Fundamental soliton

Stationary (i.e. for dA/ dz = 0) localized solutions to Eq.(3.2) represent nonlinear modes in the planar waveguide and may be found in an analytical form via matching the partial solutions of Eq.(3.2) at the core/cladding boundary. The partial solutions are Jacobi elliptical function in the core and 2l rccosh — )E]/E in the cladding (the functional dependence similar to a fundamental soliton in a uniform nonlinear medium). Here is a parameter which depends on the boundary conditions. Contrary to the modes of a linear waveguide, the transverse profile of a nonlinear mode depends on the power in the mode. 

Fig. 6.40 Comparison of spectral and time profile for the propagation of short pulses (a) linear propagation in a medium without self-phase modulation, (b) fundamental soliton for negative dispersion and positive SPM...

Introducing the refractive index n = uqn2l into the wave equation (6.15) yields stable solutions that are called solitons of order N. While the fundamental soliton N = 1) has a constant time profile I t), the higher-order solitons show an oscillatory change of their time profile I(t) the pulsewidth decreases at first and then increases again. After a path length zo which depends on the refractive index of the fiber and on the pulse intensity, the soliton recovers its initial form I t) [704, 705]. 

In the Sect. 11.1.7 we discussed the self-phase modulation of an optical pulse in a fiber because of the intensity-dependent refractive index n = no- -n2l t). While the resultant spectral broadening of the pulse leads in a medium with normal negative dispersion dno/dX < 0) to a spatial broadening of the pulse, an anomalous linear dispersion dn /dk > 0) would result in a pulse compression. Such anomalous dispersion can be found in fused quartz fibers for k> 13 p.m [11.66,11.67]. For a suitable choice of the pulse intensity the dispersion effects caused by no(k) and by n2l(t) may cancel, which means that the pulse propagates through the medium without changing its time profile. Such a pulse is named a fundamental soliton [11.68,11.69]. 

In this way, we can relate duality to quark-hadron continuity. We considered duality, which is already present at zero chemical potential, between the soliton and the vector mesons a fundamental property of the spectrum of QCD which should persists as we increase the quark chemical potential. Should be noted that differently than in [42] we have not subtracted the energy cost to excite a soliton from the Fermi sea. Since we are already considering the Lagrangian written for the excitations near the Fermi surface we would expect not to consider such a corrections. In any event this is of the order //, [42] and hence negligible with respect to Msoiiton. 

A soliton is a solitary wave that preserves its shape and speed in a collision with another solitary wave [12,13]. Soliton solutions to differential equations require complete integrability and integrable systems conserve geometric features related to symmetry. Unlike the equations of motion for conventional Maxwell theory, which are solutions of U(l) symmetry systems, solitons are solutions of SU(2) symmetry systems. These notions of group symmetry are more fundamental than differential equation descriptions. Therefore, although a complete exposition is beyond the scope of the present review, we develop some basic concepts in order to place differential equation descriptions within the context of group theory. 

When /> 877-/g2-l, the breather breaks up into a soliton and antisoliton. The fundamental state of the breather is F = 1, and 2ES - b(1) is the activation energy needed to dissociate the breather and create a soliton-antisoliton pair. When g2/8v > 1, the breather does not exist. 

Optical Signatures of Solitons, Polarons, and Bipolarons The formation of these excitations generates energy levels corresponding to optical transitions below the fundamental absorption that between the valence and the conduction band in one-electron models (but see Section... 

Questions that had been of fundamental importance to quantum chemistry for many decades were addressed. When the existence of bond alternation in trans-polyacetylene was been demonstrated [14,15], a fundamental issue that dates to the beginnings of quantum chemistry was resolved. The relative importance of the electron-electron and electron-lattice interactions in Ti-electron macromolecules quickly emerged as an issue and continues to be vigorously debated even today. Aspects of the theory of one-dimensional electronic structures were applied to these real systems. The important role of disorder on the electronic structure and properties of these low dimensional metals and semiconductors was immediately evident. The importance of structural relaxation in the excited state (solitons, polarons and bipolarons) quickly emerged. 

Because CPs conduct current without having a partially empty or partially filled band, concepts from solid state physics are used to explain the electronic phenomenon in these polymers. Thus, chemists refer to solitons, polarons, and bipolarons when they discuss the fundamentals of CPs. And, Fig. 1 shows the energy level diagram for an undoped, slightly doped, and heavily doped polymer to further illustrate the concept of doping. 

Polyacetylene (PA) is the simplest conjugated polymer. It can exist in cis- and transforms (cis- and trans-PA isomers). The latter is thermodynamically more stable. The transition between C-C and C = C bonds in tmns-PA does not require energy change, so the Peierls distortion opens up a substantial gap in the Fermi level. This twofold degeneration leads to the formation of mobile solitons with a length of == 15 C-C units and spin 5 = on trans-PA chains. These correspond to a break in the pattern of bond alternation," and thus determine the fundamental properties of the polymer. 

In the general case, the classes of homotopic mappings of the line y threaded through a planar soliton form the relative homotopy group 7Ti(iR, 91), where 91 is the OP space far from the core of the soliton, shrunk (as compared to the complete OP space 91) by additional interactions (external field, boundary conditions, etc.). If 91 consists of a single point, as in Figure 5.18, 7Ti(9I, 91) coincides with the fundamental group 7Ti(91) [77], [78]. 

Biopolymers, like nucleic acids (DNA and RNA), proteins, polysaccharides, lipids, and so on. have fundamental significance in life processes. Since the mid-1980s, highly conducting polymers, like doped different polyacetylenes (1) and (2) (SN) (3), or the TCNQ-TTF system (4) and (5), with quasi-one-dimensional alternating stacks of the two different types of molecule embedded in a three-dimensional molecular crystal, and PPV (6), have become objects of extensive experimental investigations because they are also candidates for the discovery of new physical phenomena, such as solitons, polarons and bipolarons, and superconductivity with higher transition temperatures (T ). 

For fundamental studies, these polarization self-switching processes could also be integrated into the spatial soliton formation discussed in the previous section, and thus a host of novel optical phenomena await the inquisitive. 

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