Abstract

We perform a linear stability analysis of stationary periodic waves in cubic–quintic nonlinear media and show that weak χ(5) nonlinearity can lead to stabilization of cnoidal and destabilization of snoidal periodic wave patterns existing in focusing and defocusing χ(3) media, respectively. Direct computer simulations confirm results of the linear stability analysis. The stabilization of periodic waves is expected to be a common phenomenon in physical systems where focusing–defocusing, attractive–repulsive, nonlinear self-actions compete with each other.

© 2004 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
  33. Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89, 273902 (2002).
    [CrossRef]
  34. L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  39. F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
    [CrossRef]
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    [CrossRef]
  44. I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
    [CrossRef]
  45. A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
    [CrossRef]
  46. D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
    [CrossRef]

2004 (1)

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92, 033901 (2004).
[CrossRef] [PubMed]

2003 (12)

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[CrossRef]

V. A. Aleshkevich, A. A. Egorov, Y. V. Kartashov, V. A. Vysloukh, and A. S. Zelenina, “Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media,” Phys. Rev. E 67, 066605 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media,” Phys. Rev. E 68, 015603(R) (2003).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, “Stable spatiotemporal spinning solitons in a bimodal cubic–quintic medium,” Phys. Rev. E 67, 056608 (2003).
[CrossRef]

L.-C. Crasovan, J. P. Torres, D. Mihalache, and L. Torner, “Arresting wave collapse by wave self-rectification,” Phys. Rev. Lett. 91, 063904 (2003).
[CrossRef] [PubMed]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity,” Phys. Rev. E 67, 036613 (2003).
[CrossRef]

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

J. Petter, J. Schroder, D. Trager, and C. Denz, “Optical control of arrays of photorefractive screening solitons,” Opt. Lett. 28, 438–440 (2003).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Transverse modulational instability of (2+1)-dimensional cnoidal waves in media with cubic nonlinearity,” J. Opt. Soc. Am. B 20, 1273–1284 (2003).
[CrossRef]

2002 (7)

C. Zhan, D. Zhang, D. Zhu, D. Wang, Y. Li, D. Li, Z. Lu, L. Zhao, and Y. Nie, “Third- and fifth-order optical nonlinearities in a new stilbazolium derivative,” J. Opt. Soc. Am. B 19, 369–375 (2002).
[CrossRef]

A. Apolinar-Iribe, N. Korneev, V. Vysloukh, and C. M. Gomez-Sarabia, “Transverse modulational instability of periodic light patterns in photorefractive strontium barium niobate crystal,” Opt. Lett. 27, 2088–2090 (2002).
[CrossRef]

S. Minardi, G. Arrighi, P. Di Trapani, A. Varanavicius, and A. Piskarskas, “Solitonic all-optical switch based on the fractional Talbot effect,” Opt. Lett. 27, 2097–2099 (2002).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic–quintic medium,” Physica D 161, 187–201 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

2001 (5)

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, and J. N. Kutz, “Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. Lett. 86, 1402–1405 (2001).
[CrossRef] [PubMed]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

A. Bramati, W. Chinaglia, S. Minardi, and P. Di Trapani, “Reconstruction of blurred images by controlled formation of spatial solitons,” Opt. Lett. 26, 1409–1411 (2001).
[CrossRef]

2000 (3)

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

H. W. Schurmann and V. S. Serov, “Criteria for existence and stability of soliton solutions of the cubic–quintic nonlinear Schrödinger equation,” Phys. Rev. E 62, 2821–2826 (2000).
[CrossRef]

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[CrossRef]

1999 (1)

J. M. Arnold, “Stability of solitary wave trains in Hamiltonian wave systems,” Phys. Rev. E 60, 979–986 (1999).
[CrossRef]

1998 (4)

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[CrossRef]

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearities,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

1996 (2)

R. W. Micallef, V. V. Afanasjev, Y. S. Kivshar, and J. D. Love, “Optical solitons with power-law asymptotics,” Phys. Rev. E 54, 2936–2942 (1996).
[CrossRef]

H. W. Schurmann, “Traveling-wave solutions of the cubic–quintic nonlinear Schrödinger equation,” Phys. Rev. E 54, 4312–4320 (1996).
[CrossRef]

1989 (1)

1987 (1)

S. E. Fil’chenkov, G. M. Fraiman, and A. D. Yunakovskii, “Instability of periodic solutions of the nonlinear Schrödinger equation,” Sov. J. Plasma Phys. 13, 554–557 (1987).

