Abstract

The description of shock waves beyond the shock point is a challenge in nonlinear physics and optics. Finding solutions to the global dynamics of dispersive shock waves is not always possible due to the lack of integrability. Here we propose a new method based on the eigenstates (Gamow vectors) of a reversed harmonic oscillator in a rigged Hilbert space. These vectors allow analytical formulation for the development of undular bores of shock waves in a nonlinear nonlocal medium. Experiments by a photothermal induced nonlinearity confirm theoretical predictions: the undulation period as a function of power and the characteristic quantized decays of Gamow vectors. Our results demonstrate that Gamow vectors are a novel and effective paradigm for describing extreme nonlinear phenomena.

© 2016 Optical Society of America

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References

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2015 (4)

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015).
[Crossref]

V. Arrizon, F. Soto-Eguibar, A. Zuniga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. A 32, 1140–1145 (2015).

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Physical realization of the Glauber quantum oscillator,” Sci. Rep. 50, 15816 (2015).
[Crossref]

2014 (2)

2013 (1)

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, “Shock wave in thermal lensing,” Phys. Rev. A 87, 053811 (2013).
[Crossref]

2012 (2)

M. Crosta, S. Trillo, and A. Fratalocchi, “The Whitham approach to dispersive shocks in systems with cubicquintic nonlinearities,” New J. Phys. 14, 093019 (2012).
[Crossref]

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

2010 (1)

C. Conti, S. Stark, P. S. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A,  82, 013838 (2010).
[Crossref]

2009 (2)

2008 (3)

A. Chong, W. H. Renninger, and F. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140–148 (2008).
[Crossref]

M. A. Hoefer, M. J. Ablowitz, and P. Engels, “Piston dispersive shock wave problem,” Phys. Rev. Lett. 100, 084504 (2008).
[Crossref] [PubMed]

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

2007 (4)

G. A. El, A. Gammal, E. G. Khamis, R.A. Kraenkel, and A.M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media,” Phys. Rev. A 76, 053813 (2007).
[Crossref]

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
[Crossref] [PubMed]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett.,  99, 043903 (2007).
[Crossref] [PubMed]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 4651 (2007).
[Crossref]

2006 (2)

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express oe14, 10095 (2006).

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

2005 (1)

G. A. El, “Resolution of a shock in hyperbolic systems modified by weak dispersion,” Chaos 15, 037103 (2005).
[Crossref]

2004 (2)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

D. Chruscinski, “Quantum mechanics of damped systems. II. Damping and parabolic potential barrier,” J. Mat. Phys. 45, 841–854 (2004).
[Crossref]

2002 (1)

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, “Stationary shocks in perioAsymptotic soliton train solutions of the defocusing nonlinear Schrdinger equation,” Phys. Rev. E 66, 036609 (2002).
[Crossref]

1974 (1)

A. V. Gurevich and L. P. Pitaevskii, “Nonstationary structure of a collisionless shock wave,” Sov. Phys. JEPT 38, 291 (1974).

1970 (1)

R. J. Taylor, D. R. Baker, and H. Ikezi, “Observation of collisionless electrostatic shocks,” Phys. Rev. Lett. 24, 206209 (1970).
[Crossref]

1957 (1)

L. A. Khalfin, “Monotonicity of the decay of unstable particles corresponding to an n-order,” Zh. Eksp. Teor. Fiz. 33, 1371 (1957).

1928 (1)

G. Gamow, “The quantum theory of nuclear disintegration,” Nature 122, 805–806 (1928).
[Crossref]

Ablowitz, M. J.

M. A. Hoefer, M. J. Ablowitz, and P. Engels, “Piston dispersive shock wave problem,” Phys. Rev. Lett. 100, 084504 (2008).
[Crossref] [PubMed]

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

Alberucci, A.

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” arXiv:1602.01722 [physics.optics], (2016).

Anderson, D. V.

M. D. Maiden, N. K. Lowman, D. V. Anderson, M. E. Schubert, and M. A. Hoefer, “Frictionless dispersive hydrodynamics of Stokes flows,” Arxiv, 1512.09240 (2015).

Arrizon, V.

