## Solitons supported by spatially inhomogeneous nonlinear losses |

Optics Express, Vol. 20, Issue 3, pp. 2657-2667 (2012)

http://dx.doi.org/10.1364/OE.20.002657

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### Abstract

We uncover that, in contrast to the common belief, stable dissipative solitons exist in media with uniform gain in the presence of nonuniform cubic losses, whose local strength grows with coordinate

© 2012 OSA

## 1. Introduction

2. V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B **18**(14), 627 (2004). [CrossRef]

3. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. **78**(1), 179–215 (2006). [CrossRef]

4. L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. **11**(10), 1268–1270 (1999). [CrossRef]

10. Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. **83**(1), 247–306 (2011). [CrossRef]

11. H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **72**(4), 046610 (2005). [CrossRef] [PubMed]

22. H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A **81**(1), 013624 (2010). [CrossRef]

2. V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B **18**(14), 627 (2004). [CrossRef]

3. O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. **78**(1), 179–215 (2006). [CrossRef]

23. V. Pérez-García and R. Pardo, “Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates,” Physica D **238**(15), 1352–1361 (2009). [CrossRef]

24. B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. **63**(5), 642–648 (2003). [CrossRef]

28. Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express **14**(21), 9832–9837 (2006). [CrossRef] [PubMed]

29. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. **36**(16), 3088–3090 (2011). [CrossRef] [PubMed]

31. Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. **36**(23), 4587–4589 (2011). [CrossRef] [PubMed]

32. W.-P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A **84**(4), 043801 (2011). [CrossRef]

33. B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D **29**(1-2), 155–172 (1987). [CrossRef]

38. P. Marcq, H. Chate, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation,” Physica D **73**(4), 305–317 (1994). [CrossRef]

39. C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Spec. Top. **173**(1), 233–243 (2009). [CrossRef]

48. C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **84**(6), 066609 (2011). [CrossRef]

*absorption*can be used for generation and stabilization of new types of dissipative solitons was not addressed yet.

*uniform gain*, provided that the gain is combined with non-uniform nonlinear absorption, whose local strength grows rapidly enough toward the periphery of the system.

## 2. The model

50. J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D Appl. Phys. **36**(3), R1–R16 (2003). [CrossRef]

29. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. **36**(16), 3088–3090 (2011). [CrossRef] [PubMed]

31. Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. **36**(23), 4587–4589 (2011). [CrossRef] [PubMed]

*single-well*absorption profile). By means of rescaling we fix

*exact*dipole-soliton solution,with

*stable exact*fundamental soliton solution,with propagation constant

## 3. Numerical results

*asymmetric*profiles, i.e., to the effect of the spontaneous symmetry breaking (SSB), if the conservative part of cubic nonlinearity is focusing

51. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A **55**(6), 4318–4324 (1997). [CrossRef]

54. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction,” Phys. Rev. Lett. **95**(1), 010402 (2005). [CrossRef] [PubMed]

55. A. Sigler and B. A. Malomed, “Solitary pulses in linearly coupled cubic-quintic Ginzburg-Landau equations,” Physica D **212**(3-4), 305–316 (2005). [CrossRef]

45. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A **83**(4), 041806 (2011). [CrossRef]

44. D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. **36**(7), 1200–1202 (2011). [CrossRef] [PubMed]

29. O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. **36**(16), 3088–3090 (2011). [CrossRef] [PubMed]

31. Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. **36**(23), 4587–4589 (2011). [CrossRef] [PubMed]

*supercritical*type (in other words, it is a phase transition of the second kind; note that the above-mentioned SSB bifurcation of solitons reported in Ref [45

45. Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A **83**(4), 041806 (2011). [CrossRef]

*completely stable*, featuring no SSB. The energy flow of such modes increases almost linearly with the gain, while the propagation constant decreases, always staying negative.

