Skip to main content
SpringerLink
Log in

Computing with Classical Soliton Collisions

  • Chapter
  • First Online:
Advances in Unconventional Computing

Part of the book series: Emergence, Complexity and Computation ((ECC,volume 23))

Abstract

We review work on computing with solitons, from the discovery of solitons in cellular automata, to an abstract model for particle computation, information transfer in collisions of optical solitons, state transformations in collisions of vector solitons, a proof of the universality of blinking spatial solitons, and the demonstration of multistable collision cycles and their application to state-restoring logic. We conclude by discussing open problems and the prospects for practical computing applications using optical soliton collisions in photo-refractive crystals and fibers.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We assume here that there is sufficient separation between collisions to ensure that this equality is true.

References

  1. Wolfram, S. (ed.): Theory and Application of Cellular Automata. World Scientific, Singapore (1986)

    Google Scholar 

  2. Wolfram, S.: Universality and complexity in cellular automata. Phys. D 10D, 1–35 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Park, J.K., Steiglitz, K., Thurston, W.P.: Soliton-like behavior in automata. Phys. D 19D(3), 423–432 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  4. Parks, T.W., Burrus, C.S.: Digital Filter Design. Wiley, New York (1987)

    MATH  Google Scholar 

  5. Steiglitz, K., Kamal, I., Watson, A.: Embedding computation in one-dimensional automata by phase coding solitons. IEEE Trans. Comput. 37(2), 138–145 (1988)

    Article  MathSciNet  Google Scholar 

  6. Goldberg, C.H.: Parity filter automata. Complex Syst. 2, 91–141 (1988)

    MathSciNet  MATH  Google Scholar 

  7. Fokas, A.S., Papadopoulou, E., Saridakis, Y.: Particles in soliton cellular automata. Complex Syst. 3, 615–633 (1989)

    MathSciNet  MATH  Google Scholar 

  8. Fokas, A.S., Papadopoulou, E., Saridakis, Y.: Coherent structures in cellular automata. Phys. Lett. 147A(7), 369–379 (1990)

    MathSciNet  MATH  Google Scholar 

  9. Ablowitz, M.J., Keiser, J.M., Takhtajan, L.A.: Class of stable multistate time-reversible cellular automata with rich particle content. Phys. Rev. A 44A(10), 6909–6912 (1991)

    Article  Google Scholar 

  10. Bruschi, M., Santini, P.M., Ragnisco, O.: Integrable cellular automata. Phys. Lett. A 169, 151–160 (1992)

    Article  MathSciNet  Google Scholar 

  11. Siwak, P.: On automata of some discrete recursive filters that support filtrons. In: Domek, S., Kaszynski, R., Tarasiejski, L. (eds.) Proceedings of Fifth International Symposium on Methods and Models in Automation and Robotics, vol. 3 (Discrete Processes), pp. 1069–1074, Miȩdzyzdroje, Poland, Aug. 25–29. Wydawnictwo Politechniki Szczecińskiej (1998)

    Google Scholar 

  12. Siwak, P.: Filtrons and their associated ring computations. Int. J. Gen. Syst. 27(1–3), 181–229 (1998)

    Article  MATH  Google Scholar 

  13. Siwak, P.: Iterons, fractals and computations of automata. In: Dubois, D.M. (ed.) Second International Conference on Computing Anticipatory Systems, CASYS ’98, Conference Proceedings 465, pp. 45–63. Amer. Inst. Phys, Woodbury, New York (1999)

    Google Scholar 

  14. Squier, R.K., Steiglitz, K.: Programmable parallel arithmetic in cellular automata using a particle model. Complex Syst. 8, 311–323 (1994)

    MathSciNet  MATH  Google Scholar 

  15. Kung, H.T.: Why systolic architectures? IEEE Comput. 15(1), 37–46 (1982). January

    Article  Google Scholar 

  16. Frisch, U., d’Humie’res, D., Hasslacher, B., Lallemand, P., Pomeau, Y., Rivet, J.P.: Lattice gas hydrodynamics in two and three dimensions. Complex Syst. 1, 649–707 (1987)

