Supersymmetric Quantum Mechanics and Solitons of the sine-Gordon and Nonlinear Schr\"{o}dinger Equations

TL;DR

This paper reveals how supersymmetric quantum mechanics links reflectionless Hamiltonians to soliton solutions of integrable PDEs like sine-Gordon and nonlinear Schrödinger equations, providing a systematic way to generate multi-soliton solutions and explaining a laser physics phenomenon.

Contribution

It uncovers the algebraic mechanism connecting supersymmetric chains to reflectionless potentials and soliton solutions, advancing understanding of integrable systems and their physical applications.

Findings

Reflectionless Hamiltonians are connected via SUSY chains to potential-free Hamiltonians.

Multi-soliton solutions can be systematically generated through these SUSY chains.

The work explains a laser physics effect involving atom state preservation under specific pulse shapes.

Abstract

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.