Shock wave far-field in ordered and disordered nonlocal media

doi:10.1364/OE.20.027369

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Shock wave far-field in ordered and disordered nonlocal media

S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti »View Author Affiliations

Optics Express, Vol. 20, Issue 24, pp. 27369-27375 (2012)
http://dx.doi.org/10.1364/OE.20.027369

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▸ Article Outline
– Abstract
1. Introduction
2. Sample preparation and experimental set-up
3. Shock wave far field in the ordered case
4. Shock wave far field in the disordered case
5. Conclusion
– References and links

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Abstract

We investigate the far field of a spatial dispersive shock wave generated from a Gaussian beam propagating in nonlinear nonlocal colloidal disordered media. The interplay between nonlinearity and structural randomness is quantified in terms of the threshold power for the occurrence of the shock wave.

1. Introduction

Recent experimental investigations have shown that the wavevector content of an optical beam in the presence of pronounced nonlinear effects may signal the occurrence of specific dynamic phases and self organization for light [1

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

, 2

W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007). [CrossRef]

]. The way these phenomena are affected by disorder is not yet investigated. Following recent theoretical investigations concerning the complexity of nonlinear waves in the presence of structural randomness [3

C. Conti and L. Leuzzi, “Complexity of waves in nonlinear disordered media,” Phys. Rev. B 83, 134204 (2011). [CrossRef]

–9

I. V. Shadrivov, K. Y. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of Anderson localized states in nonlinear random media,” Phys. Rev. Lett. 104, 123902 (2010). [CrossRef] [PubMed]

], experimental efforts are focused on finding a coherent and nonlinearly driven effect that can be largely affected even by a small amount of disorder, which must be in some way controllable [10

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in onedimensional disorded photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008). [CrossRef] [PubMed]

–13

L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, and M. Segev, “Disorder-enhanced transport in photonics quasicrystals,” Science 332, 1541–1544 (2011). [CrossRef] [PubMed]

Here we consider the generation and evolution of dispersive shock waves (DSWs) in a thermal nonlocal nonlinear liquid medium [14

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

–16

N. Ghofraniha, L. Santamaria Amato, V. Folli, C. Conti, and E. DelRe, “Formation and propagation of shock waves in nonlocal media,” Poster number JM5A.44 presented at Nonlinear Photonics Conference, Colorado Springs, CO, USA, 17–21 June 2012.

] into which we introduce controllable disorder using a colloidal dispersion of low index contrast dielectric particles. DSWs, or undular bores, are observed in nonlinear optics in systems described by universal models, such as the nonlinear Schrödinger (NLS) equation when the hydrodynamical approximation holds true [17

A. Gurevich and L. P. Pitaevskii, “Stationary structure of a collisionless shock wave,” Sov.Phys. JETP 38, 291–297 (1973).

–21

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

]. In the spatial case here considered, this corresponds to the regime when the nonlinear length is much smaller than the diffraction and loss (absorption and scattering) lengths.

Previous experiments in nonlinear nonlocal colloidal samples have allowed the direct detection of the shock formation point through fluorescence imaging [15

N. Ghofraniha, L. Santamaria Amato, V. Folli, S. Trillo, E. DelRe, and C. Conti, “Measurement of scaling laws for shock waves in thermal nonlocal media,” Opt. Lett. 37, 2325–2327 (2012). [CrossRef] [PubMed]

, 16

]. Direct imaging of the beam after the shock also indicates the formation of post-shock annular features that change dynamically during propagation. This suggests that the shock alters in a non-trivial way the wavevector spectrum of the beam. To detect the dynamics of the wavevector spectrum generated by the shock and its relation to the underlying physics of the nonlocal nonlinearity, we performed far field experiments on post-shock propagation beams. We consider the DSW generated by a circular Gaussian beam, and investigate the far field distribution after propagation in the nonlinear nonlocal colloidal samples. We first show that in the absence of disorder, the far field can be used to determine a threshold for the occurrence of the DSW, and then we investigate how this threshold is affected by increasing the strength of randomness.

