Abstract

We study the interaction of temporal incoherent solitons sustained by a highly noninstantaneous (Raman-like) nonlinear response. The incoherent solitons exhibit a nonmutual interaction, which can be either attractive or repulsive depending on their relative initial distance. The analysis reveals that incoherent solitons exhibit a long-range interaction in frequency space, which is in contrast with the expected spectral short-range interaction described by the usual approach based on the Raman-like spectral gain curve. Both phenomena of anomalous interaction and spectral long-range behavior of incoherent solitons are described in detail by a long-range Vlasov equation.

© 2014 Optical Society of America

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2013

E. Turitsyna, S. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. Babin, E. Podivilov, D. Churkin, G. Falkovich, and S. Turitsyn, Nat. Photonics 7, 783 (2013).
[CrossRef]

B. Barviau, J. Garnier, G. Xu, B. Kibler, G. Millot, and A. Picozzi, Phys. Rev. A 87, 035803 (2013).
[CrossRef]

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

2012

S. Derevyanko and E. Small, Phys. Rev. A 85, 053816 (2012).
[CrossRef]

M. Erkintalo, M. Surakka, J. Turunen, A. T. Friberg, and G. Genty, Opt. Lett. 37, 169 (2012).
[CrossRef]

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, Opt. Lett. 37, 2472 (2012).
[CrossRef]

J. Garnier, M. Lisak, and A. Picozzi, J. Opt. Soc. Am. B 29, 2229 (2012).
[CrossRef]

J. Laurie, U. Bortolozzo, S. Nazarenko, and S. Residori, Phys. Rep. 514, 121 (2012).
[CrossRef]

C. Michel, B. Kibler, J. Garnier, and A. Picozzi, Phys. Rev. A 86, 041801(R) (2012).
[CrossRef]

2011

A. Picozzi and J. Garnier, Phys. Rev. Lett. 107, 233901 (2011).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, Phys. Rev. A 83, 033838 (2011).
[CrossRef]

2010

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

J. Garnier and A. Picozzi, Phys. Rev. A 81, 033831 (2010).
[CrossRef]

A. Piskarskas, V. Pyragaite, and A. Stabinis, Phys. Rev. A 82, 053817 (2010).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

C. Conti, M. Schmidt, P. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

2009

A. Hause and F. Mitschke, Phys. Rev. A 80, 063824 (2009).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, Phys. Rev. A 79, 063840 (2009).
[CrossRef]

Y. Silberberg, Y. Lahini, Y. Bromberg, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

2008

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, Nat. Photonics 2, 371 (2008).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. E. Semenov, Opt. Commun. 281, 3919 (2008).
[CrossRef]

2007

2006

2005

T.-S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

2004

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

W. Królikowski, O. Bang, and J. Wyller, Phys. Rev. E 70, 036617 (2004).
[CrossRef]

2003

M. Onorato, A. Osborne, R. Fedele, and M. Serio, Phys. Rev. E 67, 046305 (2003).
[CrossRef]

2002

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

2000

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

1997

A. Schwache and F. Mitschke, Phys. Rev. E 55, 7720 (1997).
[CrossRef]

1996

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

1975

A. Hasegawa, Phys. Fluids 18, 77 (1975).
[CrossRef]

Anderson, D.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. E. Semenov, Opt. Commun. 281, 3919 (2008).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Angelani, L.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Aschieri, P.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, Phys. Rev. A 83, 033838 (2011).
[CrossRef]

Babin, S.

E. Turitsyna, S. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. Babin, E. Podivilov, D. Churkin, G. Falkovich, and S. Turitsyn, Nat. Photonics 7, 783 (2013).
[CrossRef]

S. Babin, D. Churkin, A. Ismagulov, S. Kablukov, and E. Podivilov, J. Opt. Soc. Am. B 24, 1729 (2007).
[CrossRef]

Bang, O.

W. Królikowski, O. Bang, and J. Wyller, Phys. Rev. E 70, 036617 (2004).
[CrossRef]

Barviau, B.

B. Barviau, J. Garnier, G. Xu, B. Kibler, G. Millot, and A. Picozzi, Phys. Rev. A 87, 035803 (2013).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, Phys. Rev. A 79, 063840 (2009).
[CrossRef]

Biancalana, F.

