Abstract

We consider transverse modulational instability of (2+1)-dimensional cnoidal waves of cn, dn, and sn, types that are periodic in one direction and are uniform in the other direction. The new method of stability analysis of periodic waves presented here is based on the construction of a translation matrix for a perturbation vector and on the evolution of the eigenvalues of the matrix with changes in modulation frequency and Jacobi parameter that define the degree of energy localization of the corresponding cnoidal waves. We show that the dn wave is subject to the influence of both neck and snake instabilities, the cn wave is affected by neck instability, and the sn wave suffers from snake instability in (2+1) dimensions.

© 2003 Optical Society of America

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  1. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beam in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
    [CrossRef] [PubMed]
  2. A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
    [CrossRef] [PubMed]
  3. V. Tikhonenko, J. Christou, B. Luther-Davies, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129–1131 (1996).
    [CrossRef] [PubMed]
  4. R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
    [CrossRef]
  5. R. A. Fuerst, B. L. Lawrence, W. E. Torruellas, and G. I. Stegeman, “Beam reshaping by use of spatial solitons in the quadratic nonlinear medium KTP,” Opt. Lett. 22, 19–21 (1997).
    [CrossRef] [PubMed]
  6. A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. 22, 868–870 (1997).
    [CrossRef] [PubMed]
  7. A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Symmetry-breaking instabilities of parametric spatial solitons,” Phys. Rev. E 56, R4959–R4962 (1997).
    [CrossRef]
  8. V. Tikhonenko, J. Christou, and B. Luther-Davies, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B 12, 2046–2052 (1995).
    [CrossRef]
  9. D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
    [CrossRef]
  10. S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, “Soliton algebra by vortex beam splitting,” Opt. Lett. 26, 1004–1006 (2001).
    [CrossRef]
  11. L. M. Degtyarev, V. E. Zakharov, and L. I. Rudakov, “Two examples of Langmuir wave collapse,” Sov. Phys. JETP 41, 57–61 (1975).
  12. E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Sov. Phys. JETP 67, 1583–1588 (1988).
  13. G. S. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
    [CrossRef]
  14. C. T. Law and G. A. Swartzlander, “Optical vortex solitons and the stability of dark soliton stripes,” Opt. Lett. 18, 586–588 (1993).
    [CrossRef] [PubMed]
  15. D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. S. Kivshar, “Self-focusing of plane dark solitons in nonlinear defocusing media,” Phys. Rev. E 51, 5016–5026 (1995).
    [CrossRef]
  16. K. Rypdal and J. J. Rasmussen, “Stability of solitary structures in the nonlinear Schrödinger equation,” Phys. Scr. 40, 192–201 (1989).
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  17. E. Infeld and R. Rowlands, Nonlinear Waves. Solitons and Chaos (Cambridge U. Press, Cambridge, 1990).
  18. E. A. Kuznetsov, S. L. Musher, and A. V. Shafarenko, “Collapse of acoustic waves in media with positive dispersion,” JETP Lett. 37, 241–245 (1983).
  19. Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117–195 (2000).
    [CrossRef]
  20. D. V. Skryabin and W. J. Firth, “Modulational instability of bright solitary waves in incoherently coupled nonlinear Schrödinger equations,” Phys. Rev. E 60, 1019–1029 (1999).
    [CrossRef]
  21. E. M. Wright, B. L. Lawrence, W. Torruellas, and G. Stegeman, “Stable self-trapping and ring formation in polydiacetylene para-toluene sulfonate,” Opt. Lett. 20, 2481–2483 (1995).
    [CrossRef]
  22. C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
    [CrossRef]
  23. E. Infeld and T. Lenkowska-Czerwinska, “Analysis of stability of light beams in nonlinear photorefractive media,” Phys. Rev. E 55, 6101–6106 (1997).
    [CrossRef]
  24. D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun. 148, 79–84 (1998).
    [CrossRef]
  25. A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
    [CrossRef]
  26. D.-M. Baboiu and G. I. Stegeman, “Modulational instability of a strip beam in a bulk type I quadratic medium,” Opt. Lett. 23, 31–33 (1998).
    [CrossRef]
  27. D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
    [CrossRef]
  28. A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, “Modulational instability of continuous waves and one-dimensional temporal solitons in fiber arrays,” Opt. Lett. 19, 1186–1188 (1994).
    [CrossRef] [PubMed]
  29. A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
    [CrossRef] [PubMed]
  30. V. M. Petnikova, V. V. Shuvalov, and V. A. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization, and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
    [CrossRef]
  31. F. T. Hioe, “Solitary waves for N coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 82, 1152–1155 (1999).
    [CrossRef]
  32. L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity,” Phys. Rev. A 62, 063610 (2000).
    [CrossRef]
  33. L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity,” Phys. Rev. A 62, 063611 (2000).
    [CrossRef]
  34. V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
    [CrossRef]
  35. V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Propagation of cnoidal waves in a medium with a saturable nonlinear response,” Quantum Electron. 31, 257–262 (2001).
    [CrossRef]
  36. N. Korneev, A. Apolinar-Iribe, V. A. Vysloukh, and M. A. Basurto-Pensado, “Self-compression of 1+1D cnoidal wave in photorefractive BTO crystal: an experimental evidence,” Opt. Commun. 197, 209–215 (2001).
    [CrossRef]
  37. V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Self-frequency shift of cnoidal waves in a medium with delayed nonlinear response,” J. Opt. Soc. Am. B 18, 1127–1136 (2001).
    [CrossRef]
  38. V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal waves compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
    [CrossRef]
  39. V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “On the possibility of suppression of the self-frequency shift of the cnoidal waves in the medium with delayed nonlinear response by bandwidth-limited amplification,” Opt. Commun. 190, 373–383 (2001).
    [CrossRef]
  40. V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Stimulated Raman scattering of cnoidal waves,” Quantum Electron. 31, 327–332 (2001).
    [CrossRef]
  41. L. D. Carr, J. N. Kutz, W. P. Reinhardt, “Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose–Einstein condensate,” Phys. Rev. E 63, 066604 (2001).
    [CrossRef]
  42. L. Berge, T. J. Alexander, and Y. S. Kivshar, “Stability criterion for attractive Bose–Einstein condensates,” Phys. Rev. A 62, 023607 (2000).
    [CrossRef]
  43. Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications—the inverse scattering transform,” Phys. Rep. 298, 81–197 (1998).
    [CrossRef]
  44. A. Ankiewicz, W. Krolikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
    [CrossRef]
  45. E. Infeld, “Self-focusing of waves on the surface of deep water,” JETP Lett. 32, 87–89 (1980).
  46. V. P. Pavlenko and V. I. Petviashvili, “Stability and kinetic effects of a standing Langmuir wave,” JETP Lett. 26, 200–202 (1977).
  47. V. P. Kudashev and A. B. Mikhailovsky, “Instability of periodic waves described by the nonlinear Schrödinger equation,” Sov. Phys. JETP 63, 972–979 (1986).