    MathSciNet  MATH  Google Scholar 

  17. Jakubowski, M.H., Steiglitz, K., Squier, R.K.: Implementation of parallel arithmetic in a cellular automaton. In: Cappello, P., et al. (eds.) 1995 International Conference on Application Specific Array Processors, Strasbourg, France, Los Alamitos, CA, July 24–26. IEEE Computer Society Press (1995)

    Google Scholar 

  18. Jakubowski, M.H.:: Computing with Solitons in Bulk Media (Ph.D. Thesis). Princeton University, Princeton, NJ (1998)

    Google Scholar 

  19. Leighton, F.T.: Introduction to Parallel Algorithms and Architectures. Morgan Kaufman Publishers, San Mateo (1992)

    MATH  Google Scholar 

  20. Liu, H.-H., Fu, K.-S.: VLSI arrays for minimum-distance classifications. In: Fu, K.S. (ed.) VLSI for Pattern Recognition and Image Processing. Springer, Berlin (1984)

    Google Scholar 

  21. Hordijk, W., Crutchfield, J.P., Mitchell, M.: Embedded-particle computation in evolved cellular automata. In: Toffoli, T., Biafore, M., Le\(\tilde{\text{a}}\)o, J. (eds.) Proceedings of Fourth Workshop on Physics and Computation (PhysComp96), Boston, Mass., Nov. 22–24, pp. 153–158. New England Complex Systems Institute (1996)

    Google Scholar 

  22. Boccara, N., Nasser, J., Roger, M.: Particlelike structures and their interactions in spatiotemporal patterns generated by one-dimensional deterministic cellular-automaton rules. Phys. Rev. A 44(2), 866–875 (1991)

    Article  Google Scholar 

  23. Takahashi, D.: On a fully discrete soliton system. In: Boiti, M., Martina, L., Pempinelli, P. (eds.) Proceedings of 7th Workshop on Nonlinear Evolution Equations and Dynamical Systems (NEEDS’91), pp. 245–249. World Scientific, Singapore (1991)

    Google Scholar 

  24. Santini, P.M.: Integrability for algebraic equations, functional equations and cellular automata. In: Makhankov, V., Puzynin, I., Pashev, O. (eds.) Proceedings of8th Workshop on Nonlinear Evolution Equations and Dynamical Systems (NEEDS ’92), pp. 214–221. World Scientific, Singapore (1992)

    Google Scholar 

  25. Adamatzky, A.I.: On the particle-like waves in the discrete model of excitable medium. Neural Network World, pp. 3–10 (1996)

    Google Scholar 

  26. Adamatzky, A.I.: Computing in Nonlinear Media & Automata Collectives. Taylor & Francis (2001)

    Google Scholar 

  27. Fredkin, E., Toffoli, T.: Conservative logic. Int. J. Theor. Phys. 21(3/4), 219–253 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  28. Margolus, N.: Physics-like models of computation. Phys. D 10D, 81–95 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. Adamatzky, A.I.: Universal dynamical computation in multidimensional excitable lattices. Int. J. Theor. Phys. 37(12), 3069–3108 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. Atrubin, A.J.: An iterative one-dimensional real-time multiplier. IEEE Trans. Electron. Comput., EC-14:394–399 (1965)

    Google Scholar 

  31. Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for Your Mathematical Plays, vol. 2. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London (1982)

    MATH  Google Scholar 

  32. Serizawa, T.: Three-state Neumann neighbor cellular automata capable of constructing self-reproducing machines. Syst. Comput. Jpn 18(4), 33–40 (1987)

    Article  Google Scholar 

  33. Wolfram, S.: A New Kind of Science. Wolfram Media (2002)

    Google Scholar 

  34. Scott, A.C., Chu, F.Y.F., McLaughlin, D.W.: The soliton: a new concept in applied science. Proc. IEEE 61(10), 1443–1483 (1973)