1986 (2)

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

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

1982 (1)

V. P. Pavlenko and V. I. Petviashvili, “Band theory for the stability of nonlinear periodic waves in plasmas,” Sov. J. Plasma Phys. 8, 117–120 (1982).

1980 (1)

D. U. Martin, H. C. Yuen, and P. G. Saffman, “Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation,” Wave Motion 2, 215–229 (1980).
[CrossRef]

Afanasjev, V. V.

R. W. Micallef, V. V. Afanasjev, Y. S. Kivshar, and J. D. Love, “Optical solitons with power-law asymptotics,” Phys. Rev. E 54, 2936–2942 (1996).
[CrossRef]

Aleshkevich, V. A.

V. A. Aleshkevich, A. A. Egorov, Y. V. Kartashov, V. A. Vysloukh, and A. S. Zelenina, “Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media,” Phys. Rev. E 67, 066605 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Transverse modulational instability of (2+1)-dimensional cnoidal waves in media with cubic nonlinearity,” J. Opt. Soc. Am. B 20, 1273–1284 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity,” Phys. Rev. E 67, 036613 (2003).
[CrossRef]

Apolinar-Iribe, A.

Arnold, J. M.

J. M. Arnold, “Stability of solitary wave trains in Hamiltonian wave systems,” Phys. Rev. E 60, 979–986 (1999).
[CrossRef]

Arrighi, G.

Bang, O.

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearities,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

Barashenkov, I. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Barthelemy, A.

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[CrossRef]

Boudebs, G.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Bramati, A.

Bronski, J. C.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, and J. N. Kutz, “Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. Lett. 86, 1402–1405 (2001).
[CrossRef] [PubMed]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612 (2001).
[CrossRef]

Buryak, A. V.

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearities,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

Carr, L. D.

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, and J. N. Kutz, “Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. Lett. 86, 1402–1405 (2001).
[CrossRef] [PubMed]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

Carretero-Gonzalez, R.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

Cherukulappurath, S.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Chinaglia, W.

Christodoulides, D. N.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Conti, C.

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

Couderc, V.

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[CrossRef]

Crasovan, L.-C.

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

L.-C. Crasovan, J. P. Torres, D. Mihalache, and L. Torner, “Arresting wave collapse by wave self-rectification,” Phys. Rev. Lett. 91, 063904 (2003).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic–quintic medium,” Physica D 161, 187–201 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

De Rossi, A.

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearities,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

Deconinck, B.

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, and J. N. Kutz, “Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. Lett. 86, 1402–1405 (2001).
[CrossRef] [PubMed]

Denz, C.

Di Trapani, P.

Efremidis, N. K.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Egorov, A. A.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92, 033901 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[CrossRef]

V. A. Aleshkevich, A. A. Egorov, Y. V. Kartashov, V. A. Vysloukh, and A. S. Zelenina, “Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media,” Phys. Rev. E 67, 066605 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Transverse modulational instability of (2+1)-dimensional cnoidal waves in media with cubic nonlinearity,” J. Opt. Soc. Am. B 20, 1273–1284 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity,” Phys. Rev. E 67, 036613 (2003).
[CrossRef]

Fil’chenkov, S. E.

S. E. Fil’chenkov, G. M. Fraiman, and A. D. Yunakovskii, “Instability of periodic solutions of the nonlinear Schrödinger equation,” Sov. J. Plasma Phys. 13, 554–557 (1987).

Fleischer, J. W.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Fraiman, G. M.

S. E. Fil’chenkov, G. M. Fraiman, and A. D. Yunakovskii, “Instability of periodic solutions of the nonlinear Schrödinger equation,” Sov. J. Plasma Phys. 13, 554–557 (1987).

Gagnon, L.

Gomez-Sarabia, C. M.

Kartashov, Y. V.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92, 033901 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media,” Phys. Rev. E 68, 015603(R) (2003).
[CrossRef]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

V. A. Aleshkevich, A. A. Egorov, Y. V. Kartashov, V. A. Vysloukh, and A. S. Zelenina, “Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media,” Phys. Rev. E 67, 066605 (2003).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Transverse modulational instability of (2+1)-dimensional cnoidal waves in media with cubic nonlinearity,” J. Opt. Soc. Am. B 20, 1273–1284 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity,” Phys. Rev. E 67, 036613 (2003).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

Kivshar, Y. S.