C. Conti, M. Schmidt, P. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Bortolozzo, U.

J. Laurie, U. Bortolozzo, S. Nazarenko, and S. Residori, Phys. Rep. 514, 121 (2012).
[CrossRef]

Bromberg, Y.

Y. Silberberg, Y. Lahini, Y. Bromberg, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Chen, Z.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

Christodoulides, D. N.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

Churkin, D.

E. Turitsyna, S. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. Babin, E. Podivilov, D. Churkin, G. Falkovich, and S. Turitsyn, Nat. Photonics 7, 783 (2013).
[CrossRef]

S. Babin, D. Churkin, A. Ismagulov, S. Kablukov, and E. Podivilov, J. Opt. Soc. Am. B 24, 1729 (2007).
[CrossRef]

Cohen, O.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, Nat. Photonics 2, 371 (2008).
[CrossRef]

Conti, C.

C. Conti, M. Schmidt, P. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Coskun, T.

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

Derevyanko, S.

S. Derevyanko and E. Small, Phys. Rev. A 85, 053816 (2012).
[CrossRef]

Doya, V.

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, Phys. Rev. A 83, 033838 (2011).
[CrossRef]

Dylov, D.

D. Dylov and J. Fleischer, in Localized States in Physics: Solitons and Patterns, Part 1, (Springer, 2011), pp. 17–34.

Erkintalo, M.

Falkovich, G.

E. Turitsyna, S. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. Babin, E. Podivilov, D. Churkin, G. Falkovich, and S. Turitsyn, Nat. Photonics 7, 783 (2013).
[CrossRef]

Fedele, R.

M. Onorato, A. Osborne, R. Fedele, and M. Serio, Phys. Rev. E 67, 046305 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Fischer, B.

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

Fleischer, J.

D. Dylov and J. Fleischer, in Localized States in Physics: Solitons and Patterns, Part 1, (Springer, 2011), pp. 17–34.

Fratalocchi, A.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Friberg, A. T.

Garnier, J.

B. Barviau, J. Garnier, G. Xu, B. Kibler, G. Millot, and A. Picozzi, Phys. Rev. A 87, 035803 (2013).
[CrossRef]

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, Opt. Lett. 37, 2472 (2012).
[CrossRef]

J. Garnier, M. Lisak, and A. Picozzi, J. Opt. Soc. Am. B 29, 2229 (2012).
[CrossRef]

C. Michel, B. Kibler, J. Garnier, and A. Picozzi, Phys. Rev. A 86, 041801(R) (2012).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, Phys. Rev. A 83, 033838 (2011).
[CrossRef]

A. Picozzi and J. Garnier, Phys. Rev. Lett. 107, 233901 (2011).
[CrossRef]

J. Garnier and A. Picozzi, Phys. Rev. A 81, 033831 (2010).
[CrossRef]

Gat, O.

R. Weill, B. Fischer, and O. Gat, Phys. Rev. Lett. 104, 173901 (2010).
[CrossRef]

Genty, G.

Haelterman, M.

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

Hall, B.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Phys. Fluids 18, 77 (1975).
[CrossRef]

Hause, A.

A. Hause and F. Mitschke, Phys. Rev. A 80, 063824 (2009).
[CrossRef]

Helczynski-Wolf, L.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. E. Semenov, Opt. Commun. 281, 3919 (2008).
[CrossRef]

Ismagulov, A.

Jauslin, H. R.

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

Kablukov, S.

Kibler, B.

B. Barviau, J. Garnier, G. Xu, B. Kibler, G. Millot, and A. Picozzi, Phys. Rev. A 87, 035803 (2013).
[CrossRef]

C. Michel, B. Kibler, J. Garnier, and A. Picozzi, Phys. Rev. A 86, 041801(R) (2012).
[CrossRef]

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, Opt. Lett. 37, 2472 (2012).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, Phys. Rev. A 79, 063840 (2009).
[CrossRef]

Kivshar, Y. S.

T.-S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

Knight, J. C.

Królikowski, W.

W. Królikowski, O. Bang, and J. Wyller, Phys. Rev. E 70, 036617 (2004).
[CrossRef]

Ku, T.-S.

