Modulation instability threshold for the mode versus number of oscillators in two-dimensional array. The solid line is given by the corresponds to the estimate obtained from the nonlinear Schrödinger equation in large limit [see formula (43) ] and the diamonds are results of numerical simulations.
Time evolution of the local energy (44) . In panel (a), the horizontal axis indicates lattice sites and the vertical axis is time. The grey scale goes from (white) to the maximum -value (black). The lower rectangle corresponds to and the upper one to . (b), (c); and (d) show the instantaneous local energy along the chain at three different times. Remark the difference in vertical amplitude in panel (c), when the chaotic breather is present. The initial -mode amplitude is .
Panel (a) presents the evolution of of formula (45) for the one-dimensional FPU lattice with oscillators, initialized on the -mode with an amplitude . The dashed line indicates the equilibrium value . Panel (b) presents the corresponding finite time largest Lyapunov exponent. Panel (c) shows for the two-dimensional FPU lattice with oscillators, initialized on the -mode with an amplitude . Panel (d) presents the finite time largest Lyapunov exponent for two dimensions.
Local energy (46) surface plots for the two-dimensional FPU lattice with oscillators, initialized on the -mode with an amplitude . Snapshots at four different times are shown. Breathers form after a coalescence process similarly to the one-dimensional case. The mobility of the breathers is evident and one also observes in the last panel the final decrease.
Log–log plot of versus time for an FPU chain with free-ends boundary conditions. In this representation, an exponential is a straight line with slope one. Symbols are the results of numerical simulations averaged over 20 initial conditions. The dashed line is a plot of the theoretical result (47) (Ref. 33 ) for a harmonic chain. The arrow indicates the crossover time . Parameters are and .
Space–time contour plot of the site energies (44) . Time flows up and the horizontal axis is the site index. Parameters are , , and .
Surface plot of the time-dependent spatial spectrum of particle velocities for the one-dimensional FPU lattice. (a) Fixed-ends boundary conditions. (b) Free-ends boundary conditions. Parameters are , , and .
Damping rates for an harmonic chain with and . Free-ends boundary conditions (stars) and fixed-ends boundary conditions (diamonds).
2D FPU lattice, site energies in the residual state state. Parameters are: , , .
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