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

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

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearities,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

R. W. Micallef, V. V. Afanasjev, Y. S. Kivshar, and J. D. Love, “Optical solitons with power-law asymptotics,” Phys. Rev. E 54, 2936–2942 (1996).
[CrossRef]

Korneev, N.

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.

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, and J. N. Kutz, “Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. Lett. 86, 1402–1405 (2001).
[CrossRef] [PubMed]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612 (2001).
[CrossRef]

Kuznetsov, E. A.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

Leblond, H.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, “Stable spatiotemporal spinning solitons in a bimodal cubic–quintic medium,” Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

Li, D.

Li, Y.

Love, J. D.

R. W. Micallef, V. V. Afanasjev, Y. S. Kivshar, and J. D. Love, “Optical solitons with power-law asymptotics,” Phys. Rev. E 54, 2936–2942 (1996).
[CrossRef]

Lu, Z.

Malomed, B. A.

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, “Stable spatiotemporal spinning solitons in a bimodal cubic–quintic medium,” Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic–quintic medium,” Physica D 161, 187–201 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

Martin, D. U.

D. U. Martin, H. C. Yuen, and P. G. Saffman, “Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation,” Wave Motion 2, 215–229 (1980).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, “Stable spatiotemporal spinning solitons in a bimodal cubic–quintic medium,” Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[CrossRef]

Micallef, R. W.

R. W. Micallef, V. V. Afanasjev, Y. S. Kivshar, and J. D. Love, “Optical solitons with power-law asymptotics,” Phys. Rev. E 54, 2936–2942 (1996).
[CrossRef]

Mihalache, D.

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, “Stable spatiotemporal spinning solitons in a bimodal cubic–quintic medium,” Phys. Rev. E 67, 056608 (2003).
[CrossRef]

L.-C. Crasovan, J. P. Torres, D. Mihalache, and L. Torner, “Arresting wave collapse by wave self-rectification,” Phys. Rev. Lett. 91, 063904 (2003).
[CrossRef] [PubMed]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic–quintic medium,” Physica D 161, 187–201 (2002).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[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.

Nie, Y.

Pavlenko, V. P.

V. P. Pavlenko and V. I. Petviashvili, “Band theory for the stability of nonlinear periodic waves in plasmas,” Sov. J. Plasma Phys. 8, 117–120 (1982).

Pelinovsky, D. E.

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

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Perez-Garcia, V. M.

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

Petter, J.

Petviashvili, V. I.

V. P. Pavlenko and V. I. Petviashvili, “Band theory for the stability of nonlinear periodic waves in plasmas,” Sov. J. Plasma Phys. 8, 117–120 (1982).

Piskarskas, A.

Promislow, K.

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

Quemard, C.

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[CrossRef]

Rubenchik, A. M.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

Saffman, P. G.

D. U. Martin, H. C. Yuen, and P. G. Saffman, “Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation,” Wave Motion 2, 215–229 (1980).
[CrossRef]

Sammut, R. A.

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

Sanchez, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Schroder, J.

Schurmann, H. W.

H. W. Schurmann and V. S. Serov, “Criteria for existence and stability of soliton solutions of the cubic–quintic nonlinear Schrödinger equation,” Phys. Rev. E 62, 2821–2826 (2000).
[CrossRef]

H. W. Schurmann, “Traveling-wave solutions of the cubic–quintic nonlinear Schrödinger equation,” Phys. Rev. E 54, 4312–4320 (1996).
[CrossRef]

Segev, M.

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Serov, V. S.

H. W. Schurmann and V. S. Serov, “Criteria for existence and stability of soliton solutions of the cubic–quintic nonlinear Schrödinger equation,” Phys. Rev. E 62, 2821–2826 (2000).
[CrossRef]

Smektala, F.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[CrossRef]

Torner, L.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92, 033901 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media,” Phys. Rev. E 68, 015603(R) (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

L.-C. Crasovan, J. P. Torres, D. Mihalache, and L. Torner, “Arresting wave collapse by wave self-rectification,” Phys. Rev. Lett. 91, 063904 (2003).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[CrossRef]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[CrossRef]

Torres, J. P.

