Science Bulletin

, Volume 61, Issue 7, pp 570–575 | Cite as

Localization and shock waves in curved manifolds

Article Physics & Astronomy
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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.

Keywords

Nonlinear waves Shock waves Nonlinear optics  Curvature Bose–Einstein condensation 

References

  1. 1.
    Rayleigh JWS (1877) The theory of sound. Mcmillan and co, New YorkGoogle Scholar
  2. 2.
    Einstein A (1936) Lens-like action of a star by the deviation of light in the gravitational field. Science 84:506–507CrossRefGoogle Scholar
  3. 3.
    Hawking SW (1974) Black hole explosions? Nature 248:30–31CrossRefGoogle Scholar
  4. 4.
    Unruh WG (1976) Notes on black-hole evaporation. Phys Rev D 14:870–892CrossRefGoogle Scholar
  5. 5.
    Longhi S (2007) Topological optical Bloch oscillations in a deformed slab waveguide. Opt Lett 32:2647–2649CrossRefGoogle Scholar
  6. 6.
    Philbin TG, Kuklewicz C, Robertson S et al (2008) Fiber-optical analog of the event horizon. Science 319:1367–1370CrossRefGoogle Scholar
  7. 7.
    Wong GKL, Kang MS, Lee HW et al (2012) Excitation of orbital angular momentum resonances in helically twisted photonic crystal fiber. Science 337:446–449CrossRefGoogle Scholar
  8. 8.
    Barcel C, Liberati S, Visser M (2003) Probing semiclassical analog gravity in Bose–Einstein condensates with widely tunable interactions. Phys Rev A 68:053613CrossRefGoogle Scholar
  9. 9.
    Larre PE, Recati A, Carusotto I et al (2012) Quantum fluctuations around black hole horizons in Bose–Einstein condensates. Phys Rev A 85:013621CrossRefGoogle Scholar
  10. 10.
    Gorbach AV, Skryabin DV (2007) Light trapping in gravity-like potentials and expansion of supercontinuum spectra in photonic-crystal fibres. Nat Photonics 1:653–657CrossRefGoogle Scholar
  11. 11.
    Schultheiss VH, Batz S, Szameit A et al (2010) Optics in curved space. Phys Rev Lett 105:143901CrossRefGoogle Scholar
  12. 12.
    Faccio D (2012) Laser pulse analogues for gravity and analogue Hawking radiation. Contemp Phys 53:97–112CrossRefGoogle Scholar
  13. 13.
    Conti C (2014) Quantum gravity simulation by nonparaxial nonlinear optics. Phys Rev A 89:061801CrossRefGoogle Scholar
  14. 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 293:663–668CrossRefGoogle Scholar
  15. 15.
    Batz S, Peschel U (2008) Linear and nonlinear optics in curved space. Phys Rev A 78:043821CrossRefGoogle Scholar
  16. 16.
    Batz S, Peschel U (2010) Solitons in curved space of constant curvature. Phys Rev A 81:053806CrossRefGoogle Scholar
  17. 17.
    Wan W, Muenzel S, Fleischer JW (2010) Wave tunneling and hysteresis in nonlinear junctions. Phys Rev Lett 104:073903CrossRefGoogle Scholar
  18. 18.
    Cao X, Li T, Li X et al (2014) Shock wave on materials. Chin Sci Bull 59:5302–5308CrossRefGoogle Scholar
  19. 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 57:1977–1986CrossRefGoogle Scholar
  20. 20.
    da Costa RCT (1981) Quantum mechanics of a constrained particle. Phys Rev A 23:1982–1987CrossRefGoogle Scholar
  21. 21.
    Whitham GB (1999) Linear and nonlinear waves. Wiley, New YorkCrossRefGoogle Scholar
  22. 22.
    Damski B (2004) Formation of shock waves in a Bose–Einstein condensate. Phys Rev A 69:043610CrossRefGoogle Scholar
  23. 23.
    Gentilini S, Ghofraniha N, DelRe E et al (2012) Shock wave far-field in ordered and disordered nonlocal media. Opt Express 20:27369–27375CrossRefGoogle Scholar
  24. 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 74:023623CrossRefGoogle Scholar
  25. 25.
    El GA, Gammal A, Khamis EG et al (2007) Theory of optical dispersive shock waves in photorefractive media. Phys Rev A 76:053813CrossRefGoogle Scholar
  26. 26.
    Wan W, Jia S, Fleischer JW (2007) Dispersive superfluid-like shock waves in nonlinear optics. Nat Phys 3:46–51CrossRefGoogle Scholar
  27. 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 AmericaGoogle Scholar
  28. 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 82:013838CrossRefGoogle Scholar
  29. 29.
    Dalfovo F, Giorgini S, Pitaevskii L et al (1999) Theory of Bose–Eistein condesation in trapped gases. Rev Mod Phys 71:463–512CrossRefGoogle Scholar
  30. 30.
    Gentilini S, Braidotti MC, Marcucci G et al (2015) Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media. Phys Rev A 92:023801CrossRefGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute for Complex SystemsNational Research Council (ISC-CNR)RomeItaly
  2. 2.Department of PhysicsUniversity SapienzaRomeItaly

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