2. Sample preparation and experimental set-up

As previously reported [14

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

–16

], an ordered nonlocal medium is realized by aqueous suspensions of RhodamineB (RhB). We consider different RhB concentrations, c_RhB, ranging from 0.05 to 0.5 mM (samples series A). The disordered samples (samples series B) are obtained by adding mono-dispersed 1μm diameter silica (SiO₂) spheres with concentrations, c_SiO₂, ranging from 0.005 w/w to 0.035 w/w, in units of weight of silica particles over suspension weight. By varying c_RhB and c_SiO₂ we tailor the amount of nonlinearity, as detailed below, and the strength of disorder.

The samples are put in 1mm×1cm×3cm glass cells with propagation direction z along the 1 mm vertical direction, which is set parallel to the gravity vector to moderate the effect of heat convection.

A continuous-wave solid state laser at wavelength λ = 532 nm is focused on the input facet of the cell at the beam radius waist w₀ = 50μm. A lens of focal length f = 75mm, placed at the same distance f from the output face of the cell and from the detector, allows the measurements of the spatial Fourier transform of the transmitted beam by imaging on a 1024 × 1032 pixel CCD camera. Having fixed the geometrical configuration of the experiment, we proceeded to calibrate the CCD by fitting with the Airy function the experimentally obtained Fourier transform of a 500μm diameter pinhole, placed on the exit face of the cell. In this way we were able to obtain the conversion from pixel to radians. The angular spreading θ was then translated in terms of transverse (with respect to z) wavevectors k_T as k_T = (2π/λ)sin(θ).

The DSW generation in nonlocal media was studied in previous works [14

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

–16

]. In Fig. 1, we show images of top fluorescence emission for three different input powers as retrieved by imaging top fluorescence emission through an MZ16 Leica microscope placed above the sample top surface and recorded by the CCD camera.

Fig. 1 Beam propagation as observed from top fluorescence emission for three different input powers in the ordered case, c_SiO₂ = 0 (c_RhB = 0.1mM), (a–c) and disordered case, c_SiO₂ = 0.03w/w (c_RhB = 0.1mM), (d–f). Note that the propagation regime reported in panel (a) is linear since the laser power is too low to activate the nonlinear effect.

Upper panels (a–c) show the propagation of the beam in the ordered sample, i.e. at c_SiO₂ = 0 (c_RhB = 0.1mM), while the bottom panels (d–f) the propagation within a disordered sample at c_SiO₂ = 0.03w/w (c_RhB = 0.1mM). The shock formation is characterized by the fact that the laser beam displays regularizing oscillation. The panels (a–c) show the shock formation as the oscillations are clearly visible with the increase of power, nevertheless another phenomenon can be noticed to accompanying the oscillations: the diffraction enhancement resulting in a funnel shape with the increase of input power. The far field manifests the generation of characteristic rings (Fig. 2), whose size depends on dye concentration, strength of disorder and laser power.

Fig. 2 Far field intesity of the transmitted beam for different input powers P and different concentrations c_RhB (ordered case). Superimposed curves give the k_y = 0 section profile. Similarly to Fig. 1(a), here the top panels (a–c) show a linear regime since the laser power is too low to activate the nonlinear effect.

3. Shock wave far field in the ordered case

We first consider the ordered case: we investigate the occurrence of DSW when varying input laser power P for different concentrations of RhB, c_RhB, in the pure dye solutions (samples series A, c_SiO₂ = 0).

In Fig. 2 we show images of the intensity of the far field of the transmitted beam. The spectra display a progressive enlargement in the wavevector plane with P. In order to quantify such an effect we make a radial average of the 2-D collected profiles and we measure the angular aperture θ as the full width at half maximum of the peak, as long as a single peak is observed; and as the distance between the two leading peaks, when these become distinguishable after the formation of the shock wave.

We stress that when increasing c_RhB, the amount of nonlinearity increases because of the enhanced absorption. Figure 3(a) shows the angular aperture θ versus P for different RhB concentrations. Two effects are evident: 1) by increasing P, θ increases because of the speedup of the shock formation by the augmented nonlinearity. Moreover the slope of the curves increases due to the fact that nonlinearity grows with c_RhB, hence higher angular aperture is obtained at the same power P. 2) There exists a threshold power, P_th, above which θ becomes linear with P, and this threshold decreases when increasing c_RhB (Fig. 3(a)).

Fig. 3 (a) Measured aperture angle vs laser power P in the ordered case for different c_RhB concentrations; (b) P vs c_RhB: the filled circles are the experimental threshold power as retrieved from panel (a) (dot-dashed lines is a guide for the eye).