2001 (7)

S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, “Soliton algebra by vortex beam splitting,” Opt. Lett. 26, 1004–1006 (2001).
[CrossRef]

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Propagation of cnoidal waves in a medium with a saturable nonlinear response,” Quantum Electron. 31, 257–262 (2001).
[CrossRef]

N. Korneev, A. Apolinar-Iribe, V. A. Vysloukh, and M. A. Basurto-Pensado, “Self-compression of 1+1D cnoidal wave in photorefractive BTO crystal: an experimental evidence,” Opt. Commun. 197, 209–215 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Self-frequency shift of cnoidal waves in a medium with delayed nonlinear response,” J. Opt. Soc. Am. B 18, 1127–1136 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “On the possibility of suppression of the self-frequency shift of the cnoidal waves in the medium with delayed nonlinear response by bandwidth-limited amplification,” Opt. Commun. 190, 373–383 (2001).
[CrossRef]

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Stimulated Raman scattering of cnoidal waves,” Quantum Electron. 31, 327–332 (2001).
[CrossRef]

L. D. Carr, J. N. Kutz, W. P. Reinhardt, “Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose–Einstein condensate,” Phys. Rev. E 63, 066604 (2001).
[CrossRef]

2000 (6)

L. Berge, T. J. Alexander, and Y. S. Kivshar, “Stability criterion for attractive Bose–Einstein condensates,” Phys. Rev. A 62, 023607 (2000).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal waves compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
[CrossRef]

Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117–195 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity,” Phys. Rev. A 62, 063610 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity,” Phys. Rev. A 62, 063611 (2000).
[CrossRef]

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

1999 (4)

A. Ankiewicz, W. Krolikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Modulational instability of bright solitary waves in incoherently coupled nonlinear Schrödinger equations,” Phys. Rev. E 60, 1019–1029 (1999).
[CrossRef]

V. M. Petnikova, V. V. Shuvalov, and V. A. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization, and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

F. T. Hioe, “Solitary waves for N coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 82, 1152–1155 (1999).
[CrossRef]

1998 (5)

D.-M. Baboiu and G. I. Stegeman, “Modulational instability of a strip beam in a bulk type I quadratic medium,” Opt. Lett. 23, 31–33 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun. 148, 79–84 (1998).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications—the inverse scattering transform,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[CrossRef]

1997 (6)

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

R. A. Fuerst, B. L. Lawrence, W. E. Torruellas, and G. I. Stegeman, “Beam reshaping by use of spatial solitons in the quadratic nonlinear medium KTP,” Opt. Lett. 22, 19–21 (1997).
[CrossRef] [PubMed]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. 22, 868–870 (1997).
[CrossRef] [PubMed]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Symmetry-breaking instabilities of parametric spatial solitons,” Phys. Rev. E 56, R4959–R4962 (1997).
[CrossRef]

C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
[CrossRef]

E. Infeld and T. Lenkowska-Czerwinska, “Analysis of stability of light beams in nonlinear photorefractive media,” Phys. Rev. E 55, 6101–6106 (1997).
[CrossRef]

1996 (3)

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beam in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

V. Tikhonenko, J. Christou, B. Luther-Davies, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129–1131 (1996).
[CrossRef] [PubMed]

1995 (5)

V. Tikhonenko, J. Christou, and B. Luther-Davies, “Spiraling bright spatial solitons formed by the breakup of an optical vortex in a saturable self-focusing medium,” J. Opt. Soc. Am. B 12, 2046–2052 (1995).
[CrossRef]

E. M. Wright, B. L. Lawrence, W. Torruellas, and G. Stegeman, “Stable self-trapping and ring formation in polydiacetylene para-toluene sulfonate,” Opt. Lett. 20, 2481–2483 (1995).
[CrossRef]

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. S. Kivshar, “Self-focusing of plane dark solitons in nonlinear defocusing media,” Phys. Rev. E 51, 5016–5026 (1995).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (2)

G. S. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[CrossRef]

C. T. Law and G. A. Swartzlander, “Optical vortex solitons and the stability of dark soliton stripes,” Opt. Lett. 18, 586–588 (1993).
[CrossRef] [PubMed]

1989 (1)

K. Rypdal and J. J. Rasmussen, “Stability of solitary structures in the nonlinear Schrödinger equation,” Phys. Scr. 40, 192–201 (1989).
[CrossRef]

1988 (1)

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Sov. Phys. JETP 67, 1583–1588 (1988).

