Determination of the onset of spatiotemporal complexity induced by soliton competition in a perturbed sine-Gordon chain

Ariyasu, J.; Bishop, A. R.

doi:10.1103/physreva.39.6409

Cited by 15 publications

(15 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Here, sn(u; m) is the Jacobian elliptic function (JEF) of parameter m (K(m) is the complete elliptic integral of the first kind) used solely as an example to illustrate the reshaping-induced order-chaos route. When m = 0, then sn½4Kðm ¼ 0Þt=T ; m ¼ 0 ¼ sinð2pt=T Þ, i.e., one recovers the previously studied case of harmonic excitation [4]. This is relevant in comparing the structural stability of the sG system when solely the excitation shape is varied.…”

Section: Introductionmentioning

confidence: 83%

“…Following Ref. [4], one assumes the ansatz that the solution of the perturbed sine-Gordon equation has the same form as the unperturbed breather (2), but now allowing h and v 0 to be temporal functions…”

Section: Collective Coordinatesmentioning

confidence: 99%

“…The organization of the paper is as follows. In Section 2, a collective coordinate approach [4] is used to obtain an effective equation of motion for the collective variables. In Section 3, Melnikov's method is applied to the resulting effective equation of motion to estimate the parameter-space regions of the elliptic ac force where the onset of chaotic instabilities is expected.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Reshaping-induced spatiotemporal chaos in driven, damped sine-Gordon systems

Chacón

2007

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

Spatiotemporal chaos arising from the competition between sine-Gordon-breather and kink-antikink-pair solitons by reshaping an ac force is demonstrated. After introducing soliton collective coordinates, MelnikovÕs method is applied to the resulting effective equation of motion to estimate the parameter-space regions of the ac force where homoclinic bifurcations are induced. The analysis reveals that the chaos-order threshold exhibits sensitivity to small changes in the force shape. Computer simulations of the sine-Gordon system show good agreement with these theoretical predictions.

show abstract

Section: Introductionmentioning

confidence: 83%

Section: Collective Coordinatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Reshaping-induced spatiotemporal chaos in driven, damped sine-Gordon systems

Chacón

2007

Chaos, Solitons & Fractals

View full text Add to dashboard Cite

show abstract

“…This method is based on the expansion of the solution of the perturbed problem in the complete set of eigenfunctions of the Sturm-Liouville problem associated with the linearized partial differential equation (PDE) around a kink solution. Later, the dynamic of the breather and kink-antikink solution in the perturbed sG equation was also considered using a new perturbative analysis [5,7,8], where the radiation field was also included [5,7]. All these methods involve cumbersome calculations, even more so if we extend them to the φ 4 equation where the internal mode is present.…”

Section: Introductionmentioning

confidence: 99%

Lagrangian formalism in perturbed nonlinear Klein–Gordon equations

Quintero¹,

Zamora-Sillero²

2004

Physica D: Nonlinear Phenomena

View full text Add to dashboard Cite

We develop an alternative approach to study the effect of the generic perturbation (in addition to explicitly considering the loss term) in the nonlinear Klein-Gordon equations. By a change of the variables that cancel the dissipation term we are able to write the Lagrangian density and then, calculate the Lagrangian as a function of collective variables. We use the Lagrangian formalism together with the Rice Ansatz to derive the equations of motion of the collective coordinates (CCs) for the perturbed sine-Gordon (sG) and φ 4 systems. For the N collective coordinates, regardless of the Ansatz used, we show that, for the nonlinear Klein-Gordon equations, this approach is equivalent to the Generalized Traveling Wave Ansatz (GTWA).

show abstract

“…In the work of Refs. [4,5] it has been demonstrated that a damped SG chain with additive dc driving also exhibits length-scale competition.…”

Section: Introductionmentioning

confidence: 99%

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

1993

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Determination of the onset of spatiotemporal complexity induced by soliton competition in a perturbed sine-Gordon chain

Cited by 15 publications

References 8 publications

Reshaping-induced spatiotemporal chaos in driven, damped sine-Gordon systems

Reshaping-induced spatiotemporal chaos in driven, damped sine-Gordon systems

Lagrangian formalism in perturbed nonlinear Klein–Gordon equations

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

Contact Info

Product

Resources

About