L.-C. Crasovan, J. P. Torres, D. Mihalache, and L. Torner, “Arresting wave collapse by wave self-rectification,” Phys. Rev. Lett. 91, 063904 (2003).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Towers, I.

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, “Stable spatiotemporal spinning solitons in a bimodal cubic–quintic medium,” Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

Trager, D.

Trillo, S.

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearities,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

Troles, J.

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Varanavicius, A.

Vysloukh, V.

Vysloukh, V. A.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92, 033901 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media,” Phys. Rev. E 68, 015603(R) (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity,” Phys. Rev. E 67, 036613 (2003).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[CrossRef]

V. A. Aleshkevich, A. A. Egorov, Y. V. Kartashov, V. A. Vysloukh, and A. S. Zelenina, “Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media,” Phys. Rev. E 67, 066605 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Transverse modulational instability of (2+1)-dimensional cnoidal waves in media with cubic nonlinearity,” J. Opt. Soc. Am. B 20, 1273–1284 (2003).
[CrossRef]

Wang, D.

Yuen, H. C.

D. U. Martin, H. C. Yuen, and P. G. Saffman, “Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation,” Wave Motion 2, 215–229 (1980).
[CrossRef]

Yunakovskii, A. D.

S. E. Fil’chenkov, G. M. Fraiman, and A. D. Yunakovskii, “Instability of periodic solutions of the nonlinear Schrödinger equation,” Sov. J. Plasma Phys. 13, 554–557 (1987).

Zakharov, V. E.

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

Zelenina, A. S.

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92, 033901 (2004).
[CrossRef] [PubMed]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity,” Phys. Rev. E 67, 036613 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Transverse modulational instability of (2+1)-dimensional cnoidal waves in media with cubic nonlinearity,” J. Opt. Soc. Am. B 20, 1273–1284 (2003).
[CrossRef]

V. A. Aleshkevich, A. A. Egorov, Y. V. Kartashov, V. A. Vysloukh, and A. S. Zelenina, “Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media,” Phys. Rev. E 67, 066605 (2003).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[CrossRef]

Zemlyanaya, E. V.

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

Zhan, C.

Zhang, D.

Zhao, L.

Zhu, D.

J. Non-Cryst. Solids (1)

F. Smektala, C. Quemard, V. Couderc, and A. Barthelemy, “Non-linear optical properties of chalcogenide glasses measured by Z-scan,” J. Non-Cryst. Solids 274, 232–237 (2000).
[CrossRef]

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

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

Nature (1)

J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices,” Nature 422, 147–150 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

G. Boudebs, S. Cherukulappurath, H. Leblond, J. Troles, F. Smektala, and F. Sanchez, “Experimental and theoreticalstudy of higher-order nonlinearities in chalcogenide glasses,” Opt. Commun. 219, 427–433 (2003).
[CrossRef]

Opt. Lett. (4)

Phys. Lett. A (1)

I. Towers, A. V. Buryak, R. A. Sammut, B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of spinning ring solitons of the cubic–quintic nonlinear Schrödinger equation,” Phys. Lett. A 288, 292–298 (2001).
[CrossRef]

Phys. Rep. (2)

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

E. A. Kuznetsov, A. M. Rubenchik, and V. E. Zakharov, “Soliton stability in plasmas and hydrodynamics,” Phys. Rep. 142, 103–165 (1986).
[CrossRef]

Phys. Rev. E (15)

Y. V. Kartashov, V. A. Aleshkevich, V. A. Vysloukh, A. A. Egorov, and A. S. Zelenina, “Stability analysis of (1+1)-dimensional cnoidal waves in media with cubic nonlinearity,” Phys. Rev. E 67, 036613 (2003).
[CrossRef]

V. A. Aleshkevich, A. A. Egorov, Y. V. Kartashov, V. A. Vysloukh, and A. S. Zelenina, “Stability analysis of spatiotemporal cnoidal waves in cubic nonlinear media,” Phys. Rev. E 67, 066605 (2003).
[CrossRef]

Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Two-dimensional cnoidal waves in Kerr-type saturable nonlinear media,” Phys. Rev. E 68, 015603(R) (2003).
[CrossRef]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stabilization of one-dimensional periodic waves by saturation of the nonlinear response,” Phys. Rev. E 68, 065605(R) (2003).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of repulsive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 63, 036612 (2001).
[CrossRef]