1986 (1)

V. P. Kudashev and A. B. Mikhailovsky, “Instability of periodic waves described by the nonlinear Schrödinger equation,” Sov. Phys. JETP 63, 972–979 (1986).

1983 (1)

E. A. Kuznetsov, S. L. Musher, and A. V. Shafarenko, “Collapse of acoustic waves in media with positive dispersion,” JETP Lett. 37, 241–245 (1983).

1980 (1)

E. Infeld, “Self-focusing of waves on the surface of deep water,” JETP Lett. 32, 87–89 (1980).

1977 (1)

V. P. Pavlenko and V. I. Petviashvili, “Stability and kinetic effects of a standing Langmuir wave,” JETP Lett. 26, 200–202 (1977).

1975 (1)

L. M. Degtyarev, V. E. Zakharov, and L. I. Rudakov, “Two examples of Langmuir wave collapse,” Sov. Phys. JETP 41, 57–61 (1975).

Aceves, A. B.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, “Modulational instability of continuous waves and one-dimensional temporal solitons in fiber arrays,” Opt. Lett. 19, 1186–1188 (1994).
[CrossRef] [PubMed]

Akhmediev, N. N.

A. Ankiewicz, W. Krolikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

Aleshkevich, V.

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Self-frequency shift of cnoidal waves in a medium with delayed nonlinear response,” J. Opt. Soc. Am. B 18, 1127–1136 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “On the possibility of suppression of the self-frequency shift of the cnoidal waves in the medium with delayed nonlinear response by bandwidth-limited amplification,” Opt. Commun. 190, 373–383 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal waves compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
[CrossRef]

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

Aleshkevich, V. A.

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Propagation of cnoidal waves in a medium with a saturable nonlinear response,” Quantum Electron. 31, 257–262 (2001).
[CrossRef]

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Stimulated Raman scattering of cnoidal waves,” Quantum Electron. 31, 327–332 (2001).
[CrossRef]

Alexander, T. J.

L. Berge, T. J. Alexander, and Y. S. Kivshar, “Stability criterion for attractive Bose–Einstein condensates,” Phys. Rev. A 62, 023607 (2000).
[CrossRef]

Anderson, D. Z.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

Ankiewicz, A.

A. Ankiewicz, W. Krolikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

Apolinar-Iribe, A.

N. Korneev, A. Apolinar-Iribe, V. A. Vysloukh, and M. A. Basurto-Pensado, “Self-compression of 1+1D cnoidal wave in photorefractive BTO crystal: an experimental evidence,” Opt. Commun. 197, 209–215 (2001).
[CrossRef]

Baboiu, D.-M.

D.-M. Baboiu and G. I. Stegeman, “Modulational instability of a strip beam in a bulk type I quadratic medium,” Opt. Lett. 23, 31–33 (1998).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Basurto-Pensado, M. A.

N. Korneev, A. Apolinar-Iribe, V. A. Vysloukh, and M. A. Basurto-Pensado, “Self-compression of 1+1D cnoidal wave in photorefractive BTO crystal: an experimental evidence,” Opt. Commun. 197, 209–215 (2001).
[CrossRef]

Berge, L.

L. Berge, T. J. Alexander, and Y. S. Kivshar, “Stability criterion for attractive Bose–Einstein condensates,” Phys. Rev. A 62, 023607 (2000).
[CrossRef]

Buryak, A. V.

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. 22, 868–870 (1997).
[CrossRef] [PubMed]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Symmetry-breaking instabilities of parametric spatial solitons,” Phys. Rev. E 56, R4959–R4962 (1997).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

Carr, L. D.

L. D. Carr, J. N. Kutz, W. P. Reinhardt, “Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose–Einstein condensate,” Phys. Rev. E 63, 066604 (2001).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity,” Phys. Rev. A 62, 063611 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity,” Phys. Rev. A 62, 063610 (2000).
[CrossRef]

Christou, J.

Clark, C. W.

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity,” Phys. Rev. A 62, 063610 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity,” Phys. Rev. A 62, 063611 (2000).
[CrossRef]

Cojocaru, C.

De Angelis, C.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, “Modulational instability of continuous waves and one-dimensional temporal solitons in fiber arrays,” Opt. Lett. 19, 1186–1188 (1994).
[CrossRef] [PubMed]

De Rossi, A.

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Symmetry-breaking instabilities of parametric spatial solitons,” Phys. Rev. E 56, R4959–R4962 (1997).
[CrossRef]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. 22, 868–870 (1997).
[CrossRef] [PubMed]

Degtyarev, L. M.

