Definition of the Subject
Solitons are localized, shape-preserving waves characterized by robust collisions. First observed as water waves by John Scott Russell (1844) in the Union Canal near Edinburgh and subsequently recreated in the laboratory, solitons arise in a variety of physical systems, as both temporal pulses which counteract dispersion and spatial beams which counteract diffraction.
Solitons with two components, vector solitons, are computationally universal due to their remarkable collision properties. In this article, we describe in detail the characteristics of Manakov solitons, a specific type of vector soliton, and their applications in computing.
Introduction
In this section, we review the basic principles of soliton theory and spotlight relevant experimental results. Interestingly, the phenomena of soliton propagation and collision occur in many physical systems despite the diversity of mechanisms that bring about their existence. For this reason, the discussion in...
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Abbreviations
- Integrable:
-
This term is generally used in more than one way and in different contexts. For the purposes of this article, a partial differential equation or system of partial differential equations is integrable if it can be solved explicitly to yield solitons (qv).
- Manakov system:
-
A system of two cubic Schrödinger equations where the self- and cross-phase modulation terms have equal weight.
- Nonlinear Schrodinger equation:
-
A partial differential equation that has the same form as the Schrodinger equation of quantum mechanics, with a term nonlinear in the dependent variable, and, for the purposes of this article, is interpreted classically.
- Self- and cross-phase modulation:
-
Any terms in a nonlinear Schrödinger equation that involve nonlinear functions of the dependent variable of the equation or nonlinear functions of a dependent variable of another (coupled) equation, respectively.
- Solitary wave:
-
A solitary wave is a wave characterized by undistorted propagation. Solitary waves do not in general maintain their shape under perturbations or collisions.
- Soliton:
-
A soliton is a solitary wave which is also robust under perturbations and collisions.
- Turing equivalent:
-
Capable of simulating any Turing machine, and hence by Turing’s thesis capable of performing any computation that can be carried out by a sequence of effective instructions on a finite amount of data. A machine that is Turing equivalent is therefore as powerful as any digital computer. Sometimes a device that is Turing equivalent is called “universal.”
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Rand, D., Steiglitz, K. (2015). Computing with Solitons. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27737-5_92-3
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DOI: https://doi.org/10.1007/978-3-642-27737-5_92-3
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