T.-S. Ku, M.-F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

Lahini, Y.

Y. Silberberg, Y. Lahini, Y. Bromberg, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Laurie, J.

J. Laurie, U. Bortolozzo, S. Nazarenko, and S. Residori, Phys. Rep. 514, 121 (2012).
[CrossRef]

Leonetti, M.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Lisak, M.

J. Garnier, M. Lisak, and A. Picozzi, J. Opt. Soc. Am. B 29, 2229 (2012).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. E. Semenov, Opt. Commun. 281, 3919 (2008).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602 (2002).
[CrossRef]

Luan, F.

Michel, C.

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, Opt. Lett. 37, 2472 (2012).
[CrossRef]

C. Michel, B. Kibler, J. Garnier, and A. Picozzi, Phys. Rev. A 86, 041801(R) (2012).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, Phys. Rev. A 83, 033838 (2011).
[CrossRef]

Millot, G.

B. Barviau, J. Garnier, G. Xu, B. Kibler, G. Millot, and A. Picozzi, Phys. Rev. A 87, 035803 (2013).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

Mitchell, M.

M. Mitchell, Z. Chen, M. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef]

Mitschke, F.

A. Hause and F. Mitschke, Phys. Rev. A 80, 063824 (2009).
[CrossRef]

A. Schwache and F. Mitschke, Phys. Rev. E 55, 7720 (1997).
[CrossRef]

Morandotti, R.

Y. Silberberg, Y. Lahini, Y. Bromberg, E. Small, and R. Morandotti, Phys. Rev. Lett. 102, 233904 (2009).
[CrossRef]

Nazarenko, S.

J. Laurie, U. Bortolozzo, S. Nazarenko, and S. Residori, Phys. Rep. 514, 121 (2012).
[CrossRef]

Onorato, M.

M. Onorato, A. Osborne, R. Fedele, and M. Serio, Phys. Rev. E 67, 046305 (2003).
[CrossRef]

Osborne, A.

M. Onorato, A. Osborne, R. Fedele, and M. Serio, Phys. Rev. E 67, 046305 (2003).
[CrossRef]

Picozzi, A.

B. Barviau, J. Garnier, G. Xu, B. Kibler, G. Millot, and A. Picozzi, Phys. Rev. A 87, 035803 (2013).
[CrossRef]

J. Garnier, G. Xu, S. Trillo, and A. Picozzi, Phys. Rev. Lett. 111, 113902 (2013).
[CrossRef]

C. Michel, B. Kibler, J. Garnier, and A. Picozzi, Phys. Rev. A 86, 041801(R) (2012).
[CrossRef]

J. Garnier, M. Lisak, and A. Picozzi, J. Opt. Soc. Am. B 29, 2229 (2012).
[CrossRef]

B. Kibler, C. Michel, J. Garnier, and A. Picozzi, Opt. Lett. 37, 2472 (2012).
[CrossRef]

P. Aschieri, J. Garnier, C. Michel, V. Doya, and A. Picozzi, Phys. Rev. A 83, 033838 (2011).
[CrossRef]

A. Picozzi and J. Garnier, Phys. Rev. Lett. 107, 233901 (2011).
[CrossRef]

J. Garnier and A. Picozzi, Phys. Rev. A 81, 033831 (2010).
[CrossRef]

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

B. Barviau, B. Kibler, and A. Picozzi, Phys. Rev. A 79, 063840 (2009).
[CrossRef]

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

A. Picozzi and M. Haelterman, Phys. Rev. Lett. 92, 103901 (2004).
[CrossRef]

Piskarskas, A.

A. Piskarskas, V. Pyragaite, and A. Stabinis, Phys. Rev. A 82, 053817 (2010).
[CrossRef]

Pitois, S.

A. Picozzi, S. Pitois, and G. Millot, Phys. Rev. Lett. 101, 093901 (2008).
[CrossRef]

Podivilov, E.

E. Turitsyna, S. Smirnov, S. Sugavanam, N. Tarasov, X. Shu, S. Babin, E. Podivilov, D. Churkin, G. Falkovich, and S. Turitsyn, Nat. Photonics 7, 783 (2013).
[CrossRef]

S. Babin, D. Churkin, A. Ismagulov, S. Kablukov, and E. Podivilov, J. Opt. Soc. Am. B 24, 1729 (2007).
[CrossRef]

Pyragaite, V.