L. M. Degtyarev, V. E. Zakharov, and L. I. Rudakov, “Two examples of Langmuir wave collapse,” Sov. Phys. JETP 41, 57–61 (1975).

Di Trapani, P.

Firth, W. J.

D. V. Skryabin and W. J. Firth, “Modulational instability of bright solitary waves in incoherently coupled nonlinear Schrödinger equations,” Phys. Rev. E 60, 1019–1029 (1999).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun. 148, 79–84 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
[CrossRef]

G. S. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[CrossRef]

Fuerst, R. A.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

R. A. Fuerst, B. L. Lawrence, W. E. Torruellas, and G. I. Stegeman, “Beam reshaping by use of spatial solitons in the quadratic nonlinear medium KTP,” Opt. Lett. 22, 19–21 (1997).
[CrossRef] [PubMed]

Hioe, F. T.

F. T. Hioe, “Solitary waves for N coupled nonlinear Schrödinger equations,” Phys. Rev. Lett. 82, 1152–1155 (1999).
[CrossRef]

Infeld, E.

E. Infeld and T. Lenkowska-Czerwinska, “Analysis of stability of light beams in nonlinear photorefractive media,” Phys. Rev. E 55, 6101–6106 (1997).
[CrossRef]

E. Infeld, “Self-focusing of waves on the surface of deep water,” JETP Lett. 32, 87–89 (1980).

Josserand, C.

C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
[CrossRef]

Kartashov, Y.

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “On the possibility of suppression of the self-frequency shift of the cnoidal waves in the medium with delayed nonlinear response by bandwidth-limited amplification,” Opt. Commun. 190, 373–383 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Self-frequency shift of cnoidal waves in a medium with delayed nonlinear response,” J. Opt. Soc. Am. B 18, 1127–1136 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal waves compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
[CrossRef]

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

Kartashov, Y. V.

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Propagation of cnoidal waves in a medium with a saturable nonlinear response,” Quantum Electron. 31, 257–262 (2001).
[CrossRef]

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Stimulated Raman scattering of cnoidal waves,” Quantum Electron. 31, 327–332 (2001).
[CrossRef]

Kivshar, Y. S.

L. Berge, T. J. Alexander, and Y. S. Kivshar, “Stability criterion for attractive Bose–Einstein condensates,” Phys. Rev. A 62, 023607 (2000).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications—the inverse scattering transform,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Kivshar, Yu. S.

Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117–195 (2000).
[CrossRef]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. 22, 868–870 (1997).
[CrossRef] [PubMed]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Symmetry-breaking instabilities of parametric spatial solitons,” Phys. Rev. E 56, R4959–R4962 (1997).
[CrossRef]

V. Tikhonenko, J. Christou, B. Luther-Davies, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129–1131 (1996).
[CrossRef] [PubMed]

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. S. Kivshar, “Self-focusing of plane dark solitons in nonlinear defocusing media,” Phys. Rev. E 51, 5016–5026 (1995).
[CrossRef]

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

Korneev, N.

N. Korneev, A. Apolinar-Iribe, V. A. Vysloukh, and M. A. Basurto-Pensado, “Self-compression of 1+1D cnoidal wave in photorefractive BTO crystal: an experimental evidence,” Opt. Commun. 197, 209–215 (2001).
[CrossRef]

Krolikowski, W.

A. Ankiewicz, W. Krolikowski, and N. N. Akhmediev, “Partially coherent solitons of variable shape in a slow Kerr-like medium: exact solutions,” Phys. Rev. E 59, 6079–6087 (1999).
[CrossRef]

Kudashev, V. P.

V. P. Kudashev and A. B. Mikhailovsky, “Instability of periodic waves described by the nonlinear Schrödinger equation,” Sov. Phys. JETP 63, 972–979 (1986).

Kutz, J. N.

L. D. Carr, J. N. Kutz, W. P. Reinhardt, “Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose–Einstein condensate,” Phys. Rev. E 63, 066604 (2001).
[CrossRef]

Kuznetsov, E. A.

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Sov. Phys. JETP 67, 1583–1588 (1988).

E. A. Kuznetsov, S. L. Musher, and A. V. Shafarenko, “Collapse of acoustic waves in media with positive dispersion,” JETP Lett. 37, 241–245 (1983).

Law, C. T.

Lawrence, B.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Lawrence, B. L.

Lenkowska-Czerwinska, T.

E. Infeld and T. Lenkowska-Czerwinska, “Analysis of stability of light beams in nonlinear photorefractive media,” Phys. Rev. E 55, 6101–6106 (1997).
[CrossRef]

Luther, G. G.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, “Modulational instability of continuous waves and one-dimensional temporal solitons in fiber arrays,” Opt. Lett. 19, 1186–1188 (1994).
[CrossRef] [PubMed]

Luther-Davies, B.

Mamaev, A. V.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beam in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

Martorell, J.

McDonald, G. S.

G. S. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[CrossRef]

Mikhailovsky, A. B.

V. P. Kudashev and A. B. Mikhailovsky, “Instability of periodic waves described by the nonlinear Schrödinger equation,” Sov. Phys. JETP 63, 972–979 (1986).

Minardi, S.

Molina-Terriza, G.

Musher, S. L.

E. A. Kuznetsov, S. L. Musher, and A. V. Shafarenko, “Collapse of acoustic waves in media with positive dispersion,” JETP Lett. 37, 241–245 (1983).

Pavlenko, V. P.

