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 H
amiltonians, namely, Akulins 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 Akulins 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 $V(t) = (nhbar/tau)/cosh(t/tau)$, 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.
Two zero-range-interacting atoms in a circular, transversely harmonic waveguide are used as a test-bench for a quantitative description of the crossover between integrability and chaos in a quantum system with no selection rules. For such systems we
show that the expectation value after relaxation of a generic observable is given by a linear interpolation between its initial and thermal expectation values. The variable of this interpolation is universal; it governs this simple law to cover the whole spectrum of the chaotic behavior from integrable regime through the well- developed quantum chaos. The predictions are confirmed for the waveguide system, where the mode occupations and the trapping energy were used as the observables of interest; a variety of the initial states and a full range of the interaction strengths have been tested.