*stable*multipoles (e.g., dipoles) can be found in the system with a double-well nonlinear-absorption profile, described, e.g., by functionin Eqs. (1) and (2), see examples in Fig. 4 . Generic results can be adequately demonstrated for

*stable*within a limited interval of gain parameters,

*defocusing*medium with the double-well absorption landscape. This finding resembles a well-known fact that, in dual-core conservative systems, the SSB of spatially odd modes occurs under the action of defocusing nonlinearities (in contrast to the spatially even ground state, which undergoes the symmetry breaking under the action of focusing nonlinearities [51

51. G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A **55**(6), 4318–4324 (1997). [CrossRef]

54. M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction,” Phys. Rev. Lett. **95**(1), 010402 (2005). [CrossRef] [PubMed]

*supercritical*character of the bifurcation in Figs. 2(c) and 2(d)]. As shown in Fig. 3(c), the symmetric dipoles in the two-well absorption landscapes, with the defocusing nonlinearity, are stable in the domain of

*dynamical restoration*of the symmetry [Fig. 5(f)].

## 4. Analytical results

40. D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A **81**(1), 013606 (2010). [CrossRef]

56. L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R. Soc. Lond. A Math. Phys. Sci. **326**(1566), 289–313 (1972). [CrossRef]

58. J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A **246**(5), 412–422 (1998). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | Y. S. Kivshar and G. P. Agrawal, |

2. | V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B |

3. | O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. |

4. | L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. |

5. | I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. |

6. | F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A |

7. | H. Saito and M. Ueda, “Dynamically stabilized bright solitons in a two-dimensional bose-einstein condensate,” Phys. Rev. Lett. |

8. | M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. |

9. | B. A. Malomed, |

10. | Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. |

11. | H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

12. | G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D |

13. | J. Garnier and F. K. Abdullaev, “Transmission of matter wave solitons through nonlinear traps and barriers,” Phys. Rev. A |

14. | D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis, A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, “Statics and dynamics of an inhomogeneously nonlinear lattice,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

15. | Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. |

16. | J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. |

17. | P. Niarchou, G. Theocharis, P. G. Kevrekidis, P. Schmelcher, and D. J. Frantzeskakis, “Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates,” Phys. Rev. A |

18. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. |

19. | Y. Sivan, G. Fibich, and B. Ilan, “Drift instability and tunneling of lattice solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

20. | F. K. Abdullaev, A. Gammal, M. Salerno, and L. Tomio, “Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices,” Phys. Rev. A |

21. | H. A. Cruz, V. A. Brazhnyi, and V. V. Konotop, “Partial delocalization of two-component condensates in optical lattices,” J. Phys. B |

22. | H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A |

23. | V. Pérez-García and R. Pardo, “Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates,” Physica D |

24. | B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. |

25. | D. Neshev, A. Sukhorukov, Y. Kivshar, E. Ostrovskaya, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. |

26. | J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. |

27. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express |

28. | Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express |

29. | O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. |

30. | O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, “Bright solitons from defocusing nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

31. | Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. |

32. | W.-P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A |

33. | B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D |

34. | O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. France |

35. | W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. |

36. | V. Hakim, P. Jakobsen, and Y. Pomeau, “Fronts vs solitary waves in non equilibrium systems,” Europhys. Lett. |

37. | B. A. Malomed and A. A. Nepomnyashchy, “Kinks and solitons in the generalized Ginzburg-Landau equation,” Phys. Rev. A |

38. | P. Marcq, H. Chate, and R. Conte, “Exact solutions of the one-dimensional quintic complex Ginzburg-Landau equation,” Physica D |

39. | C.-K. Lam, B. A. Malomed, K. W. Chow, and P. K. A. Wai, “Spatial solitons supported by localized gain in nonlinear optical waveguides,” Eur. Phys. J. Spec. Top. |

40. | D. A. Zezyulin, G. L. Alfimov, and V. V. Konotop, “Nonlinear modes in a complex parabolic potential,” Phys. Rev. A |

41. | C. H. Tsang, B. A. Malomed, C. K. Lam, and K. W. Chow, “Solitons pinned to hot spots,” Eur. Phys. J. D |

42. | F. Kh. Abdullaev, V. V. Konotop, M. Salerno, and A. V. Yulin, “Dissipative periodic waves, solitons, and breathers of the nonlinear Schrödinger equation with complex potentials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

