Solitons

Inverse scattering techniques are used to investigate the
propagation in an amplifier of a coherent optical pulse
whose initial profile is a left-truncated hyperbolic secant
of area A such that 0< A< 4pi. A two-level nondegenerate
medium is assumed and dissipative effects are neglected. In the limit of extreme inhomogeneous broadening, the exact
solution for the pulse is obtained, and it is found to
evolve into a pi-pulse (for A< 2pi) at the light cone which
exhibits the expected amplification and compression, and
(for 2pi< A<.... 4pi) the above Ti-pulse plus a phase-shifted 2pi
soliton of lesser velocity whose amplitude and width is
related to a zero of the transmission coefficient in the
associated scattering problem. This final area of 3pi is in
agreement with the area theorem of McCall and Hahn.

The long-time behavior of the solution for x >> t^1/3 of
the Korteweg-de Vries equation is found when the initial
data consists of a right or left-truncated soliton. The
initial data in either case is found to evolve into a
complete soliton of smaller amplitude. The amplitude,
velocity, and phase shift of the resultant soliton is explicitly
given, and the emergence of this soliton from
the initial disturbance is described in both cases.