Enigmas of optical and matter-wave soliton nonlinear tunneling

Abstract

The method of mathematical analogies opens the possibility to study optical and matter-wave solitons in parallel and, due to the evident complexity of experiments with matter wave solitons, offers remarkable possibilities in studies of the BEC systems by performing experiments in the nonlinear optical systems and vice versa. We discuss previous misinterpretations of the binding soliton energy and investigate all possible scenarios of nonlinear soliton tunneling of Schrödinger solitons including soliton splitting on a potential barrier, leading to partial reflection and partial transmission, sub-barrier soliton tunneling and over-barrier soliton reflection. We reveal the most significant modifications of soliton tunneling scenarios due to increasing the binding energy, and show that the self-compressing soliton resembles more the classical particle case than a quantum mechanical behavior. The enhancement of the soliton binding energy is of decisive importance in the testing of a long-standing theoretical result which predicts that an optical soliton can tunnel between two regions of anomalous dispersion across a forbidden region of normal dispersion (enhanced soliton spectral tunneling effect).

Keywords:

• optical solitons
• nonlinear Schrödinger equation
• matter-wave solitons
• Gross–Pitaevskii equation
• soliton tunneling effect
• self-compressing solitons
• colored solitons
• soliton spectral tunneling effect

Acknowledgements

This investigation is a natural follow up of the works 33 performed in collaboration with Professor Akira Hasegawa and the authors would like to thank him for this collaboration. The study of the soliton spectral tunneling effect was initiated following a discussion between one of the authors (VNS) and Professor Alan Newell many years ago at the Soliton Workshop in Kyoto organized by Akira Hasegawa. The authors would like to thank Professor Nail Akhmediev for stimulating discussions and sending the copy of the pioneering paper published by Kosevich. Special thanks are due to Professor Roy Taylor for many fruitful discussions concerning different puzzle aspects of nonlinear soliton dynamics and for carefully checking the manuscript. The work reported in this paper was, in part, supported by BUAP and CONACyT Grants.