In previous works [14

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

–16

], DSWs has been investigated by defining the point of shock formation along z, as the point of the maximum steepness in intensity profiles; in that case the threshold of the shock was defined as the power level when the shock point becomes substantially dependent of the input fluence. Conversely, here, we define a threshold power, P_th as the power at which the angular aperture starts to linearly increase (indicated by arrows in Fig. 3(a)). We report the threshold power values in the diagram of P versus c_RhB of Fig. 3(b).

In previous works [14

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

, 15

] nonlocality has been shown to hamper shock formation; however in our case, as shown in Fig. 2, nonlocality does not prevent this phenomenon and can be retained as a perturbation if compared with to disorder introduced by silica beads, which can completely inhibit DSW as discussing in the following section.

4. Shock wave far field in the disordered case

We consider the interplay between disorder and nonlinearity in the DSW formation by dispersing the 1μm diameter SiO₂ spheres in two different aqueous solutions of RhB at c_RhB = 0.05 mM and c_RhB = 0.1 mM (samples series B).

Figure 4 shows the spectral profiles for different c_SiO₂ and c_RhB at fixed power P = 40 mW. At this input laser power, no shock formation emerges from the spectra at the lowest c_RhB (panels (a–c)). On the other hand, the spectra reported in Fig. 4(d–f) show the characteristic rings signature of DSW formation. Note that the circular symmetry is lost because of the inhomogeneous refractive index and that DSW is more evident in some directions than in others. In other words, the shock wave has a partially randomized spatial distribution. The DSW is more evident at the second higher considered RhB concentration, where the nonlinearity is enhanced.

Fig. 4 Spectra of transmitted beam at fixed input power P=42 mW and different c_RhB and c_SiO₂.

In Fig. 5 we show the case P = 140 mW. Even at the lowest c_RhB there are evidences of rings, top panels in Fig. 5. DSW formation is more pronounced in the case of c_RhB = 0.1 mM (Fig. 5(d–f)).

Fig. 5 The same of Fig. 4 at input power P=130 mW

The angular aperture θ vs power is shown in Fig. 6(a) and 6(b) for different c_SiO₂, when c_RhB = 0.05 mM and 0.1 mM respectively. The curves provide information on the competition between nonlinearity and disorder. For the lowest concentration of dye (c_RhB = 0.05mM), as shown in Fig. 6(a), disorder prevails over nonlinearity, preventing the occurrence of DSW and no threshold is observed (compare with the ordered case, circles in Fig. 6(a)). In Fig. 6(b) we show the results at higher dye concentration c_RhB = 0.1 mM; we observe the threshold behavior as in the ordered case. However, the hampering effect of randomness is revealed by the fact that the aperture angle at a given power is smaller than the case c_SiO₂ = 0; in addition, when c_SiO₂ is greater than 0.017 w/w no shock formation is observed as shown for c_SiO₂ = 0.035 w/w in Fig. 6(b). Similarly to the ordered case, we extract from Fig. 6(b), a diagram of the threshold power P_th versus c_SiO₂ (Fig. 6(c)).

Fig. 6 Measured aperture angle θ vs laser power P for different SiO₂ dispersions in two RhB solutions: (a) c_RhB = 0.05mM and (b) c_RhB = 0.1mM; (c) threshold power P vs disorder concentrations c_SiO₂ at fixed c_RhB = 0.1mM: dots are the experimental data, dot dashed lines is a guide for the eye, dashed line is the boundary between shock and no shock regimes.

In order to furnish quantitative arguments to support the reported results, we estimate the ratio between the nonlinear and the random contribution to the refractive index. We expect that disorder completely washes out the shock phenomenon when the volume average of the index perturbation due to the silica spheres is comparable with the index perturbation due to nonlinearity. The nonlinear perturbation can be estimated of the order 10⁻³. By considering the refractive index of the silica particles (n_SiO₂ ≃ 1.4) and their concentration, this index perturbation is found at the concentration c_SiO₂ ≃ 0.03w/w, consistently with data of Fig. 6(c).