V. P. Pavlenko and V. I. Petviashvili, “Stability and kinetic effects of a standing Langmuir wave,” JETP Lett. 26, 200–202 (1977).

Pelinovsky, D. E.

Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117–195 (2000).
[CrossRef]

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. S. Kivshar, “Self-focusing of plane dark solitons in nonlinear defocusing media,” Phys. Rev. E 51, 5016–5026 (1995).
[CrossRef]

Petnikova, V. M.

V. M. Petnikova, V. V. Shuvalov, and V. A. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization, and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

Petrov, D. V.

Petviashvili, V. I.

V. P. Pavlenko and V. I. Petviashvili, “Stability and kinetic effects of a standing Langmuir wave,” JETP Lett. 26, 200–202 (1977).

Rasmussen, J. J.

K. Rypdal and J. J. Rasmussen, “Stability of solitary structures in the nonlinear Schrödinger equation,” Phys. Scr. 40, 192–201 (1989).
[CrossRef]

Reinhardt, W. P.

L. D. Carr, J. N. Kutz, W. P. Reinhardt, “Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose–Einstein condensate,” Phys. Rev. E 63, 066604 (2001).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity,” Phys. Rev. A 62, 063610 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity,” Phys. Rev. A 62, 063611 (2000).
[CrossRef]

Rica, S.

C. Josserand and S. Rica, “Coalescence and droplets in the subcritical nonlinear Schrödinger equation,” Phys. Rev. Lett. 78, 1215–1218 (1997).
[CrossRef]

Rubenchik, A. M.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, “Modulational instability of continuous waves and one-dimensional temporal solitons in fiber arrays,” Opt. Lett. 19, 1186–1188 (1994).
[CrossRef] [PubMed]

Rudakov, L. I.

L. M. Degtyarev, V. E. Zakharov, and L. I. Rudakov, “Two examples of Langmuir wave collapse,” Sov. Phys. JETP 41, 57–61 (1975).

Rypdal, K.

K. Rypdal and J. J. Rasmussen, “Stability of solitary structures in the nonlinear Schrödinger equation,” Phys. Scr. 40, 192–201 (1989).
[CrossRef]

Saffman, M.

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beam in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
[CrossRef] [PubMed]

Shafarenko, A. V.

E. A. Kuznetsov, S. L. Musher, and A. V. Shafarenko, “Collapse of acoustic waves in media with positive dispersion,” JETP Lett. 37, 241–245 (1983).

Shuvalov, V. V.

V. M. Petnikova, V. V. Shuvalov, and V. A. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization, and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

Skryabin, D. V.

D. V. Skryabin and W. J. Firth, “Modulational instability of bright solitary waves in incoherently coupled nonlinear Schrödinger equations,” Phys. Rev. E 60, 1019–1029 (1999).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun. 148, 79–84 (1998).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Modulational instability of solitary waves in nondegenerate three-wave mixing: the role of phase symmetries,” Phys. Rev. Lett. 81, 3379–3382 (1998).
[CrossRef]

Stegeman, G.

Stegeman, G. I.

D.-M. Baboiu and G. I. Stegeman, “Modulational instability of a strip beam in a bulk type I quadratic medium,” Opt. Lett. 23, 31–33 (1998).
[CrossRef]

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

R. A. Fuerst, B. L. Lawrence, W. E. Torruellas, and G. I. Stegeman, “Beam reshaping by use of spatial solitons in the quadratic nonlinear medium KTP,” Opt. Lett. 22, 19–21 (1997).
[CrossRef] [PubMed]

Stepanyants, Yu. A.

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. S. Kivshar, “Self-focusing of plane dark solitons in nonlinear defocusing media,” Phys. Rev. E 51, 5016–5026 (1995).
[CrossRef]

Swartzlander, G. A.

Syed, K. S.

G. S. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[CrossRef]

Tikhonenko, V.

Torner, L.

Torres, J. P.

Torruellas, W.

Torruellas, W. E.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

R. A. Fuerst, B. L. Lawrence, W. E. Torruellas, and G. I. Stegeman, “Beam reshaping by use of spatial solitons in the quadratic nonlinear medium KTP,” Opt. Lett. 22, 19–21 (1997).
[CrossRef] [PubMed]

Trillo, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. 22, 868–870 (1997).
[CrossRef] [PubMed]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Symmetry-breaking instabilities of parametric spatial solitons,” Phys. Rev. E 56, R4959–R4962 (1997).
[CrossRef]

Turitsyn, S. K.

A. B. Aceves, G. G. Luther, C. De Angelis, A. M. Rubenchik, and S. K. Turitsyn, “Energy localization in nonlinear fiber arrays: collapse-effect compressor,” Phys. Rev. Lett. 75, 73–76 (1995).
[CrossRef] [PubMed]

E. A. Kuznetsov and S. K. Turitsyn, “Instability and collapse of solitons in media with a defocusing nonlinearity,” Sov. Phys. JETP 67, 1583–1588 (1988).

Vilaseca, R.

Vysloukh, V.

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Self-frequency shift of cnoidal waves in a medium with delayed nonlinear response,” J. Opt. Soc. Am. B 18, 1127–1136 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “On the possibility of suppression of the self-frequency shift of the cnoidal waves in the medium with delayed nonlinear response by bandwidth-limited amplification,” Opt. Commun. 190, 373–383 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal waves compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
[CrossRef]

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

Vysloukh, V. A.