J. C. Bronski, L. D. Carr, R. Carretero-Gonzalez, B. Deconinck, J. N. Kutz, and K. Promislow, “Stability of attractive Bose-Einstein condensates in a periodic potential,” Phys. Rev. E 64, 056615 (2001).
[CrossRef]

D. Mihalache, D. Mazilu, I. Towers, B. A. Malomed, and F. Lederer, “Stable spatiotemporal spinning solitons in a bimodal cubic–quintic medium,” Phys. Rev. E 67, 056608 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, B. A. Malomed, A. V. Buryak, L. Torner, and F. Lederer, “Stable three-dimensional spinning optical solitons supported by competing quadratic and cubic nonlinearities,” Phys. Rev. E 66, 016613 (2002).
[CrossRef]

O. Bang, Y. S. Kivshar, A. V. Buryak, A. De Rossi, and S. Trillo, “Two-dimensional solitary waves in media with quadratic and cubic nonlinearities,” Phys. Rev. E 58, 5057–5069 (1998).
[CrossRef]

L.-C. Crasovan, Y. V. Kartashov, D. Mihalache, L. Torner, Y. S. Kivshar, and V. M. Perez-Garcia, “Soliton molecules: Robust clusters of optical spatiotemporal solitons,” Phys. Rev. E 67, 046610 (2003).
[CrossRef]

D. Mihalache, D. Mazilu, L.-C. Crasovan, B. A. Malomed, F. Lederer, and L. Torner, “Robust soliton clusters in media with competing cubic and quintic nonlinearities,” Phys. Rev. E 68, 046612 (2003).
[CrossRef]

R. W. Micallef, V. V. Afanasjev, Y. S. Kivshar, and J. D. Love, “Optical solitons with power-law asymptotics,” Phys. Rev. E 54, 2936–2942 (1996).
[CrossRef]

H. W. Schurmann and V. S. Serov, “Criteria for existence and stability of soliton solutions of the cubic–quintic nonlinear Schrödinger equation,” Phys. Rev. E 62, 2821–2826 (2000).
[CrossRef]

J. M. Arnold, “Stability of solitary wave trains in Hamiltonian wave systems,” Phys. Rev. E 60, 979–986 (1999).
[CrossRef]

H. W. Schurmann, “Traveling-wave solutions of the cubic–quintic nonlinear Schrödinger equation,” Phys. Rev. E 54, 4312–4320 (1996).
[CrossRef]

Phys. Rev. Lett. (8)

I. V. Barashenkov, D. E. Pelinovsky, and E. V. Zemlyanaya, “Vibrations and oscillatory instabilities of gap solitons,” Phys. Rev. Lett. 80, 5117–5120 (1998).
[CrossRef]

A. De Rossi, C. Conti, and S. Trillo, “Stability, multistability and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

D. Mihalache, D. Mazilu, and L. Torner, “Stability of walking vector solitons,” Phys. Rev. Lett. 81, 4353–4356 (1998).
[CrossRef]

L.-C. Crasovan, J. P. Torres, D. Mihalache, and L. Torner, “Arresting wave collapse by wave self-rectification,” Phys. Rev. Lett. 91, 063904 (2003).
[CrossRef] [PubMed]

Y. V. Kartashov, L.-C. Crasovan, D. Mihalache, and L. Torner, “Robust propagation of two-color soliton clusters supported by competing nonlinearities,” Phys. Rev. Lett. 89, 273902 (2002).
[CrossRef]

J. C. Bronski, L. D. Carr, B. Deconinck, and J. N. Kutz, “Bose-Einstein condensates in standing waves: The cubic nonlinear Schrödinger equation with a periodic potential,” Phys. Rev. Lett. 86, 1402–1405 (2001).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, L.-C. Crasovan, I. Towers, A. V. Buryak, B. A. Malomed, L. Torner, J. P. Torres, and F. Lederer, “Stable spinning optical solitons in three dimensions,” Phys. Rev. Lett. 88, 073902 (2002).
[CrossRef] [PubMed]

Y. V. Kartashov, A. A. Egorov, A. S. Zelenina, V. A. Vysloukh, and L. Torner, “Stable multicolor periodic-wave arrays,” Phys. Rev. Lett. 92, 033901 (2004).
[CrossRef] [PubMed]

Physica D (1)

B. A. Malomed, L.-C. Crasovan, and D. Mihalache, “Stability of vortex solitons in the cubic–quintic medium,” Physica D 161, 187–201 (2002).
[CrossRef]

Sov. J. Plasma Phys. (2)

S. E. Fil’chenkov, G. M. Fraiman, and A. D. Yunakovskii, “Instability of periodic solutions of the nonlinear Schrödinger equation,” Sov. J. Plasma Phys. 13, 554–557 (1987).

