Encyclopedia of Complexity and Systems Science

Living Edition
| Editors: Robert A. Meyers

Computing with Solitons

Living reference work entry
First Online:
DOI: https://doi.org/10.1007/978-3-642-27737-5_92-3
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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...

Keywords

Linear Fractional Transformation Spatial Soliton Photorefractive Crystal NAND Gate Data Pulse 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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Bibliography

  1. Ablowitz MJ, Prinari B, Trubatch AD (2004) Soliton interactions in the vector nls equation. Inverse Prob 20(4):1217–1237MathSciNetCrossRefzbMATHADSGoogle Scholar
  2. Ablowitz MJ, Prinari B, Trubatch AD (2006) Discrete vector solitons: composite solitons, Yang- Baxter maps and computation. Stud Appl Math 116:97–133MathSciNetCrossRefzbMATHGoogle Scholar
  3. Agrawal GP (2001) Nonlinear fiber optics, 3rd edn. Academic, San DiegozbMATHGoogle Scholar
  4. Anastassiou C, Segev M, Steiglitz K, Giordmaine JA, Mitchell M, Shih MF, Lan S, Martin J (1999) Energy-exchange interactions between colliding vector solitons. Phys Rev Lett 83(12):2332–2335CrossRefADSGoogle Scholar
  5. Anastassiou C, Fleischer JW, Carmon T, Segev M, Steiglitz K (2001) Information transfer via cascaded collisions of vector solitons. Opt Lett 26(19):1498–1500CrossRefADSGoogle Scholar
  6. Barad Y, Silberberg Y (1997) Phys Rev Lett 78:3290CrossRefADSGoogle Scholar
  7. Berlekamp ER, Conway JH, Guy RK (1982) Winning ways for your mathematical plays, vol 2. Academic [Harcourt Brace Jovanovich Publishers], LondonzbMATHGoogle Scholar
  8. Cao XD, McKinstrie CJ (1993) J Opt Soc Am B 10:1202CrossRefADSGoogle Scholar
  9. Chen ZG, Segev M, Coskun TH, Christodoulides DN (1996) Observation of incoherently coupled photorefractive spatial soliton pairs. Opt Lett 21(18):1436–1438CrossRefADSGoogle Scholar
  10. Christodoulides DN, Joseph RI (1988) Discrete self-focusing in nonlinear arrays of coupled waveguides. Opt Lett 13:794–796CrossRefADSGoogle Scholar
  11. Christodoulides DN, Singh SR, Carvalho MI, Segev M (1996) Incoherently coupled soliton pairs in biased photorefractive crystals. Appl Phys Lett 68(13):1763–1765CrossRefADSGoogle Scholar
  12. Cundiff ST, Collings BC, Akhmediev NN, Soto-Crespo JM, Bergman K, Knox WH (1999) Phys Rev Lett 82:3988CrossRefADSGoogle Scholar
  13. Fredkin E, Toffoli T (1982) Conservative logic. Int J Theor Phys 21(3/4):219–253MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hasegawa A, Tappert F (1973) Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers 1: anomalous dispersion. Appl Phys Lett 23(3):142–144CrossRefADSGoogle Scholar
  15. Islam MN, Poole CD, Gordon JP (1989) Opt Lett 14:1011CrossRefADSGoogle Scholar
  16. Jakubowski MH, Steiglitz K, Squier R (1998) State transformations of colliding optical solitons and possible application to computation in bulk media. Phys Rev E 58(5):6752–6758CrossRefADSGoogle Scholar
  17. Kang JU, Stegeman GI, Aitchison JS, Akhmediev N (1996) Observation of Manakov spatial solitons in AlGaAs planar waveguides. Phys Rev Lett 76(20):3699–3702CrossRefADSGoogle Scholar
  18. Korolev AE, Nazarov VN, Nolan DA, Truesdale CM (2005) Opt Lett 14:132CrossRefADSGoogle Scholar
  19. Manakov SV (1973) On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Zh Eksp Teor Fiz 65(2):505–516 [Sov. Phys. JETP 38, 248 (1974)]ADSGoogle Scholar
  20. Mano MM (1972) Computer logic design. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
  21. Menyuk CR (1987) Opt Lett 12:614CrossRefADSGoogle Scholar
  22. Menyuk CR (1988) J Opt Soc Am B 5:392CrossRefADSGoogle Scholar
  23. Menyuk CR (1989) Pulse propagation in an elliptically birefringent Kerr medium. IEEE J Quantum Electron 25(12):2674–2682CrossRefADSGoogle Scholar
  24. Mollenauer LF, Stolen RH, Gordon JP (1980) Experimental observation of picosecond pulse narrowing and solitons in optical fibers. Phys Rev Lett 45(13):1095–1098CrossRefADSGoogle Scholar
  25. Nishizawa N, Goto T (2002) Opt Express 10:1151–1160CrossRefADSGoogle Scholar
  26. Radhakrishnan R, Lakshmanan M, Hietarinta J (1997) Inelastic collision and switching of coupled bright solitons in optical fibers. Phys Rev E 56(2):2213–2216CrossRefADSGoogle Scholar
  27. Rand D, Steiglitz K, Prucnal PR (2005) Signal standardization in collision-based soliton computing. Int J Unconv Comput 1:31–45Google Scholar
  28. Rand D, Glesk I, Bres CS, Nolan DA, Chen X, Koh J, Fleischer JW, Steiglitz K, Prucnal PR (2007) Observation of temporal vector soliton propagation and collision in birefringent fiber. Phys Rev Lett 98(5):053902CrossRefADSGoogle Scholar
  29. Russell JS (1844) Report on waves. In: Report of the 14th meeting of the British Association for the Advancement of Science. Taylor and Francis, London, pp 331–390Google Scholar
  30. Shih MF, Segev M (1996) Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons. Opt Lett 21(19):1538–1540CrossRefADSGoogle Scholar
  31. Shor PW (1994) Algorithms for quantum computation: discrete logarithms and factoring. In: 35th IEEE Press, Piscataway, pp 124–134Google Scholar
  32. Squier RK, Steiglitz K (1993) 2-d FHP lattice gasses are computation universal. Complex Syst 7:297–307zbMATHGoogle Scholar
  33. Steblina VV, Buryak AV, Sammut RA, Zhou DY, Segev M, Prucnal P (2000) Stable self-guided propagation of two optical harmonics coupled by a microwave or a terahertz wave. J Opt Soc Am B 17(12):2026–2031CrossRefADSGoogle Scholar
  34. Steiglitz K (2000) Time-gated Manakov spatial solitons are computationally universal. Phys Rev E 63(1):016608CrossRefADSGoogle Scholar
  35. Steiglitz K (2001) Multistable collision cycles of Manakov spatial solitons. Phys Rev E 63(4):046607CrossRefADSGoogle Scholar
  36. Yang J (1997) Physica D 108:92–112MathSciNetCrossRefADSGoogle Scholar
  37. Yang J (1999) Multisoliton perturbation theory for the Manakov equations and its applications to nonlinear optics. Phys Rev E 59(2):2393–2405MathSciNetCrossRefADSGoogle Scholar
  38. Zakharov VE, Shabat AB (1971) Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh Eksp Teor Fiz 61(1):118–134 [Sov. Phys. JETP 34, 62 (1972)]MathSciNetGoogle Scholar

Authors and Affiliations

  1. 1.Lincoln LaboratoryMassachusetts Institute of TechnologyLexingtonUSA
  2. 2.Computer Science DepartmentPrinceton UniversityPrincetonUSA