    Article  MathSciNet  Google Scholar 

  35. Drazin, P.G., Johnson, R.S.: Solitons: An Introduction. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  36. Ablowitz, M.J., Clarkson, P.A.: Solitons, Nonlinear Evolution Equations, and Inverse Scattering. Cambridge University Press, Cambridge (1991)

    Book  MATH  Google Scholar 

  37. Rebbi, C., Soliani, G.: Solitons and Particles. World Scientific, Singapore (1984)

    Book  MATH  Google Scholar 

  38. Makhankov, V.G.: Soliton Phenomenology. Kluwer Academic Publishers, Norwell (1990)

    Book  Google Scholar 

  39. Jakubowski, M.H., Steiglitz, K., Squier, R.K.: When can solitons compute? Complex Syst. 10(1), 1–21 (1996)

    MathSciNet  MATH  Google Scholar 

  40. Segev, M., Valley, G.C., Crosignani, B., DiPorto, P., Yariv, A.: Steady-state spatial screening solitons in photorefractive materials with external applied field. Phys. Rev. Lett. 73(24), 3211–3214 (1994)

    Article  Google Scholar 

  41. Shih, M., Segev, M., Valley, G.C., Salamo, G., Crosignani, B., DiPorto, P.: Observation of two-dimensional steady-state photorefractive screening solitons. Electron. Lett. 31(10), 826–827 (1995)

    Article  Google Scholar 

  42. Shih, M.F., Segev, M.: Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons. Opt. Lett. 21(19), 1538–1540 (1996)

    Article  Google Scholar 

  43. Tikhonenko, V., Christou, J., Luther-Davies, B.: Three-dimensional bright spatial soliton collision and fusion in a saturable nonlinear medium. Phys. Rev. Lett. 76, 2698–2702 (1996)

    Article  Google Scholar 

  44. Jakubowski, M.H., Steiglitz, K., Squier, R.K.: Information transfer between solitary waves in the saturable Schrödinger equation. Phys. Rev. E 56, 7267–7273 (1997)

    Article  Google Scholar 

  45. Radhakrishnan, R., Lakshmanan, M., Hietarinta, J.: Inelastic collision and switching of coupled bright solitons in optical fibers. Phys. Rev. E 56(2), 2213–2216 (1997)

    Article  Google Scholar 

  46. Manakov, S.V.: On the theory of two-dimensional stationary self-focusing of electromagnetic waves. Sov. Phys. JETP 38(2), 248–253 (1974)

    MathSciNet  Google Scholar 

  47. Kang, J.U., Stegeman, G.I., Aitchison, J.S., Akhmediev, N.: Observation of Manakov spatial solitons in AlGaAs planar waveguides. Phys. Rev. Lett. 76(20), 3699–3702 (1996)

    Article  Google Scholar 

  48. Jakubowski, M.H., Steiglitz, K., Squier, R.: State transformations of colliding optical solitons and possible application to computation in bulk media. Phys. Rev. E 58(5), 6752–6758 (1998)

    Article  Google Scholar 

  49. Yariv, A., Yeh, P.: Optical Waves in Crystals. Wiley, New York (1984)

    Google Scholar 

  50. Bennett, C.H.: Logical reversibility of computation. IBM J. Res. Dev. 17(6), 525–532 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  51. Steiglitz, K.: Time-gated Manakov spatial solitons are computationally universal. Phys. Rev. E 63(1), 016608 (2000)

    Article  Google Scholar 

  52. Stegeman, G.I., Segev, M.: Optical spatial solitons and their interactions: Universality and diversity. Science 286(5444), 1518–1523 (1999)

    Article  Google Scholar 

  53. Mano, M.M.: Computer Logic Design. Prentice-Hall, Englewood Cliffs (1972)

    MATH  Google Scholar 

  54. Mowle, F.J.: A Systematic Approach to Digital Logic Design. Addison-Wesley, Reading (1976)

    Google Scholar 

  55. Squier, R.K., Steiglitz, K.: 2-d FHP lattice gasses are computation universal. Complex Syst. 7, 297–307 (1993)

    MATH  Google Scholar 

  56. Steiglitz, K.: Multistable collision cycles of Manakov spatial solitons. Phys. Rev. E 63(4), 046607 (2001)

    Article  Google Scholar 

  57. Rand, D., Steiglitz, K., Prucnal, P.R.: Noise-immune universal computation using Manakov soliton collision cycles. In: Proceedings of Nonlinear Guided Waves and Their Applications. Optical Society of America, x (2004)

