  We present a case demonstrating the connection between supersymmetric quantum
mechanics (SUSY--QM), reflectionless scattering, and soliton solutions of
integrable partial differential equations. We show that the members of a class
of reflectionless Hamiltonians, namely, Akulin's Hamiltonians, are connected
via supersymmetric chains to a potential-free Hamiltonian, explaining their
reflectionless nature. While the reflectionless property in question has been
mentioned in the literature for over two decades, the enabling algebraic
mechanism was previously unknown. Our results indicate that the multi-solition
solutions of the sine-Gordon and nonlinear Schr\"{o}dinger equations can be
systematically generated via the supersymmetric chains connecting Akulin's
Hamiltonians. Our findings also explain a well-known but little-understood
effect in laser physics: when a two-level atom, initially in the ground state,
is subjected to a laser pulse of the form $V(t) = (n\hbar/\tau)/\cosh(t/\tau)$,
with $n$ being an integer and $\tau$ being the pulse duration, it remains in
the ground state after the pulse has been applied, for {\it any} choice of the
laser detuning.