V. Arrizon, F. Soto-Eguibar, A. Zuniga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. A 32, 1140–1145 (2015).

Assanto, G.

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” arXiv:1602.01722 [physics.optics], (2016).

Audebert, P.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Baker, D. R.

R. J. Taylor, D. R. Baker, and H. Ikezi, “Observation of collisionless electrostatic shocks,” Phys. Rev. Lett. 24, 206209 (1970).
[Crossref]

Ballarini, D.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Bang, O.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

Baronio, F.

M. Conforti, F. Baronio, and S. Trillo, “Resonant radiation shed by dispersive shock waves,” Phys. Rev. A 89, 013807 (2014).
[Crossref]

Barsi, C.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
[Crossref] [PubMed]

Biancalana, F.

C. Conti, S. Stark, P. S. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A,  82, 013838 (2010).
[Crossref]

Borghesi, M.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Braidotti, M. C.

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015).
[Crossref]

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Physical realization of the Glauber quantum oscillator,” Sci. Rep. 50, 15816 (2015).
[Crossref]

Bramati, A.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Bronski, J. C.

J. C. Bronski and D. W. McLaughlin, Singular Limits of Dispersive Waves (Plenum, 1994).

Buckley, J.

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express oe14, 10095 (2006).

Bulanov, S. V.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Cancellieri, E.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Chong, A.

Chruscinski, D.

D. Chruscinski, “Quantum mechanics of damped systems. II. Damping and parabolic potential barrier,” J. Mat. Phys. 45, 841–854 (2004).
[Crossref]

Coddington, I.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

Colas, D.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Conforti, M.

M. Conforti, F. Baronio, and S. Trillo, “Resonant radiation shed by dispersive shock waves,” Phys. Rev. A 89, 013807 (2014).
[Crossref]

Conti, C.

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015).
[Crossref]

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Physical realization of the Glauber quantum oscillator,” Sci. Rep. 50, 15816 (2015).
[Crossref]

C. Lecaplain, J. M. Soto-Crespo, P. Grelu, and C. Conti, “Dissipative shock waves in all-normal-dispersion mode-locked fiber lasers,” Opt. Lett. 39, 263–266 (2014).
[Crossref] [PubMed]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, “Shock wave in thermal lensing,” Phys. Rev. A 87, 053811 (2013).
[Crossref]

C. Conti, S. Stark, P. S. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A,  82, 013838 (2010).
[Crossref]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett.,  99, 043903 (2007).
[Crossref] [PubMed]

Cornell, E. A.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

Crosta, M.

M. Crosta, S. Trillo, and A. Fratalocchi, “The Whitham approach to dispersive shocks in systems with cubicquintic nonlinearities,” New J. Phys. 14, 093019 (2012).
[Crossref]

Daraio, C.

A. Molinari and C. Daraio, “Stationary shocks in periodic highly nonlinear granular chains,” Phys. Rev. E 80, 056602 (2009).
[Crossref]

De Giorgi, M.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

DelRe, E.

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Physical realization of the Glauber quantum oscillator,” Sci. Rep. 50, 15816 (2015).
[Crossref]

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015).
[Crossref]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, “Shock wave in thermal lensing,” Phys. Rev. A 87, 053811 (2013).
[Crossref]

Dominici, L.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Edmundson, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

El, G. A.

G. A. El, A. Gammal, E. G. Khamis, R.A. Kraenkel, and A.M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media,” Phys. Rev. A 76, 053813 (2007).
[Crossref]

G. A. El, “Resolution of a shock in hyperbolic systems modified by weak dispersion,” Chaos 15, 037103 (2005).
[Crossref]

Engels, P.

M. A. Hoefer, M. J. Ablowitz, and P. Engels, “Piston dispersive shock wave problem,” Phys. Rev. Lett. 100, 084504 (2008).
[Crossref] [PubMed]

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

Fleischer, J. M.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

Fleischer, J. W.

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
[Crossref] [PubMed]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 4651 (2007).
[Crossref]

Fratalocchi, A.

M. Crosta, S. Trillo, and A. Fratalocchi, “The Whitham approach to dispersive shocks in systems with cubicquintic nonlinearities,” New J. Phys. 14, 093019 (2012).
[Crossref]

Gammal, A.

