Analytical calculation of the Peierls–Nabarro pinning barrier for one-dimensional parametric double-well models

Dikandé, Alain M.

doi:10.1016/s0375-9601(02)01386-5

Cited by 8 publications

(5 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…which is the φ 4 potential [2,18]. As established in previous works [21][22][23][24][25], none of the three PDWP models admits exact nonlinear solutions in the discrete regime. However, in the continuum limit, exact single and periodic kink solutions are accessible.…”

Section: Family Of Completely Integrable Discrete Static Hamiltoniansmentioning

confidence: 66%

“…Given that the last model reduces exactly to the φ 4 one for a particular value of its parameter, the resulting discrete integrable Hamiltonian provides a new rich family of spatial chaos-free 2D maps. However, besides the model studied in [21], there exist two other variants [21][22][23][24][25] that form with the first a family of non-integrable double-well potentials whose shapes are parameterized distinctively.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A class of conservative Hamiltonians with exactly integrable discrete two-dimensional parametric maps

Dikandé¹,

Njumbe²

2010

Phys. Scr.

Self Cite

View full text Add to dashboard Cite

A class of discrete conservative Hamiltonians with completely integrable two-dimensional (2D) mappings is constructed whose generic models are three families of non-integrable discrete Hamiltonians with on-site potentials whose double-well shapes vary. Unlike the discrete 2D mappings associated with the generic models, which all display pitchfork bifurcations towards randomly pinned states with chaotic features, for the derived models the pitchfork bifurcation leads to fixed points always surrounded by periodic trajectories. A nonlinear stability analysis reveals a finite crossover on the bifurcation line at which the pitchfork transition takes the maps from regular real periodic trajectories towards a regime dominated by a cluster of periodic point trajectories representing the allowed real solutions. The rich variety of structures displayed by the new class of discrete maps, combined with their complete integrability, offer rich perspectives for theoretical modelling of a wide class of systems undergoing structural instabilities without noticeable chaotic precursors.

show abstract

Section: Family Of Completely Integrable Discrete Static Hamiltoniansmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 99%

A class of conservative Hamiltonians with exactly integrable discrete two-dimensional parametric maps

Dikandé¹,

Njumbe²

2010

Phys. Scr.

Self Cite

View full text Add to dashboard Cite

show abstract

“…To effectively deal with a parametrized DW potential that can be reduced smoothly to the ϕ 4 model, in some previous works, we proposed a family of hyperbolic potentials whose DW shapes could be tuned distinctively, admitting the canonical ϕ 4 potential as a specific limit [31][32][33][34]. Peculiar features of some of these parametrized DW potentials (hereafter referred to as the Dikandé-Kofané double-well (DKDW) potentials) have been pointed out in recent studies [35][36][37][38][39].…”

Section: Introductionmentioning

confidence: 99%

Route to Chaos in Discrete Two-dimensional Maps with Hyperbolic Double-well Potentials

Dikandé¹

2022

Acta Phys. Pol. B

View full text Add to dashboard Cite

The texture of phase-space and bifurcation diagrams of two-dimensional discrete maps describing a lattice of interacting oscillators, confined in bistable potentials with deformable double-well shapes, are examined. There are considered two bistable potentials that belong to a family of hyperbolic nonlinear on-site potentials whose double-well shapes can be tuned differently: one has a variable barrier height and the other has variable minima positions. However, the two hyperbolic double-well potentials reduce to the well-known canonical ϕ 4 field, familiar in the studies of structural phase transitions in centro-symmetric crystals and bistable processes in biophysics. It is shown that although the parametric maps are area-preserving, their routes to chaos display different characteristic features: the first map exhibits a cascade of period-doubling bifurcations with respect to the potential amplitude, but period-halving bifurcations with respect to the shape deformability parameter. On the other hand, the first bifurcation of the second map always coincides with the first pitchfork bifurcation of the ϕ 4 map. However, an increase of the deformability parameter shrinks the region between successive period-doubling bifurcations. The two opposite bifurcation cascades characterizing the first map, and the shrinkage of regions between successive bifurcation cascades which is characteristic of the second map, suggest a non-universal character of the Feigenbaum-number sequences associated with the two discrete parametric double-well maps.

show abstract

“…To effectively deal with a parametrized DW potential that can be reduced smoothly to the φ 4 model, in some previous works we proposed a family of hyperbolic potentials whose DW shapes could be tuned distinctively, admitting the canonical φ 4 potential as a specific limit [31,32,33,34]. Peculiar features of some of these parametrized DW potentials (hereafter referred to as Dikandé-Kofané double-well (DKDW) potentials) have been pointed out in recent studies [35,36,37,38].…”

Section: Introductionmentioning

confidence: 99%

Route to chaos in two-dimensional discrete parametric maps with bistable potentials

Dikandé¹

2021

Preprint

View full text Add to dashboard Cite

The texture of phase space and bifurcation diagrams of two-dimensional discrete maps describing a lattice of interacting oscillators, confined in on-site potentials with deformable double-well shapes, are examined. The two double-well potentials considered belong to a family proposed by Dikandé and Kofané (A. M. Dikandé and T. C. Kofané, Solid State Commun. vol. 89, p. 559, 1994), whose shapes can be tuned distinctively: one has a variable barrier height and the other has variable minima positions. However the two parametrized double-well potentials reduce to the φ 4 substrate, familiar in the studies of structural phase transitions in centro-symmetric crystals or bistable processes in biophysics. It is shown that although the parametric maps are area preserving their routes to chaos display different characteristic features: the first map exhibits a cascade of period-doubling bifurcations with respect to the potential amplitude, but period-halving bifurcations with respect to the shape deformability parameter. On the other hand the first bifurcation of the second map always coincides with the first pitchfork bifurcation of the φ 4 map. However, an increase of the deformability parameter shrinks the region between successive perioddoubling bifurcations. The two opposite bifurcation cascades characterizing the first map, and the shrinkage of regions between successive bifurcation cascades which is characteristic of the second map, suggest a non-universal character of the Feigenbaumnumber sequences associate with the two discrete parametric double-well maps.

show abstract

Analytical calculation of the Peierls–Nabarro pinning barrier for one-dimensional parametric double-well models

Abstract: Lattice effects on the kink families of two models for one-dimensional nonlinear Klein-Gordon systems with double-well on-site potentials are considered. The analytical expression of the generalized Peierls-Nabarro pinning potential is obtained and confronted with numerical simulations.

Cited by 8 publications

References 24 publications

A class of conservative Hamiltonians with exactly integrable discrete two-dimensional parametric maps

A class of conservative Hamiltonians with exactly integrable discrete two-dimensional parametric maps

Route to Chaos in Discrete Two-dimensional Maps with Hyperbolic Double-well Potentials

Route to chaos in two-dimensional discrete parametric maps with bistable potentials

Contact Info

Product

Resources

About