V. P. Pavlenko and V. I. Petviashvili, “Band theory for the stability of nonlinear periodic waves in plasmas,” Sov. J. Plasma Phys. 8, 117–120 (1982).

Sov. Phys. JETP (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).

Wave Motion (1)

D. U. Martin, H. C. Yuen, and P. G. Saffman, “Stability of plane wave solutions of the two-space-dimensional nonlinear Schrödinger equation,” Wave Motion 2, 215–229 (1980).
[CrossRef]

Other (5)

N. N. Akhmediev and A. Ankiewicz, Solitons (Chapman & Hall, London, 1997).

Yu. S. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, New York, 2003).

A. M. Kamchatnov, Nonlinear Periodic Waves and Their Modulations (Kluwer Academic, Dordrecht, The Netherlands, 2000).

E. Infeld and R. Rowlands, Nonlinear Waves, Solitons and Chaos (Cambridge U. Press, Cambridge, UK, 1990).

H. C. Yuen and B. M. Lake, Nonlinear Dynamics of Deep-Water Gravity Waves (Academic, New York, 1982).

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Figures (4)

Fig. 1
Fig. 1

(a) Energy flow of sn-type wave versus propagation constant. (b) Integral width versus energy flow. (c) Profiles of sn waves with various energy flows at σ5=-0.1. Profiles shown correspond to points marked by circles in panel (a). (d) Areas of existence of finite perturbations at σ5=-0.25. Shaded area corresponds to complex growth rates, whereas white area corresponds to real growth rates. (e) Profile of perturbation with growth rate δ=0.53134 corresponding to sn wave from upper branch with b=0 at σ5=-0.25. (f) Area of existence of stable sn waves. Curves for critical value of propagation constant and cutoff almost coincide and are not distinguishable in the plot.

Fig. 2
Fig. 2

(a) Energy flow of cn-type wave versus propagation constant. (b) Integral width versus energy flow. (c) Profiles of cn waves with various energy flows at σ5=0.1. Profiles shown correspond to points marked by circles in panel (a). (d) Maximum real part of complex growth rate versus propagation constant. (e) Curves at the complex plane showing possible increment values for various propagation constants at σ5=0.25. (f) Areas of existence of stable and unstable (shaded) cn waves.

Fig. 3
Fig. 3

(a) Energy flow of dn-type wave versus propagation constant. (b) Spectral width versus energy flow. (c) Profiles of dn waves with various energy flows at σ5=0.1. Profiles shown correspond to points marked by circles in panel (a). (d) Areas of existence of finite perturbations with real growth rates at σ5=0.25. Dashed lines in (d) stand for cutoff values.

Fig. 4
Fig. 4

(a) Profile of stationary sn-type wave with b=-1.2 at σ5=-0.25 and its long-term propagation in the presence of white input noise. (b) Profile of stationary cn-type wave with b=-0.5 at σ5=0.25 and its long-term propagation in the presence of white noise. Noise variance σ2=0.01.

Equations (13)

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

iqξ=-122qη2+σ3q|q|2+σ5q|q|4.
-12d2wdη2+σ3w3+σ5w5+bw=0,
U=-T/2T/2w2(η)dη
W=2-T/4T/4w2(η)η2dη1/2-T/4T/4w2(η)dη-1/2,
ΩW=2-Ω2w2(Ω)dΩ1/2-w2(Ω)dΩ-1/2,
q(η, ξ)=[w(η)+U(η, ξ)+iV(η, ξ)]exp(ibξ),
U(η, ξ)=ReC(δ)u(η, δ)exp(δξ)dδ,
V(η, ξ)=ReC(δ)v(η, δ)exp(δξ)dδ,
dΦdη=BΦ,B=OENO,
N=2b+6σ3w2+10σ5w42δ-2δ2b+2σ3w2+2σ5w4
Φ(η)=J(η, η)Φk(η)=exp(iμkη)Ψk(η, η),
Ψk(η, η)=Ψk(η+T, η),
Ψk(η, η)=S(η, η)exp(-iμkη)Φk(η).

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