Dynamics of Partial Differential Equations

Volume 1 (2004)

Number 3

Integrable nonlinear Schrödinger systems and their soliton dynamics

Pages: 239 – 299

DOI: http://dx.doi.org/10.4310/DPDE.2004.v1.n3.a1

Authors

M. J. Ablowitz (Department of Applied Mathematics, University of Colorado, Boulder, Colo., U.S.A.)

B. Prinari (Dipartimento di Fisica and Sezione INFN, Lecce, Italy)

A. D. Trubatch (Department of Mathematical Sciences, United States Military Academy, West Point, New York)

Abstract

Nonlinear Schrödinger (NLS) systems are important examples of physically-significant nonlinear evolution equations that can be solved by the inverse scattering transform (IST) method. In fact, the IST for discrete and continuous, as well as scalar and vector, NLS systems all fit into the same framework, which is reviewed here. The parallel presentation of the IST for each of these systems not only clarifies the common structure of the IST, but also highlights the key variations. Importantly, these variations manifest themselves in the dynamics of the solutions. With the IST approach, one can explicitly construct the soliton solutions of each of these systems, as well as formulas from which one can determine the dynamics of soliton interaction. In particular, vector solitons, both continuous and discrete, are partially characterized by a polarization vector, which is shifted by soliton interaction. Here, we give a complete account of the nature of this polarization shift. The polarization vector can be used to encode the value of a binary digit (“bit”) and the soliton interaction arranged so as to effect logical computations.

Keywords

nonlinear Schrödinger systems, inverse scattering transform, soliton interaction

2010 Mathematics Subject Classification

Primary 35-xx. Secondary 78-xx.

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