A. Piskarskas, V. Pyragaite, and A. Stabinis, Phys. Rev. A 82, 053817 (2010).
[CrossRef]

Randoux, S.

P. Suret, S. Randoux, H. R. Jauslin, and A. Picozzi, Phys. Rev. Lett. 104, 054101 (2010).
[CrossRef]

Residori, S.

J. Laurie, U. Bortolozzo, S. Nazarenko, and S. Residori, Phys. Rep. 514, 121 (2012).
[CrossRef]

Rotschild, C.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, Nat. Photonics 2, 371 (2008).
[CrossRef]

Ruocco, G.

C. Conti, M. Leonetti, A. Fratalocchi, L. Angelani, and G. Ruocco, Phys. Rev. Lett. 101, 143901 (2008).
[CrossRef]

Russell, P.

C. Conti, M. Schmidt, P. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Schmidt, M.

C. Conti, M. Schmidt, P. Russell, and F. Biancalana, Phys. Rev. Lett. 105, 263902 (2010).
[CrossRef]

Schwache, A.

A. Schwache and F. Mitschke, Phys. Rev. E 55, 7720 (1997).
[CrossRef]

Schwartz, T.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, Nat. Photonics 2, 371 (2008).
[CrossRef]

Segev, M.

C. Rotschild, T. Schwartz, O. Cohen, and M. Segev, Nat. Photonics 2, 371 (2008).
[CrossRef]

M. Soljacic, M. Segev, T. Coskun, D. N. Christodoulides, and A. Vishwanath, Phys. Rev. Lett. 84, 467 (2000).
[CrossRef]

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[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Simulations of the NLS Eq. (1) (a) and Vlasov Eq. (2) (b), showing the attractive interaction between two ISs, whose initial distance is τd=τR. The ISs spiral around each other and eventually fuse into a single IS. From left to right z=0, 190, 300, 380; 450, 500, 550, 600; 790, 820, 850, 880; 2140, 2150, 2160, 2170 (T0=2048). A quantitative agreement is obtained between NLS and Vlasov simulations, without adjustable parameters. The inset shows the antisymmetric spectral deformation, G(ω,z0)=zS(ω) (reminiscent of a “gain spectrum”), with a spectral long-range bandwidth Δωint1 (z0=2170).

Fig. 2.
Fig. 2.

Evolutions of the intensities N1,2(t,z) versus z showing the attractive [1st row: τd=τR, (a) NLS, (b) Vlasov] and the repulsive [2nd row: τd=1.6τR, (c) NLS, (d) Vlasov] interaction of a pair of ISs (T0=2048).

Fig. 3.
Fig. 3.

Gaussian intensity profiles Njs(τ) (black) and corresponding potentials U2s(τ) (dashed-green), U1s(τ) (green) for τd=τR (a) and τd=1.6τR (b). The dashed blue lines correspond to the IS solutions obtained from Vlasov simulations [Eq. (2)]. The arrows indicate the “particle motions” of IS2 in U1s(τ). [For visibility reasons, we plotted Njs(τ)/2].

Fig. 4.
Fig. 4.

(a) and (b) Evolutions of the temporal intensity profiles Nj(t,z) versus z for two ISs initially separated by the threshold distance, τd=τth, for the Gaussian-like response (τth1.435τR, first column) and Raman-like response (τth0.831τR, second column). (c) and (d) ISj profiles Njs(τ) and potentials Ujs(τ), U˜s(τ) (T0=4096, for visibility reasons we plotted Njs(τ)/2).

Equations (6)

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

izψ+sttψ+σψR(tt)|ψ|2(z,t)dt=0.
znω(t,z)=2ωtnω(t,z)tU(t,z)ωnω(t,z),
ξnj=2ΩτnjτU˜(τ,ξ)Ωnj.
12ξ2Tj(ξ)=ξPj(ξ)=Sj(ξ)+Xj(ξ),
12ξ2Tj(ξ)=ξPj(ξ)=τU3js(τ)Njs(τ)dτ
12ξ2T2(ξ)=ξP2(ξ)N2τU1s|τ=τd.

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