N. Korneev, A. Apolinar-Iribe, V. A. Vysloukh, and M. A. Basurto-Pensado, “Self-compression of 1+1D cnoidal wave in photorefractive BTO crystal: an experimental evidence,” Opt. Commun. 197, 209–215 (2001).
[CrossRef]

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Propagation of cnoidal waves in a medium with a saturable nonlinear response,” Quantum Electron. 31, 257–262 (2001).
[CrossRef]

V. A. Aleshkevich, V. A. Vysloukh, and Y. V. Kartashov, “Stimulated Raman scattering of cnoidal waves,” Quantum Electron. 31, 327–332 (2001).
[CrossRef]

V. M. Petnikova, V. V. Shuvalov, and V. A. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization, and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

Wabnitz, S.

R. A. Fuerst, D.-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulation instability and multisolitonlike generation in a quadratically nonlinear optical media,” Phys. Rev. Lett. 78, 2756–2759 (1997).
[CrossRef]

Wright, E. M.

Zakharov, V. E.

L. M. Degtyarev, V. E. Zakharov, and L. I. Rudakov, “Two examples of Langmuir wave collapse,” Sov. Phys. JETP 41, 57–61 (1975).

Zozulya, A. A.

A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of dark stripe beam in nonlinear media: snake instability and creation of optical vortices,” Phys. Rev. Lett. 76, 2262–2265 (1996).
[CrossRef] [PubMed]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (2)

JETP Lett. (3)

E. Infeld, “Self-focusing of waves on the surface of deep water,” JETP Lett. 32, 87–89 (1980).

V. P. Pavlenko and V. I. Petviashvili, “Stability and kinetic effects of a standing Langmuir wave,” JETP Lett. 26, 200–202 (1977).

E. A. Kuznetsov, S. L. Musher, and A. V. Shafarenko, “Collapse of acoustic waves in media with positive dispersion,” JETP Lett. 37, 241–245 (1983).

Opt. Commun. (6)

G. S. McDonald, K. S. Syed, and W. J. Firth, “Dark spatial soliton break-up in the transverse plane,” Opt. Commun. 95, 281–288 (1993).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Generation and stability of optical bullets in quadratic nonlinear media,” Opt. Commun. 148, 79–84 (1998).
[CrossRef]

V. Aleshkevich, V. Vysloukh, and Y. Kartashov, “Self-bending of cnoidal waves in photorefractive medium with drift and diffusion nonlinearity,” Opt. Commun. 173, 277–284 (2000).
[CrossRef]

N. Korneev, A. Apolinar-Iribe, V. A. Vysloukh, and M. A. Basurto-Pensado, “Self-compression of 1+1D cnoidal wave in photorefractive BTO crystal: an experimental evidence,” Opt. Commun. 197, 209–215 (2001).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “Cnoidal waves compression by means of multisoliton effect,” Opt. Commun. 185, 305–314 (2000).
[CrossRef]

V. Aleshkevich, Y. Kartashov, and V. Vysloukh, “On the possibility of suppression of the self-frequency shift of the cnoidal waves in the medium with delayed nonlinear response by bandwidth-limited amplification,” Opt. Commun. 190, 373–383 (2001).
[CrossRef]

Opt. Lett. (9)

E. M. Wright, B. L. Lawrence, W. Torruellas, and G. Stegeman, “Stable self-trapping and ring formation in polydiacetylene para-toluene sulfonate,” Opt. Lett. 20, 2481–2483 (1995).
[CrossRef]

D.-M. Baboiu and G. I. Stegeman, “Modulational instability of a strip beam in a bulk type I quadratic medium,” Opt. Lett. 23, 31–33 (1998).
[CrossRef]

A. B. Aceves, C. De Angelis, G. G. Luther, and A. M. Rubenchik, “Modulational instability of continuous waves and one-dimensional temporal solitons in fiber arrays,” Opt. Lett. 19, 1186–1188 (1994).
[CrossRef] [PubMed]

C. T. Law and G. A. Swartzlander, “Optical vortex solitons and the stability of dark soliton stripes,” Opt. Lett. 18, 586–588 (1993).
[CrossRef] [PubMed]

R. A. Fuerst, B. L. Lawrence, W. E. Torruellas, and G. I. Stegeman, “Beam reshaping by use of spatial solitons in the quadratic nonlinear medium KTP,” Opt. Lett. 22, 19–21 (1997).
[CrossRef] [PubMed]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Snake instability of one-dimensional parametric spatial solitons,” Opt. Lett. 22, 868–870 (1997).
[CrossRef] [PubMed]

D. V. Petrov, L. Torner, J. Martorell, R. Vilaseca, J. P. Torres, and C. Cojocaru, “Observation of azimuthal modulational instability and formation of patterns of optical solitons in a quadratic nonlinear crystal,” Opt. Lett. 23, 1444–1446 (1998).
[CrossRef]

S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, “Soliton algebra by vortex beam splitting,” Opt. Lett. 26, 1004–1006 (2001).
[CrossRef]

V. Tikhonenko, J. Christou, B. Luther-Davies, and Yu. S. Kivshar, “Observation of vortex solitons created by the instability of dark soliton stripes,” Opt. Lett. 21, 1129–1131 (1996).
[CrossRef] [PubMed]

Phys. Lett. A (1)

