Skip to main content
SpringerLink
Log in

Localization and shock waves in curved manifolds

  • Article
  • Physics & Astronomy
  • Published:
Science Bulletin

Abstract

The investigation of the interplay between geometry and nonlinearity may open the road to the control of extreme waves. We study three-dimensional localization and dispersive shocks in a bent cigar shaped potential by the nonlinear Schrödinger equation. At high bending and high nonlinearity, topological trapping is frustrated by the generation of curved wave-breaking. Four-dimensional parallel simulations confirm the theoretical model. This work may contribute to novel devices based on geometrically constrained highly nonlinear dynamics and tests and analogs of fundamental physical theories in curved space.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. Rayleigh JWS (1877) The theory of sound. Mcmillan and co, New York

    Google Scholar 

  2. Einstein A (1936) Lens-like action of a star by the deviation of light in the gravitational field. Science 84:506–507

    Article  Google Scholar 

  3. Hawking SW (1974) Black hole explosions? Nature 248:30–31

    Article  Google Scholar 

  4. Unruh WG (1976) Notes on black-hole evaporation. Phys Rev D 14:870–892

    Article  Google Scholar 

  5. Longhi S (2007) Topological optical Bloch oscillations in a deformed slab waveguide. Opt Lett 32:2647–2649

    Article  Google Scholar 

  6. Philbin TG, Kuklewicz C, Robertson S et al (2008) Fiber-optical analog of the event horizon. Science 319:1367–1370

    Article  Google Scholar 

  7. Wong GKL, Kang MS, Lee HW et al (2012) Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber. Science 337:446–449

    Article  Google Scholar 

  8. Barcel C, Liberati S, Visser M (2003) Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions. Phys Rev A 68:053613

    Article  Google Scholar 

  9. Larre PE, Recati A, Carusotto I et al (2012) Quantum fluctuations around black hole horizons in Bose–Einstein condensates. Phys Rev A 85:013621

    Article  Google Scholar 

  10. Gorbach AV, Skryabin DV (2007) Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres. Nat Photonics 1:653–657

    Article  Google Scholar 

  11. Schultheiss VH, Batz S, Szameit A et al (2010) Optics in curved space. Phys Rev Lett 105:143901

    Article  Google Scholar 

  12. Faccio D (2012) Laser pulse analogues for gravity and analogue Hawking radiation. Contemp Phys 53:97–112

    Article  Google Scholar 

  13. Conti C (2014) Quantum gravity simulation by nonparaxial nonlinear optics. Phys Rev A 89:061801

    Article  Google Scholar 

  14. Dutto Z, Budde M, Slowe C et al (2001) Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose–Einstein condensate. Science 293:663–668

    Article  Google Scholar 

  15. Batz S, Peschel U (2008) Linear and nonlinear optics in curved space. Phys Rev A 78:043821

    Article  Google Scholar 

  16. Batz S, Peschel U (2010) Solitons in curved space of constant curvature. Phys Rev A 81:053806

    Article  Google Scholar 

  17. Wan W, Muenzel S, Fleischer JW (2010) Wave tunneling and hysteresis in nonlinear junctions. Phys Rev Lett 104:073903

    Article  Google Scholar 

  18. Cao X, Li T, Li X et al (2014) Shock wave on materials. Chin Sci Bull 59:5302–5308

    Article  Google Scholar 

  19. Wang B, Tang Z (2014) Study on the propagation of coupling shock waves with phase transition under combined tension-torsion impact loading. Sci China Phy Mech Astron 57:1977–1986

    Article  Google Scholar 

  20. da Costa RCT (1981) Quantum mechanics of a constrained particle. Phys Rev A 23:1982–1987

    Article  Google Scholar 

  21. Whitham GB (1999) Linear and nonlinear waves. Wiley, New York

    Book  Google Scholar 

  22. Damski B (2004) Formation of shock waves in a Bose–Einstein condensate. Phys Rev A 69:043610

    Article  Google Scholar 

  23. Gentilini S, Ghofraniha N, DelRe E et al (2012) Shock wave far-field in ordered and disordered nonlocal media. Opt Express 20:27369–27375

    Article  Google Scholar 

  24. Hoefer MA, Ablowitz MJ, Coddington I et al (2006) Dispersive and classical shock waves in Bose–Einstein condensates and gas dynamics. Phys Rev A 74:023623

    Article  Google Scholar 

  25. El GA, Gammal A, Khamis EG et al (2007) Theory of optical dispersive shock waves in photorefractive media. Phys Rev A 76:053813

    Article  Google Scholar 

  26. Wan W, Jia S, Fleischer JW (2007) Dispersive superfluid-like shock waves in nonlinear optics. Nat Phys 3:46–51

    Article  Google Scholar 

  27. Karpov M, Sivan Y, Fleurov V et al (2015) Autofocusing of cylindrical caustics self-generated in a defocusing nonlinear medium. In: Nonlinear optics, NF2A.3. Optical Society of America

  28. Conti C, Stark S, Russell PSJ et al (2010) Multiple hydrodynamical shocks induced by the Raman effect in photonic crystal fibers. Phys Rev A 82:013838

    Article  Google Scholar 

  29. Dalfovo F, Giorgini S, Pitaevskii L et al (1999) Theory of Bose–Eistein condesation in trapped gases. Rev Mod Phys 71:463–512

    Article  Google Scholar 

  30. Gentilini S, Braidotti MC, Marcucci G et al (2015) Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media. Phys Rev A 92:023801

    Article  Google Scholar 

Download references

Acknowledgments

This publication was made possible through the support of a grant from the John Templeton Foundation (58277). The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation. We also acknowledge support by the European Research Council Grant ERC-POC-2014 Vanguard (664782).

Author information

Authors and Affiliations

  1. Institute for Complex Systems, National Research Council (ISC-CNR), Via dei Taurini 19, 00185, Rome, IT, Italy

    Claudio Conti

  2. Department of Physics, University Sapienza, Piazzale Aldo Moro 5, 00185, Rome, IT, Italy

    Claudio Conti

Authors
  1. Claudio Conti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claudio Conti.

Ethics declarations

Conflict of interest

authors declare that they have no conflict of interest.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Conti, C. Localization and shock waves in curved manifolds. Sci. Bull. 61, 570–575 (2016). https://doi.org/10.1007/s11434-016-1040-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11434-016-1040-z

Keywords

Search

Navigation