Loss of stability of a solitary wave through exciting a cnoidal wave on a Fermi-Pasta-Ulam ring

Abstract

The spatiotemporal propagation behavior of a solitary wave is investigated on a Fermi-Pasta-Ulam ring. We observe the emergence of a cnoidal wave excited by the solitary wave. The cnoidal wave may coexist with the solitary wave for a long time associated with the periodic exchange of energy between these two nonlinear waves. The module of the cnoidal wave, which is considered as an indicator of the nonlinearity, is found to oscillate with the same period of the energy exchange. After the stage of coexistence, the interaction between these two nonlinear waves leads to the destruction of the cnoidal wave by the radiation of phonons. Finally, the interaction of the solitary wave with phonons leads to the loss of stability of the solitary wave.

Transition from Solitons to Solitary Waves in the Fermi-Pasta-Ulam Lattice

“…This leads the deviation from the fitted curve of Eq. (17). So with the increase of energy, the nonlinear part in the interaction potential of Eq.…”

“…The velocity v and energy E of the solitary wave are calculated numerically: v = 1.000743, E = 0.044664, and v and E satisfy the relation of Eq. (17). Then the solitary wave may be a soliton solution.…”

“…If the energy E < 0.1, the width of solitary wave is large and the continuum approximation is applicable, so the relation is consistent with Eq. (17) [see the inset in Fig. 3 (b)] and the solitary wave is a soliton.…”

“…Previously in numerical work solitons are constructed by using analytical soliton solution directly [14] or starting with a special initial condition [4]. Here we obtain solitons in the FPU lattice for the first time with momentum excitation, which is the widely used method to excite solitary waves [12,13,15,16,17]. Soliton transforms to solitary wave when energy increases larger than the threshold.…”

Nonlinear ion-acoustic cnoidal waves in a dense relativistic degenerate magnetoplasma

“…Kartashov et al [22,23] reported, respectively, that in contradistinction with the case of localized solitons (where the spectrum of perturbations is discrete) for cnoidal waves, one has a band of possible increments at each energy flow, and, under the proper conditions of low-and high-energy flows, the two-dimensional cnoidal waves appear to be robust enough to be observable in experiments. Recently, Yuan et al [24] demonstrated that, due to the interaction of the cnoidal wave with the solitary wave, phonons can be radiated, which destroys the cnoidal wave and finally results in a loss of stability of the solitary wave. However, the basic features (amplitude and width) of cnoidal waves in a dense relativistic degenerate magnetoplasma consisting of relativistic degenerate electrons and nondegenerate cold ions is still lacking.…”