A. V. Buryak and Yu. S. Kivshar, “Solitons due to second harmonic generation,” Phys. Lett. A 197, 407–412 (1995).
[CrossRef]

Phys. Rep. (2)

Yu. S. Kivshar and D. E. Pelinovsky, “Self-focusing and transverse instabilities of solitary waves,” Phys. Rep. 331, 117–195 (2000).
[CrossRef]

Y. S. Kivshar and B. Luther-Davies, “Dark optical solitons: physics and applications—the inverse scattering transform,” Phys. Rep. 298, 81–197 (1998).
[CrossRef]

Phys. Rev. A (4)

L. Berge, T. J. Alexander, and Y. S. Kivshar, “Stability criterion for attractive Bose–Einstein condensates,” Phys. Rev. A 62, 023607 (2000).
[CrossRef]

A. V. Mamaev, M. Saffman, D. Z. Anderson, and A. A. Zozulya, “Propagation of light beams in anisotropic nonlinear media: from symmetry breaking to spatial turbulence,” Phys. Rev. A 54, 870–879 (1996).
[CrossRef] [PubMed]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. I. Case of repulsive nonlinearity,” Phys. Rev. A 62, 063610 (2000).
[CrossRef]

L. D. Carr, C. W. Clark, and W. P. Reinhardt, “Stationary solutions of the one-dimensional nonlinear Schrödinger equation. II. Case of attractive nonlinearity,” Phys. Rev. A 62, 063611 (2000).
[CrossRef]

Phys. Rev. E (7)

V. M. Petnikova, V. V. Shuvalov, and V. A. Vysloukh, “Multicomponent photorefractive cnoidal waves: stability, localization, and soliton asymptotics,” Phys. Rev. E 60, 1–10 (1999).
[CrossRef]

E. Infeld and T. Lenkowska-Czerwinska, “Analysis of stability of light beams in nonlinear photorefractive media,” Phys. Rev. E 55, 6101–6106 (1997).
[CrossRef]

D. V. Skryabin and W. J. Firth, “Modulational instability of bright solitary waves in incoherently coupled nonlinear Schrödinger equations,” Phys. Rev. E 60, 1019–1029 (1999).
[CrossRef]

A. De Rossi, S. Trillo, A. V. Buryak, and Yu. S. Kivshar, “Symmetry-breaking instabilities of parametric spatial solitons,” Phys. Rev. E 56, R4959–R4962 (1997).
[CrossRef]

D. E. Pelinovsky, Yu. A. Stepanyants, and Yu. S. Kivshar, “Self-focusing of plane dark solitons in nonlinear defocusing media,” Phys. Rev. E 51, 5016–5026 (1995).
[CrossRef]

L. D. Carr, J. N. Kutz, W. P. Reinhardt, “Stability of stationary states in the cubic nonlinear Schrödinger equation: applications to the Bose–Einstein condensate,” Phys. Rev. E 63, 066604 (2001).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

(a) Maximal modulation frequency, for which transverse modulational instability of a dn wave still occurs, versus localization parameter. (b), (c) Areas of limited perturbations of the dn wave that correspond to purely real increments for Ω=0.5 and Ω=1.5, respectively. (d) Perturbation profile that corresponds to m=0.95, Ω=0.1, and δ=0.17227. (e) Perturbation profile that corresponds to m=0.95, Ω=1.5, and δ=0.671202.

Fig. 2
Fig. 2

(a), (b) Areas of limited perturbations of a dn wave that correspond to purely imaginary increments for Ω=0.5 and Ω=1.5, respectively. (c) Perturbation profile that corresponds to m=0.95, Ω=1.5, and δ=1.52053i.

Fig. 3
Fig. 3

(a), (b) Dependence of dn-wave amplitude on propagation distance and typical wave-field distributions in the presence of perturbations depicted in Figs. 1(d) and 1(e), respectively. (c) Profile of dn-wave perturbation depicted in Fig. 2(c) and dependence of the amplitude of such a perturbation on propagation distance.

Fig. 4
Fig. 4

(a) Maximal modulation frequency, for which transverse modulational instability of a cn wave still occurs, versus localization parameter. (b), (c) Areas of limited perturbations of the cn wave that correspond to purely real increments for Ω=0.5 and Ω=1.1, respectively. (d) Perturbation profile that corresponds to m=0.95, Ω=0.25, and δ=0.17407. (e) Perturbation profile that corresponds to m=0.95, Ω=1.1, and δ=0.683439.

Fig. 5
Fig. 5

(a) Curves at the plane of complex increments where one of conditions |λ1,2| =1 is satisfied for a cn wave with m=0.95 for different modulation frequencies. (b) Same as (a) but for Ω=0.2 and different localization parameters. At the left in these curves (before points marked by circles) |λ1| =1 and |λ2| 1, whereas the parts of these curves at the right (after points marked by circles) correspond to |λ2| =1 and |λ1| 1. (c) Perturbation profile that corresponds to m=0.95, Ω=0.2, and δ=0.03458+0.14832i.

Fig. 6
Fig. 6

(a), (b) Areas of limited perturbations of a cn wave that correspond to purely imaginary increments for Ω=0.5 and Ω=1.5, respectively. (c) Perturbation profile that corresponds to m=0.95, Ω=1.5, and δ=1.39255i. (d) Perturbation profile that corresponds to m=0.95, Ω=1.5, and δ=1.59653i.

