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R. Ali and H. A. Shah
by Iye et al. [10];resistivity studies have been carried out by Raffy et al. [11] and Hao
et al. [12], and interface studies by Aarnink ꢀsee [13] and the references therein).
In this work we have attempted to explore the nonlinear wave propagation properties
of density waves in a periodically layered superconducting plasma ꢀeach layer is say a
YBCO superconductor described by the London and two fluid model [1]). We employ a
standard perturbation and scaling technique to derive the nonlinear SchroÈdinger ꢀNLS)
equation which governs the propagation of density waves in the superconducting medium.
The properties of the nonlinear wave for a layered structure consisting of alternating
superconductor layers are investigated using the Kronig-Penney model. As in the work of
Bunch and Grow [1] and Ali and Shah [4] here, too, we make use of the London model
along with the two fluid model to describe the superconducting plasma which can be
thought to be composed of both normal and superfluid electrons. The two fluid model
takes losses into account, since one fluid is related to the superconducting electrons and
the other to the semiconducting electrons. This approach is further justified since high Tc
ceramics exhibit metallic properties along the superconducting plane ꢀab-plane) and semi-
conductor characteristics along the c-plane. Bunch and Grow [1] have noted that this
approach may have applications in the use of superconductors in travelling wave devices.
The advantages of such a device are, that no electron focusing structure is needed, no
cathode is necessary. This makes possible the fabrication of millimetre and infrared de-
vices since HTS are compatible with such fabrication ꢀas opposed to conventional electron
devices which are difficult to size down at high frequencies) along with the fact that these
would be moderate power devices which are of high quality with low noise. The micro-
wave and infrared properties are of importance in superconductors because of the exis-
tence of the energy gap, which implies that photons of energy less than the energy gap are
not absorbed. For superconductors the frequencies of interest are those which lie below
the electron plasma frequency, that is, the microwave and infrared frequency range.
The layout of the paper is as follows. In Section 2 we give a general mathematical
formulation of the problem and derive the nonlinear SchroÈdinger ꢀNLS) equation,
which governs the propagation of nonlinear density waves within a single layer of the
superconducting medium. The NLS has a known soliton solution, which is used in a
self-consistent manner to relate the different parameters entering into the system. We
then investigate the modulational instability of the NLS equation.
In Section 3 we introduce the boundary conditions of the Kronig-Penney model and
derive an expression for the nonlinear dispersion relation for the propagation of den-
sity waves through a periodically layered superconducting medium. The nonlinear dis-
persion relation relates the nonlinear analog of the Bloch wave vector to the propaga-
tion frequency of the soliton. In Section 4 we give a numerical analysis of the results
obtained in the previous section.
2. Nonlinear SchroÈdinger Equation
For the propagation of charge density waves in the superconducting medium, the fol-
lowing set of equations is used [1]:
@H
r  E Àma
;
ꢀ1
ꢀ2
@t
@E
@t
r  H Ja ea
;