G. A. El, A. Gammal, E. G. Khamis, R.A. Kraenkel, and A.M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media,” Phys. Rev. A 76, 053813 (2007).
[Crossref]

Gamow, G.

G. Gamow, “The quantum theory of nuclear disintegration,” Nature 122, 805–806 (1928).
[Crossref]

Gauthier, J. C.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Gentilini, S.

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Physical realization of the Glauber quantum oscillator,” Sci. Rep. 50, 15816 (2015).
[Crossref]

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015).
[Crossref]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, “Shock wave in thermal lensing,” Phys. Rev. A 87, 053811 (2013).
[Crossref]

Ghofraniha, N.

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, “Shock wave in thermal lensing,” Phys. Rev. A 87, 053811 (2013).
[Crossref]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett.,  99, 043903 (2007).
[Crossref] [PubMed]

Gigli, G.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Glauber, R. J.

R. J. Glauber, “Amplifiers, attenuators, and Schrdinger’s cat,” in Conference on Frontiers of Quantum Optics (1985).

Grelu, P.

Gurevich, A. V.

A. V. Gurevich and L. P. Pitaevskii, “Nonstationary structure of a collisionless shock wave,” Sov. Phys. JEPT 38, 291 (1974).

Hoefer, M. A.

M. A. Hoefer, M. J. Ablowitz, and P. Engels, “Piston dispersive shock wave problem,” Phys. Rev. Lett. 100, 084504 (2008).
[Crossref] [PubMed]

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

M. D. Maiden, N. K. Lowman, D. V. Anderson, M. E. Schubert, and M. A. Hoefer, “Frictionless dispersive hydrodynamics of Stokes flows,” Arxiv, 1512.09240 (2015).

Ikezi, H.

R. J. Taylor, D. R. Baker, and H. Ikezi, “Observation of collisionless electrostatic shocks,” Phys. Rev. Lett. 24, 206209 (1970).
[Crossref]

Jia, S.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 4651 (2007).
[Crossref]

Jisha, C. P.

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” arXiv:1602.01722 [physics.optics], (2016).

Kamchatnov, A. M.

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, “Stationary shocks in perioAsymptotic soliton train solutions of the defocusing nonlinear Schrdinger equation,” Phys. Rev. E 66, 036609 (2002).
[Crossref]

Kamchatnov, A.M.

G. A. El, A. Gammal, E. G. Khamis, R.A. Kraenkel, and A.M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media,” Phys. Rev. A 76, 053813 (2007).
[Crossref]

Kavokin, A.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Khalfin, L. A.

L. A. Khalfin, “Monotonicity of the decay of unstable particles corresponding to an n-order,” Zh. Eksp. Teor. Fiz. 33, 1371 (1957).

Khamis, E. G.

G. A. El, A. Gammal, E. G. Khamis, R.A. Kraenkel, and A.M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media,” Phys. Rev. A 76, 053813 (2007).
[Crossref]

Kieu, K.

Kraenkel, R. A.

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, “Stationary shocks in perioAsymptotic soliton train solutions of the defocusing nonlinear Schrdinger equation,” Phys. Rev. E 66, 036609 (2002).
[Crossref]

Kraenkel, R.A.

G. A. El, A. Gammal, E. G. Khamis, R.A. Kraenkel, and A.M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media,” Phys. Rev. A 76, 053813 (2007).
[Crossref]

Krolikowski, W.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

Laussy, F.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Lecaplain, C.

Löwenbrück, K.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Lowman, N. K.

M. D. Maiden, N. K. Lowman, D. V. Anderson, M. E. Schubert, and M. A. Hoefer, “Frictionless dispersive hydrodynamics of Stokes flows,” Arxiv, 1512.09240 (2015).

Mackinnon, A. J.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Maiden, M. D.

M. D. Maiden, N. K. Lowman, D. V. Anderson, M. E. Schubert, and M. A. Hoefer, “Frictionless dispersive hydrodynamics of Stokes flows,” Arxiv, 1512.09240 (2015).

Marcucci, G.