    Google Scholar 

  58. Enns, R.H., Edmundson, D.E., Rangnekar, S.S., Kaplan, A.E.: Optical switching between bistable soliton states: a theoretical review. Opt. Quantum Electron. 24, S1295–1314 (1992)

    Article  Google Scholar 

  59. Kawaguchi, H.: Polarization bistability in vertical-cavity surface-emitting lasers. In: Osinski, M., Chow, W.W. (eds.) SPIE Proceedings. Physics and Simulation of Optoelectronic Devices V, volume 2994, pp. 230–241. National Labs, Sandia Park, NM, USA (1997)

    Google Scholar 

  60. Lo, A.W.: Some thoughts on digital components and circuit techniques. IRE Trans. Elect. Comp., EC-10:416–425 (1961)

    Google Scholar 

  61. Christodoulides, D.N., Singh, S.R., Carvalho, M.I., Segev, M.: Incoherently coupled soliton pairs in biased photorefractive crystals. Appl. Phys. Lett. 68(13), 1763–1765 (1996)

    Article  Google Scholar 

  62. Chen, Z., Segev, M., Coskun, T., Christodoulides, D.N.: Observation of incoherently coupled photorefractive spatial soliton pairs. Opt. Lett. 21, 1436–1439 (1996)

    Article  Google Scholar 

  63. Anastassiou, C., Segev, M., Steiglitz, K., Giordmaine, J.A., Mitchell, M., Shih, M., Lan, S., Martin, J.: Energy switching interactions between colliding vector solitons. Phys. Rev. Lett. 83, 2332–2335 (1999)

    Article  Google Scholar 

  64. Anastassiou, C., Steiglitz, K., Lewis, D., Segev, M., Giordmaine, J.A.: Bouncing of vector solitons. In Conference on Lasers and Electro-Optics, San Francisco, CA, May 8–12 (2000)

    Google Scholar 

  65. Bakaoukas, A.G., Edwards, J.: Computing in the 3NLS domain using first order solitons. Int. J. Unconv. Comput. 5, 489–522 (2009)

    Google Scholar 

  66. Bakaoukas, A.G., Edwards, J.: Computation in the 3NLS domain using first and second order solitons. Int. J. Unconv. Comput. 5, 523–545 (2009)

    Google Scholar 

Download references

Acknowledgments

We owe a debt of gratitude to colleagues and students, too many to enumerate, for useful comments and discussions over the years. Most notably, Stephen Wolfram provided an important spark three decades ago, and Mordechai Segev two decades ago. The description of state-restoring logic is based on work with Darren Rand and Paul Prucnal.

Author information

Authors and Affiliations

  1. Computer Science Department, Princeton University, Princeton, NJ, 08544, USA

    Mariusz H. Jakubowski & Ken Steiglitz

  2. Computer Science Department, Georgetown University, Washington, DC, 20057, USA

    Richard Squier

Authors
  1. Mariusz H. Jakubowski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ken Steiglitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Richard Squier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mariusz H. Jakubowski .

Editor information

Editors and Affiliations

  1. Unconventional ComputingCentre, University of the Westof England Unconventional ComputingCentre, Bristol, United Kingdom

    Andrew Adamatzky

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Jakubowski, M.H., Steiglitz, K., Squier, R. (2017). Computing with Classical Soliton Collisions. In: Adamatzky, A. (eds) Advances in Unconventional Computing. Emergence, Complexity and Computation, vol 23. Springer, Cham. https://doi.org/10.1007/978-3-319-33921-4_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-33921-4_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-33920-7

  • Online ISBN: 978-3-319-33921-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation