Soliton Model

Solitons A mathematically appealing model of real particles is that of solitons. It is known that in a dispersive linear medium, a general wave form changes its shape as it moves. In a nonlinear system, however, shape-preserving solitary solutions exist. 

Monte Carlo simulations [17, 18], the valence bond approach [19, 20], and g-ology [21-24] indicate that the Peierls instability in half-filled chains survives the presence of electron-electron interactions (at least, for some range of interaction parameters). This holds for a variety of different models, such as the Peierls-Hubbard model with the onsite Coulomb repulsion, or the Pariser-Parr-Pople model, where also long-range Coulomb interactions are taken into account ]2]. As the dimerization persists in the presence of electron-electron interactions, also the soliton concept survives. An important difference with the SSH model is that neu-... 

In a seminal paper [4], Su, Schrieffer, and Heeger discussed the energetics of solitons using a simple model of free electrons interacting with the lattice described by the Hamiltonian ... 

In Ref. [4], the soliton lattice configuration and energy within the SSH model were found numerically. Analytical expressions for these quantities can be obtained in the weak-coupling limit, when the gap 2A() is much smaller than the width of the re-electron band 4/0. At this point it is useful to define the lattice correlation length ... 

In this contribution, we review our recent work on disordered quasi-one-dimen-sional Peierls systems. In Section 3-2, we introduce the basic models and concepts. In Section 3-3, we discuss the localized electron stales in the FGM, while, in Section 3-4, we allow for lattice relaxation, leading to disorder-induced solitons. Finally, Section 3-5 contains the concluding remarks. 

The electronic band structure of a neutral polyacetylene is characterized by an empty band gap, like in other intrinsic semiconductors. Defect sites (solitons, polarons, bipolarons) can be regarded as electronic states within the band gap. The conduction in low-doped poly acetylene is attributed mainly to the transport of solitons within and between chains, as described by the intersoliton-hopping model (IHM) . Polarons and bipolarons are important charge carriers at higher doping levels and with polymers other than polyacetylene. 

Whereas the profile in linear wave equations is usually arbitrary it is important to note that a nonlinear equation will normally describe a restricted class of profiles which ensure persistence of solitons as t — oo. Any theory of ordered structures starts from the assumption that there exist localized states of nonlinear fields and that these states are stable and robust. A one-dimensional soliton is an example of such a stable structure. Rather than identify elementary particles with simple wave packets, a much better assumption is therefore to regard them as solitons. Although no general formulations of stable two or higher dimensional soliton solutions in non-linear field models are known at present, the conceptual construct is sufficiently well founded to anticipate the future development of standing-wave soliton models of elementary particles. 

Damgov, V. N, Trenchev PI. and Sheiretsky K. Oscillator-Wave Model Properties and Heuristic Instances. Chaos, Solitons and Fractals, Oxford, Vol. 17 (2003), P. 41... 

Fig. 27. Single soliton solution of the ID Frank-van der Merve model of incommensurate...

Fig. 28. Soliton lattice solution of the FvdM model with regularly spaced, distance 1, domain walls. The dashed line corresponds to the incommensurate phase with negligible potential modulation (K = 0). The plateaus represent the commensurate domains.

M. Berezowski. Effect of delay time on the generation of chaos in continuous systems. One-dimensional model. Two-dimensional model - tubular chemical reactor with recycle. Chaos, Solitons Fractals, 12(l) 83-89, 2001. 

This is accompanied by a structural distortion and thus, according to the above model, to an electronic state which, due to the symmetry of the electronic distribution lies precisely at midgap. This single state, having half occupied levels in the gap, has been termed native solitons. 

The result (173) applies to a photon model with the angular momentum h/2n of a boson, whereas the photon radius r would become half as large for the angular momentum h/4 of a fermion. Moreover, the present analysis on superposition of EMS normal modes is applicable not only to narrow linewidth wavepackets but also to a structure of short pulses and soliton-like waves. In these latter cases the radius in Eq. (173) is expected to be replaced by an average value resulting from a spectrum of broader linewidth. 

These and other results on absorption and luminescence were the reason why the soliton model was proposed for the photoconduction process in trans-polyacetylene [103-110],... 

The soliton conductivity model for rrans-(CH) was put forward by Kivelson [115]. It was shown that at low temperature phonon assisted electron hopping between soliton-bound states may be the dominant conduction process in a lightly doped one - dimensional Peierls system such as polyacetylene. The presence of disorder, as represented by a spatially random distribution of charged dopant molecules causes the hopping conduction pathway to be essentially three dimensional. At the photoexitation stage, mainly neutral solitons have to be formed. These solitons maintain the soliton bands. The transport processes have to be hopping ones with a highly expressed dispersive... 

The deficiency of this model is the assumption of only light doping. However the soliton conductivity has also been established in highly doped material and not only for polyacetylene, but for polyphenylene too, in which the solitons cannot exist... 

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