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015).
[Crossref]

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Physical realization of the Glauber quantum oscillator,” Sci. Rep. 50, 15816 (2015).
[Crossref]

Matuszewski, M.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

McLaughlin, D. W.

J. C. Bronski and D. W. McLaughlin, Singular Limits of Dispersive Waves (Plenum, 1994).

Molinari, A.

A. Molinari and C. Daraio, “Stationary shocks in periodic highly nonlinear granular chains,” Phys. Rev. E 80, 056602 (2009).
[Crossref]

Moya-Cessa, H. M.

V. Arrizon, F. Soto-Eguibar, A. Zuniga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. A 32, 1140–1145 (2015).

Neshev, D.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

Nikolov, N. I.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

Patel, P.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Petrov, M.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Picozzi, A.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

Pitaevskii, L. P.

A. V. Gurevich and L. P. Pitaevskii, “Nonstationary structure of a collisionless shock wave,” Sov. Phys. JEPT 38, 291 (1974).

Pretzler, G.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Rasmussen, J. J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

Renninger, W.

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express oe14, 10095 (2006).

Renninger, W. H.

Rica, S.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

Romagnani, L.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Ruocco, G.

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett.,  99, 043903 (2007).
[Crossref] [PubMed]

Russell, P. S. J.

C. Conti, S. Stark, P. S. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A,  82, 013838 (2010).
[Crossref]

Sanvitto, D.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Schubert, M. E.

M. D. Maiden, N. K. Lowman, D. V. Anderson, M. E. Schubert, and M. A. Hoefer, “Frictionless dispersive hydrodynamics of Stokes flows,” Arxiv, 1512.09240 (2015).

Schweikhard, V.

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

SilvaFernandez, B.

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Soto-Crespo, J. M.

Soto-Eguibar, F.

V. Arrizon, F. Soto-Eguibar, A. Zuniga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. A 32, 1140–1145 (2015).

Stark, S.

C. Conti, S. Stark, P. S. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A,  82, 013838 (2010).
[Crossref]

Sun, C.

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
[Crossref] [PubMed]

Taylor, R. J.

R. J. Taylor, D. R. Baker, and H. Ikezi, “Observation of collisionless electrostatic shocks,” Phys. Rev. Lett. 24, 206209 (1970).
[Crossref]

Toncian, T.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Trillo, S.

M. Conforti, F. Baronio, and S. Trillo, “Resonant radiation shed by dispersive shock waves,” Phys. Rev. A 89, 013807 (2014).
[Crossref]

M. Crosta, S. Trillo, and A. Fratalocchi, “The Whitham approach to dispersive shocks in systems with cubicquintic nonlinearities,” New J. Phys. 14, 093019 (2012).
[Crossref]

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett.,  99, 043903 (2007).
[Crossref] [PubMed]

Umarov, B. A.

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, “Stationary shocks in perioAsymptotic soliton train solutions of the defocusing nonlinear Schrdinger equation,” Phys. Rev. E 66, 036609 (2002).
[Crossref]

Wan, W.

C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007).
[Crossref] [PubMed]

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 4651 (2007).
[Crossref]

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1999).
[Crossref]

Willi, O.

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Wise, F.

A. Chong, W. H. Renninger, and F. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140–148 (2008).
[Crossref]

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express oe14, 10095 (2006).

Wise, F. W.

Wyller, J.

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

Zuniga-Segundo, A.

V. Arrizon, F. Soto-Eguibar, A. Zuniga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. A 32, 1140–1145 (2015).

Chaos (1)

G. A. El, “Resolution of a shock in hyperbolic systems modified by weak dispersion,” Chaos 15, 037103 (2005).
[Crossref]

J. Mat. Phys. (1)

D. Chruscinski, “Quantum mechanics of damped systems. II. Damping and parabolic potential barrier,” J. Mat. Phys. 45, 841–854 (2004).
[Crossref]

J. Opt. A (1)

V. Arrizon, F. Soto-Eguibar, A. Zuniga-Segundo, and H. M. Moya-Cessa, “Revival and splitting of a Gaussian beam in gradient index media,” J. Opt. A 32, 1140–1145 (2015).