5. Conclusion

The analysis of the far field of dispersive shock waves generated in a nonlocal nonlinear medium allows the investigation of the interplay between disorder-induced and spontaneous wave breaking phenomena. In the ordered case, we have shown that the angular aperture of the beam versus power provides a simple way to determine the threshold for the generation of the shock; this threshold power decreases with the amount of nonlinearity, controlled by dye concentrations in our samples.

In the presence of disorder, loss of transverse spatial coherence prevents the occurrence of the spontaneous wave breaking. The shock is observed above a certain amount of nonlinearity and below a critical strength of disorder. We show these phenomena can also be quantified by measuring the threshold power in terms of the strength of randomness, controlled by the concentrations of colloidal suspension which is used to introduce scattering.

Investigating the spatial spectral distribution of waves in the presence of strong nonlinearity and disorder may lead to future developments in assessing dynamic phase transitions and structured landscapes for light.

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Community’s Seventh Framework Program (FP7/2007–2013)/ERC grant agreement n. 201766, from the Italian Ministry of Research (MIUR) through the PRIN project no.2009P3K72Z and from the Italian Ministry of Education, University and Research under the Basic Research Investigation Fund (FIRB/2008) program/CINECA grant code RBFR08M3P4 and RBFR08E7VA. We thank M. Deen Islam for the technical assistance.

References and links

1.	C. Sun, J. Schu, C. Barsi, S. Rica, A. Picozzi, and J. W. Fleischer, “Observation of the kinetic condensation of classical waves,” Nat. Phys. 8, 471–475 (2012). [CrossRef]
2.	W. Wan, S. Jia, and J. W. Fleischer, “Dispersive superfluid-like shock waves in nonlinear optics,” Nat. Phys. 3, 46–51 (2007). [CrossRef]
3.	C. Conti and L. Leuzzi, “Complexity of waves in nonlinear disordered media,” Phys. Rev. B 83, 134204 (2011). [CrossRef]
4.	S. A. Gredeskul and Y. S. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep. 216, 1–61 (1992). [CrossRef]
5.	Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton percolation in random optical lattices,” Opt. Express 15, 12409–12417 (2007). [CrossRef] [PubMed]
6.	Y. S. Kivshar, S. A. Gredeskul, A. Sanchez, and L. Vazquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett. 64, 1693–1696 (1990). [CrossRef] [PubMed]
7.	A. Pikovsky and S. Fishman, “Scaling properties of weak chao in nonlinear disordered lattcies,” Phys. Rev. E 83, 025201 (2011). [CrossRef]
8.	Z. V. Vardeny and A. Nahata, “Photonic quasicrystals: disorder-enhanced light transport,” Nat. Photonics 5, 453–454 (2011). [CrossRef]
9.	I. V. Shadrivov, K. Y. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of Anderson localized states in nonlinear random media,” Phys. Rev. Lett. 104, 123902 (2010). [CrossRef] [PubMed]
10.	Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in onedimensional disorded photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008). [CrossRef] [PubMed]
11.	L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011). [CrossRef] [PubMed]
12.	T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007). [CrossRef] [PubMed]
13.	L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, and M. Segev, “Disorder-enhanced transport in photonics quasicrystals,” Science 332, 1541–1544 (2011). [CrossRef] [PubMed]
14.	N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett. 99, 043903 (2007). [CrossRef] [PubMed]
15.	N. Ghofraniha, L. Santamaria Amato, V. Folli, S. Trillo, E. DelRe, and C. Conti, “Measurement of scaling laws for shock waves in thermal nonlocal media,” Opt. Lett. 37, 2325–2327 (2012). [CrossRef] [PubMed]
16.	N. Ghofraniha, L. Santamaria Amato, V. Folli, C. Conti, and E. DelRe, “Formation and propagation of shock waves in nonlocal media,” Poster number JM5A.44 presented at Nonlinear Photonics Conference, Colorado Springs, CO, USA, 17–21 June 2012.
17.	A. Gurevich and L. P. Pitaevskii, “Stationary structure of a collisionless shock wave,” Sov.Phys. JETP 38, 291–297 (1973).
18.	J. C. Bronski and D. McLaughlin, Singular Limits of Dispersive Waves (Plenum, New York, 1994).
19.	A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, “Asymptotic soliton train solution of the defocusing nonlinear Schrödinger equation,” Phys. Rev. E 66, 036609 (2002). [CrossRef]
20.	C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009). [CrossRef] [PubMed]
21.	C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett. 32, 2930–2932 (2007). [CrossRef] [PubMed]

