Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExact solutions in the FPU oscillator chain
60
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
After a brief comprehensive review of old and new results on the well known Fermi-Pasta-Ulam (FPU) conservative system of $N$ nonlinearly coupled oscillators, we present a compact linear mode representation of the Hamiltonian of the FPU system with quartic nonlinearity and periodic boundary conditions, with explicitly computed mode coupling coefficients. The core of the paper is the proof of the existence of one-mode and two-mode exact solutions, physically representing nonlinear standing and travelling waves of small wavelength whose explicit lattice representations are obtained, and which are valid also as $N rightarrow infty$. Moreover, and more generally, we show the presence of multi-mode invariant submanifolds. Destabilization of these solutions by a parametric perturbation mechanism leads to the establishment of chaotic in time mode interaction channels, corresponding to the formation in phase space of bounded stochastic layers on submanifolds. The full mode-space stability problem of the $N/2$ zone-boundary mode is solved, showing that this mode becomes unstable through a mechanism of the modulational Benjamin-Feir type. In the thermodynamic limit the mode is always unstable but with instability growth rate linearly vanishing with energy density. The physical significance of these solutions and of their stability properties, with respect to the previously much more studied equipartition problem for long wavelength initial excitations, is briefly discussed.
rate research
Read More
We study the interaction of small amplitude, long wavelength solitary waves in the Fermi-Pasta-Ulam model with general nearest-neighbor interaction potential. We establish global-in-time existence and stability of counter-propagating solitary wave solutions. These solutions are close to the linear superposition of two solitary waves for large positive and negative values of time; for intemediate values of time these solutions describe the interaction of two counterpropagating pulses. These solutions are stable with respect to perturbations in $ell^2$ and asymptotically stable with respect to perturbations which decay exponentially at spatial $pm infty$.}
We study the exact solutions of quantum integrable model associated with the $C_n$ Lie algebra, with either a periodic or an open one with off-diagonal boundary reflections, by generalizing the nested off-diagonal Bethe ansatz method. Taking the $C_3$ as an example we demonstrate how the generalized method works. We give the fusion structures of the model and provide a way to close fusion processes. Based on the resulted operator product identities among fused transfer matrices and some necessary additional constraints such as asymptotic behaviors and relations at some special points, we obtain the eigenvalues of transfer matrices and parameterize them as homogeneous $T-Q$ relations in the periodic case or inhomogeneous ones in the open case. We also give the exact solutions of the $C_n$ model with an off-diagonal open boundary condition. The method and results in this paper can be generalized to other high rank integrable models associated with other Lie algebras.
We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. A first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: i) a Stochasticity Threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. ii) a Strong Stochasticity Threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weakly and strongly chaotic regimes. It is stable with N. A second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. The starting of this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, a third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.
Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrodinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU(1,1) group-related coherent states for the SO minimize the Robertson and Schrodinger relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.
We examine a class of exact solutions for the eigenvalues and eigenfunctions of a doubly anharmonic oscillator defined by the potential $V(x)=omega^2/2 x^2+lambda x^4/4+eta x^6/6$, $eta>0$. These solutions hold provided certain constraints on the coupling parameters $omega^2$, $lambda$ and $eta$ are satisfied.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
