Solitons in Nonlinear Systems and Eigen-states in Quantum Wells

TL;DR

This paper explores the deep connection between solitons in nonlinear Schrödinger systems and eigen-states in quantum wells, revealing new ways to understand soliton stability and quantum state evolution across various physical systems.

Contribution

It establishes a novel correspondence between nonlinear solitons and quantum eigen-states, including in non-Hermitian and asymmetric quantum wells, expanding the understanding of soliton stability and quantum state solutions.

Findings

Many solitons are derived from quantum eigen-states.

Vector solitons with different interactions correspond to identical eigen-states.

Nonlinear systems can be used to solve quantum eigen-problems.

Abstract

We study the relations between solitons of nonlinear Schr\"{o}dinger equation described systems and eigen-states of linear Schr\"{o}dinger equation with some quantum wells. Many different non-degenerated solitons are re-derived from the eigen-states in the quantum wells. We show that the vector solitons for coupled system with attractive interactions correspond to the identical eigen-states with the ones of coupled systems with repulsive interactions. The energy eigenvalues of them seem to be different, but they can be reduced to identical ones in the same quantum wells. The non-degenerated solitons for multi-component systems can be used to construct much abundant degenerated solitons in more components coupled cases. On the other hand, we demonstrate soliton solutions in nonlinear systems can be also used to solve the eigen-problems of quantum wells. As an example, we present eigenvalue and eigen-state in a complicated quantum well for which the Hamiltonian belongs to the non-Hermitian Hamiltonian having Parity-Time symmetry. We further present the ground state and the first exited state in an asymmetric quantum double-well from asymmetric solitons. Based on these results, we expect that many nonlinear physical systems can be used to observe the quantum states evolution of quantum wells, such as water wave tank, nonlinear fiber, Bose-Einstein condensate, and even plasma, although some of them are classical physical systems. These relations provide another different way to understand the stability of solitons in nonlinear Schr\"{o}dinger equation described systems, in contrast to the balance between dispersion and nonlinearity.