OCIS Codes
(190.0190) Nonlinear optics : Nonlinear optics
(190.3970) Nonlinear optics : Microparticle nonlinear optics
(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter

ToC Category:
Materials

History
Original Manuscript: September 4, 2012
Revised Manuscript: October 16, 2012
Manuscript Accepted: October 17, 2012
Published: November 19, 2012

Virtual Issues
Nonlinear Photonics (2012) Optics Express

Citation
S. Gentilini, N. Ghofraniha, E. DelRe, and C. Conti, "Shock wave far-field in ordered and disordered nonlocal media," Opt. Express 20, 27369-27375 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-24-27369

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References

C. Sun, J. Schu, C. Barsi, S. Rica, A. Picozzi, and J. W. 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, 46–51 (2007). [CrossRef]
C. Conti and L. Leuzzi, “Complexity of waves in nonlinear disordered media,” Phys. Rev. B83, 134204 (2011). [CrossRef]
S. A. Gredeskul and Y. S. Kivshar, “Propagation and scattering of nonlinear waves in disordered systems,” Phys. Rep.216, 1–61 (1992). [CrossRef]
Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton percolation in random optical lattices,” Opt. Express15, 12409–12417 (2007). [CrossRef] [PubMed]
Y. S. Kivshar, S. A. Gredeskul, A. Sanchez, and L. Vazquez, “Localization decay induced by strong nonlinearity in disordered systems,” Phys. Rev. Lett.64, 1693–1696 (1990). [CrossRef] [PubMed]
A. Pikovsky and S. Fishman, “Scaling properties of weak chao in nonlinear disordered lattcies,” Phys. Rev. E83, 025201 (2011). [CrossRef]
Z. V. Vardeny and A. Nahata, “Photonic quasicrystals: disorder-enhanced light transport,” Nat. Photonics5, 453–454 (2011). [CrossRef]
I. V. Shadrivov, K. Y. Bliokh, V. Freilikher, and Y. S. Kivshar, “Bistability of Anderson localized states in nonlinear random media,” Phys. Rev. Lett.104, 123902 (2010). [CrossRef] [PubMed]
Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in onedimensional disorded photonic lattices,” Phys. Rev. Lett.100, 013906 (2008). [CrossRef] [PubMed]
L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express19, 13636–13646 (2011). [CrossRef] [PubMed]
T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature446, 52–55 (2007). [CrossRef] [PubMed]
L. Levi, M. Rechtsman, B. Freedman, T. Schwartz, and M. Segev, “Disorder-enhanced transport in photonics quasicrystals,” Science332, 1541–1544 (2011). [CrossRef] [PubMed]
N. Ghofraniha, C. Conti, G. Ruocco, and S. Trillo, “Shocks in nonlocal media,” Phys. Rev. Lett.99, 043903 (2007). [CrossRef] [PubMed]
N. Ghofraniha, L. Santamaria Amato, V. Folli, S. Trillo, E. DelRe, and C. Conti, “Measurement of scaling laws for shock waves in thermal nonlocal media,” Opt. Lett.37, 2325–2327 (2012). [CrossRef] [PubMed]
N. Ghofraniha, L. Santamaria Amato, V. Folli, C. Conti, and E. DelRe, “Formation and propagation of shock waves in nonlocal media,” Poster number JM5A.44 presented at Nonlinear Photonics Conference, Colorado Springs, CO, USA, 17–21 June 2012.
A. Gurevich and L. P. Pitaevskii, “Stationary structure of a collisionless shock wave,” Sov.Phys. JETP38, 291–297 (1973).
J. C. Bronski and D. McLaughlin, Singular Limits of Dispersive Waves (Plenum, New York, 1994).
A. M. Kamchatnov, R. A. Kraenkel, and B. A. Umarov, “Asymptotic soliton train solution of the defocusing nonlinear Schrödinger equation,” Phys. Rev. E66, 036609 (2002). [CrossRef]
C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett.102, 083902 (2009). [CrossRef] [PubMed]
C. Barsi, W. Wan, C. Sun, and J. W. Fleischer, “Dispersive shock waves with nonlocal nonlinearity,” Opt. Lett.32, 2930–2932 (2007). [CrossRef] [PubMed]

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