Length-scale competition in the damped sine-Gordon chain with spatiotemporal periodic driving

Abstract

It is shown that there are two different regimes for the damped sine-Gordon chain driven by the spatiotemporal periodic force rsin(wt -knx) with a flat initial condition. For w > kn, the system first bifurcates at a critical re (n) to a translating two-breather excitation from a state locked to the driver. For w < kn, the excitations of the system are the locked states with the phase velocity w/kn in all the regions of r studied. In the first regime, the frequency of the breathers is controlled by w, and the velocity of the breathers, controlled by kn, is shown to be the group velocity determined from the linear dispersion relation for the sine-Gordon equation. A linear stability analysis reveals that, in addition to two competing length scales, namely, the width of the breathers and the spatial period of the driving, there is one more length scale which plays an important role in controlling the dynamics of the system at small driving. In the second regime the length scale k n controls the excitation. The above picture is further corroborated by numerical nonlinear spectral analysis. An energy-balance estimate is also presented and shown to predict the critical value of r in good agreement with the numerical simulations.

Periodic and hyperbolic soliton solutions of a number of nonlocal nonlinear equations

“…Remarkably, unlike the usual focusing AL case [13], the same nonlocal, AL Eq. (217) (which admits cn and dn solutions), also admits the periodic sn solution…”

ac driven sine-Gordon solitons: dynamics and stability

“…When using the sG equation as a model for an actual physical situation, it is often necessary to account for factors that cause deviation from the perfect system, arising, for example, from forces acting on it, thermal effects, fluc-tuations, dissipation, or spatial modulations (deterministic or random). To account for some or all of those, appropriately chosen perturbing terms have to be included in the sG equation [3,4]; to quote a few instances of such perturbations, (see also the next paragraph) let us mention the studies of forces acting over a DNA molecule, or long DNA fragments containing regions of finite size and specific structure [21], additive and multiplicative noise sources [22], spatially periodic parametric potential [23] or damping with spatiotemporal periodic driving [24]. The work reported on here belongs to this class of problems; specifically, it focuses on the action of ac (sinusoidal in time, homogeneous in space) forces on the solitonic solutions, kinks and breathers, of the sG equation.…”

Superposition of elliptic functions as solutions for a large number of nonlinear equations

“…It is well known that the celebrated Ablowitz-Ladik (AL) model [10,11], which is an integrable model, admits moving dn and cn periodic solutions [24]. We now show that the same model also admits linearly superposed moving periodic solutions.…”