Fig. 7
Fig. 7

(a), (b), (c) Dependence of cn-wave amplitude on propagation distance and typical wave-field distributions in the presence of the perturbations depicted in Figs. 4(d), 4(e), and 5(c), respectively. (d) Profile of the cn-wave perturbation depicted in Fig. 6(c) and dependence of the amplitude of such perturbation on propagation distance.

Fig. 8
Fig. 8

(a) Maximal modulation frequency, for which transverse modulational instability of an sn wave still occurs, versus localization parameter. (b), (c) Areas of limited perturbations of an sn wave that correspond to purely real increments for Ω=0.5 and Ω=0.95, respectively. (d) Perturbation profile that corresponds to m=0.95, Ω=0.25, and δ=0.12575. (e) Perturbation profile that corresponds to m=0.95, Ω=0.8, and δ=0.23379.

Fig. 9
Fig. 9

(a), (b) Areas of limited perturbations of an sn wave that correspond to purely imaginary increments for Ω=0.5 and Ω=1.5, respectively. (c) Perturbation profile that corresponds to m=0.95, Ω=1.5, and δ=0.85996i. (d) Perturbation profile that corresponds to m=0.95, Ω=1.5, and δ=2.01289i.

Fig. 10
Fig. 10

(a), (b) Dependence of sn-wave amplitude on propagation distance and typical wave-field distributions in the presence of the perturbations depicted in Figs. 8(d) and 8(e), respectively. (c) Profile of sn-wave perturbation depicted in Fig. 9(c) and dependence of the amplitude of such perturbation on propagation distance.

Equations (49)

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i qξ=-122qη2+2qζ2+σ|q|2q.
qdn(η, ζ, ξ)=χ dn(χη, m)exp[iξχ2(1-m2/2)],
qcn(η, ζ, ξ)=mχ cn(χη, m)exp[iξχ2(m2-1/2)],
qsn(η, ζ, ξ)=mχ sn(χη, m)exp[-iξχ2(1+m2)/2].
q(η, ζ, ξ)={w(η)+[U(η, ξ)+iV(η, ξ)]cos(Ωζ)}×exp(ibξ),
Uξ=-LV,
Vξ=RU.
U(η, ξ)=Re C(δ)u(η, δ)exp(δξ)dδ,
V(η, ξ)=Re C(δ)v(η, δ)exp(δξ)dδ,
δu=-Lv,
δv=Ru.
dΦdη=BΦ,B=OENO,
N=2b+Ω2+6σw22δ-2δ2b+Ω2+2σw2,
Φ(η)=J(η, η)Φ(η),
J(η, η)/η=B(η)J(η, η),
P(η)=J(η+T, η).
D(λ)=λ4+p1λ3+p2λ2+p3λ+p4.
D(P)=P4+p1P3+p2P2+p3P+p4E=O.
Tk+p1Tk-1++pk-1T1=-kpk(k=1,, 4),
p1=-T1,
p2=-½(T2-T12),
p3=-T1,
p4=1.
D(λ)=λ4-T1λ3-½(T2-T12)λ2-T1λ+1,
λ1=¼[T1+(2T2-T12+8)1/2]+{ 1 16[T1+(2T2-T12+8)1/2]2-1}1/2,
λ2=¼[T1-(2T2-T12+8)1/2]+{ 1 16[T1-(2T2-T12+8)1/2]2-1}1/2,
λ3=1/λ1,
λ4=1/λ2.
P(η)Φn(η)=λnΦn(η).
Φ(η)=n=14 CnΦn(η),
Φ(η+kT)=J(η+kT, η)Φ(η)=n=14 CnλnkΦn(η).
Φ(η)=n=1,|λn| =1r CnΦn(η),
Σ1:s1=(u1+u3)/2,s2=(v1+v3)/2,
Σ2:s3=(u2+u4)/2,s4=(v2+v4)/2,
A1:a1=i(u1-u3)/2,a2=i(v1-v3)/2,
A2:a3=i(u2-u4)/2,a4=i(v2-v4)/2.
J(η, η)=E+k=1ηηdη1ηη1dη2×ηηk-1dηkB(η1)B(η2)  B(ηk).
Pk(η+η0)=J(η+η0, η)Pk(η)J-1(η+η0, η)
J-1(η, η)=E+k=1(-1)kηηdη1ηη1dη2 ×ηηk-1dηkB(ηk)B(ηk-1)  B(η1).
Tr[J(η, η)]=4+k=1ηηdη1ηη1dη2 ×ηηk-1dηk×Tr[B(η1)B(η2)  B(ηk)],
Tr[J-1(η, η)]=4+k=1(-1)kηηdη1×ηη1dη2  η1ηk-1dηkTr[B(ηk)B(ηk-1)  B(η1)].
K=OIIO,I=100-1.
Tr[B(η1)B(η2)  B(η2n+1)]=0,
Tr[B(η1)B(η2)  B(η2n)]=Tr[N(η1)N(η3)  N(η2n-1)+N(η2)N(η4)  N(η2n)],
D(λ)=n=1r(λ-λn)mn,n=1r mn=4,
[P(η)-λnE]mnΦn(η)=O.
Φ(η)=n=1r Φn(η),
Φ(η+kT)=J(η+kT, η)Φ(η)=n=1r λnkm=0k Ckmλn-m×[P(η)-λnE]mΦn(η).
P(η)Φn(η)=λnΦn(η)

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