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Lagrangian systems described by coupled scalar fields are
gaining renewed attention recently.
Also, the coupled scalar fields play an important role in
supersymmetry. In supersymmetry,
the solution of the solitonic field equations can be categorized
by the topological defects as
BPS and non-BPS states, in accordance with the works of
Bogomol¡¦nyi (1976) and Prasad and
Somerfield (1975).
In the case of two real scalar fields, the specific class
of systems is presented in Bazeia et al. (1995, 1996a and
1996b) and Bazeia and Santos (1996). They solved corresponding
equations of motion by field configurations, which are obeyed
by the first order differential equations. Jackiw (1977),
Rajaraman (1979), Bazeia et al. (1995, 1996a and 1996b)
and Bazeia and Santos (1996), presented a general way for
investigating stability to know the spectrum of the corresponding
Schrodinger operator (Jackiw, 1977; and Bazeia et al., 1996b).
In this paper, we are going to discuss the three coupled
scalar fields. The applications of three coupled scalar
fields for hexagonal network defect presented by Morris
and Bazeia (1996), Bazeia and Brito (2000), Carroll et al.
(2000) and Bazeia et al. (2002). One of them with the domain
walls are discussed by Bazeia et al. (2002). Also, the three
fields solutions in the Einstein equations for describing
black holes with the cosmic strings is discussed by Frolov
and Fursaev (2001), and two and three coupled scalar field
play important role in the standard model Higgs in five
dimension. In the case of coupling g ->0, the solution
is tanh like, and here we also show this solution (Surujon,
2006), and also two scalar fields applied in braneworld
scenario in warped geometry (Sadeghi and Mohammadi, 2007).
In general, these give us motivation to study three scalar
fields.
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