J. Opt. B (1)

W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitons and beam propagation in spatially nonlocal nonlinear media,” J. Opt. B 6, S288 (2004).
[Crossref]

J. Opt. Soc. Am. B (1)

Nat. Commun. (1)

L. Dominici, M. Petrov, M. Matuszewski, D. Ballarini, M. De Giorgi, D. Colas, E. Cancellieri, B. SilvaFernandez, A. Bramati, G. Gigli, A. Kavokin, F. Laussy, and D. Sanvitto, “Real-space collapse of a polariton condensate,” Nat. Commun. 6, 9993 (2015).
[Crossref] [PubMed]

Nat. Phys. (2)

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 4651 (2007).
[Crossref]

C. Sun, S. Jia, C. Barsi, S. Rica, A. Picozzi, and J. M. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012).
[Crossref]

Nature (1)

G. Gamow, “The quantum theory of nuclear disintegration,” Nature 122, 805–806 (1928).
[Crossref]

New J. Phys. (1)

M. Crosta, S. Trillo, and A. Fratalocchi, “The Whitham approach to dispersive shocks in systems with cubicquintic nonlinearities,” New J. Phys. 14, 093019 (2012).
[Crossref]

Opt. Express (1)

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express oe14, 10095 (2006).

Opt. Lett. (3)

Phys. Rev. A (6)

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Nonlinear Gamow vectors, shock waves, and irreversibility in optically nonlocal media,” Phys. Rev. A 92, 023801 (2015).
[Crossref]

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, “Shock wave in thermal lensing,” Phys. Rev. A 87, 053811 (2013).
[Crossref]

C. Conti, S. Stark, P. S. J. Russell, and F. Biancalana, “Multiple hydrodynamical shocks induced by Raman effect in photonic crystal fibres,” Phys. Rev. A,  82, 013838 (2010).
[Crossref]

M. Conforti, F. Baronio, and S. Trillo, “Resonant radiation shed by dispersive shock waves,” Phys. Rev. A 89, 013807 (2014).
[Crossref]

G. A. El, A. Gammal, E. G. Khamis, R.A. Kraenkel, and A.M. Kamchatnov, “Theory of optical dispersive shock waves in photorefractive media,” Phys. Rev. A 76, 053813 (2007).
[Crossref]

M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels, and V. Schweikhard, “Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics,” Phys. Rev. A 74, 023623 (2006).
[Crossref]

Phys. Rev. E (2)

A. Molinari and C. Daraio, “Stationary shocks in periodic highly nonlinear granular chains,” Phys. Rev. E 80, 056602 (2009).
[Crossref]

A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, “Stationary shocks in perioAsymptotic soliton train solutions of the defocusing nonlinear Schrdinger equation,” Phys. Rev. E 66, 036609 (2002).
[Crossref]

Phys. Rev. Lett (1)

N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett.,  99, 043903 (2007).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

M. A. Hoefer, M. J. Ablowitz, and P. Engels, “Piston dispersive shock wave problem,” Phys. Rev. Lett. 100, 084504 (2008).
[Crossref] [PubMed]

R. J. Taylor, D. R. Baker, and H. Ikezi, “Observation of collisionless electrostatic shocks,” Phys. Rev. Lett. 24, 206209 (1970).
[Crossref]

L. Romagnani, S. V. Bulanov, M. Borghesi, P. Audebert, J. C. Gauthier, K. Löwenbrück, A. J. Mackinnon, P. Patel, G. Pretzler, T. Toncian, and O. Willi, “Observation of collisionless shocks in laser- plasma experiments,” Phys. Rev. Lett. 101, 025004 (2008).
[Crossref]

Sci. Rep. (1)

S. Gentilini, M. C. Braidotti, G. Marcucci, E. DelRe, and C. Conti, “Physical realization of the Glauber quantum oscillator,” Sci. Rep. 50, 15816 (2015).
[Crossref]

Sov. Phys. JEPT (1)

A. V. Gurevich and L. P. Pitaevskii, “Nonstationary structure of a collisionless shock wave,” Sov. Phys. JEPT 38, 291 (1974).

Zh. Eksp. Teor. Fiz. (1)

L. A. Khalfin, “Monotonicity of the decay of unstable particles corresponding to an n-order,” Zh. Eksp. Teor. Fiz. 33, 1371 (1957).

