the displacement amplitude of the center particles in the LNE. In the cubic FPUT lattice, the LNE lifetime $T$ decreases rapidly with increasing amplitude $mathcal{A}$ and is well described by the double log relationship $log_{10}log_{10}(T)approx -(0.15pm0.01)mathcal{A}sqrt{beta}+(0.62pm0.02)$.
We investigate numerically the existence and stability of higher-order recurrences (HoRs), including super-recurrences, super-super-recurrences, etc., in the alpha and beta Fermi-Pasta-Ulam-Tsingou (FPUT) lattices for initial conditions in the fundamental normal mode. Our results represent a considerable extension of the pioneering work of Tuck and Menzel on super-recurrences. For fixed lattice sizes, we observe and study apparent singularities in the periods of these HoRs, speculated to be caused by nonlinear resonances. Interestingly, these singularities depend very sensitively on the initial energy and the respective nonlinear parameters. Furthermore, we compare the mechanisms by which the super-recurrences in the two models breakdown as the initial energy and respective nonlinear parameters are increased. The breakdown of super-recurrences in the beta-FPUT lattice is associated with the destruction of the so-called metastable state and hence is associated with relaxation towards equilibrium. For the alpha-FPUT lattice, we find this is not the case and show that the super-recurrences break down while the lattice is still metastable. We close with comments on the generality of our results for different lattice sizes.
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