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By Antonio Barone

Finished insurance of the Josephson influence, from a survey of underlying actual thought to real and proposed engineering purposes. Considers many macroscopic quantum results with strength for technical improvement. Theoretical fabric is by way of correct discussions of gadget purposes. comprises a hundred unique figures and images.

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13), it is characterized by some maximum phase velocity s0 . The velocity of a soliton (if it arises) will exceed s. If a plot of the dispersion law shows convexity downwards, as in the case of a sine-Gordon equation, it is characterized by the minimum phase velocity s0 , and the velocity of the existing soliton should be less than s0 . This conclusion will be proved in the following. 15) moved along a 1D crystal, remaining undeformed and losing no velocity, i. , moves “by inertia” like a particle.

Therefore, the long-wave approximation allows us to solve problems associated with local vibrations in the frequency interval ω − ωm ωm . 11 Heavy Defects and 1D Superlattice In the previous section we analyzed the local vibrations and found that only a light isotope defect could produce such a vibration with a frequency higher than all frequencies of a defectless chain. Under the conditions m − M m and ω − ωm ωm description of the problems under consideration could be performed in the long-wave approximation.

8) is characterized by two parameters (Fig. 13): the limiting frequency ω0 and the minimum phase velocity of optical vibrations s0 . Fig. 13 The dispersion law of harmonic vibrations. To simplify the equations further, we introduce the dimensionless displacement 2πu/a and denote it by the same letter u. 6) will then change slightly: ∂2 u ∂2 u = s20 2 − ω02 sin u. 9) This equation is called the sine-Gordon equation. All its physically meaningful solutions are now systematized and studied. 9) in the form u = u(x − Vt), V = const, where the function u(x) satisfies the boundary conditions and has a plot such as in Fig.

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