Other (5)

R. J. Glauber, “Amplifiers, attenuators, and Schrdinger’s cat,” in Conference on Frontiers of Quantum Optics (1985).

J. C. Bronski and D. W. McLaughlin, Singular Limits of Dispersive Waves (Plenum, 1994).

G. B. Whitham, Linear and Nonlinear Waves (Wiley, 1999).
[Crossref]

A. Alberucci, C. P. Jisha, and G. Assanto, “Breather solitons in highly nonlocal media,” arXiv:1602.01722 [physics.optics], (2016).

M. D. Maiden, N. K. Lowman, D. V. Anderson, M. E. Schubert, and M. A. Hoefer, “Frictionless dispersive hydrodynamics of Stokes flows,” Arxiv, 1512.09240 (2015).

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Figures (3)

Fig. 1
Fig. 1 First five even Gamow eigenstates square modulus (a) and their tilt (b), calculated as xφ, (n = 0, 2, 4, 6 and 8).
Fig. 2
Fig. 2 (a) Experimental setup scheme for the laser beam transmission in a RhB sample. L1 is the cylindrical lens used to make the beam elliptical and to focus it in the sample. L2 is the spherical lens used to collect the beam with a CCD camera. The inset shows the CCD output for PMKS ≃ 1W. (b) CCD image of the light beam at laser powers (PMKS = 2W); the bottom panel shows the normalized intensity profile at the maximum waist along Y (Y = 0). (c) as in (b) for (PMKS = 4W). The red line indicates data reported in Fig. 3(a). (d) Analytical solution, Eq. (11), obtained changing the power P Gaussianly in the y direction; the bottom panel shows the slice of panel (c) at y = 0, i.e., Eq. (11) square modulus for W = 1.5 and γ ≃ 12. (e) As in (d) but for higher powers; the bottom panel shows Eq. (11) square modulus for W = 1.5 and γ ≃ 40.
Fig. 3
Fig. 3 (a) Log-scale normalized intensity as a function of power (red line in Fig. 2(a)). The slopes of the straight lines give the Gamow vectors decay rates (γ1 = −8 ± 0.4 and γ2 = 1.6 ± 0.1). Their quantized ratio is 5.0 ± 0.4 as expected from theory (see [27]). The units of axes in Fig. 3(a) correspond to scaled x and y direction in Fig. 2(a). (b) Intensity oscillations for different power values; data have been shifted on the vertical axes to allow a clearer view of the oscillations. (c) Measured undulation period T as a function of power; continuous line is the fit function T P 4 as expected by the theory; (d) as in (c) with P M K S 1 / 4 as abscissa axes.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

2 i k A Z + 2 A X 2 + 2 k 2 Δ n [ | A | 2 ] ( X ) n 0 A = 0
Δ n [ I ] ( X ) = n 2 G ( X X ) I ( X ) d X .
i ψ z + 1 2 2 ψ x 2 P K ( x ) * | ψ ( x ) | 2 ψ = 0 ,
f n ± ( x ) = e ± i π / 8 ( ± i γ 2 n n ! π ) 1 / 2 e i γ 2 x 2 H n ( ± i γ x ) ,
H ^ r h o ( p ^ , x ^ ) = p ^ 2 2 1 2 γ 2 x ^ 2
i z ϕ = H ^ r h o ϕ = n = 0 E n | f n f n + | ϕ .
ϕ ( x ) = n = 0 N Γ n f n f n + | ψ ( x , 0 ) .
γ = P / π σ 2 .
H ^ r h o ( p , i p ) = p 2 2 + 1 2 γ 2 p 2 = H ^ r h o ( i x , x ) .
ϕ G W ( x ) = n = 0 N Γ n f n + | ψ ( x , 0 ) rect W ( x ) ,
ψ ˜ ( k x ) = [ f 0 ( x ) ] = ( 1 4 + i 4 ) e i k x 2 2 γ ( i γ π ) 1 / 4 W × × { Erf [ ( 1 2 i 2 ) ( k x W γ ) γ ] + Erf [ ( 1 2 i 2 ) ( k x W γ ) γ ] } .

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