43. | D. A. Zezyulin, V. V. Konotop, and G. L. Alfimov, “Dissipative double-well potential: nonlinear stationary and pulsating modes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

44. | D. A. Zezyulin, Y. V. Kartashov, and V. V. Konotop, “Solitons in a medium with linear dissipation and localized gain,” Opt. Lett. |

45. | Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Symmetry breaking and multipeaked solitons in inhomogeneous gain landscapes,” Phys. Rev. A |

46. | Y. V. Kartashov, V. V. Konotop, and V. A. Vysloukh, “Dissipative surface solitons in periodic structures,” Europhys. Lett. |

47. | V. E. Lobanov, Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Stable radially symmetric and azimuthally modulated vortex solitons supported by localized gain,” Opt. Lett. |

48. | C. H. Tsang, B. A. Malomed, and K. W. Chow, “Multistable dissipative structures pinned to dual hot spots,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

49. | O. V. Borovkova, V. E. Lobanov, and B. A. Malomed, “Stable nonlinear amplification of solitons without gain saturation,” Europhys. Lett. in press. |

50. | J. Hukriede, D. Runde, and D. Kip, “Fabrication and application of holographic Bragg gratings in lithium niobate channel waveguides,” J. Phys. D Appl. Phys. |

51. | G. J. Milburn, J. Corney, E. M. Wright, and D. F. Walls, “Quantum dynamics of an atomic Bose-Einstein condensate in a double-well potential,” Phys. Rev. A |

52. | A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, “Quantum coherent atomic tunneling between two trapped Bose-Einstein condensates,” Phys. Rev. Lett. |

53. | S. Raghavan, A. Smerzi, S. Fantoni, and S. R. Shenoy, “Coherent oscillations between two weakly coupled Bose-Einstein condensates: Josephson effects, π oscillations, and macroscopic quantum self-trapping,” Phys. Rev. A |

54. | M. Albiez, R. Gati, J. Fölling, S. Hunsmann, M. Cristiani, and M. K. Oberthaler, “Direct observation of tunneling and nonlinear self-trapping in a single bosonic Josephson junction,” Phys. Rev. Lett. |

55. | A. Sigler and B. A. Malomed, “Solitary pulses in linearly coupled cubic-quintic Ginzburg-Landau equations,” Physica D |

56. | L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R. Soc. Lond. A Math. Phys. Sci. |

57. | N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids |

58. | J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg–Landau equations,” Phys. Lett. A |

**OCIS Codes**

(190.5940) Nonlinear optics : Self-action effects

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 20, 2011

Revised Manuscript: January 12, 2012

Manuscript Accepted: January 12, 2012

Published: January 20, 2012

**Citation**

Olga V. Borovkova, Yaroslav V. Kartashov, Victor A. Vysloukh, Valery E. Lobanov, Boris A. Malomed, and Lluis Torner, "Solitons supported by spatially inhomogeneous nonlinear losses," Opt. Express **20**, 2657-2667 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-3-2657

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### References

- Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003).
- V. A. Brazhnyi and V. V. Konotop, “Theory of nonlinear matter waves in optical lattices,” Mod. Phys. Lett. B 18(14), 627 (2004). [CrossRef]
- O. Morsch and M. Oberthaler, “Dynamics of Bose-Einstein condensates in optical lattices,” Rev. Mod. Phys. 78(1), 179–215 (2006). [CrossRef]
- L. Torner, “Walkoff-compensated dispersion-mapped quadratic solitons,” IEEE Photon. Technol. Lett. 11(10), 1268–1270 (1999). [CrossRef]
- I. Towers and B. A. Malomed, “Stable (2+1)-dimensional solitons in a layered medium with sign-alternating Kerr nonlinearity,” J. Opt. Soc. Am. 19(3), 537 (2002). [CrossRef]
- F. K. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, “Controlling collapse in Bose-Einstein condensates by temporal modulation of the scattering length,” Phys. Rev. A 67(1), 013605 (2003). [CrossRef]
- H. Saito and M. Ueda, “Dynamically stabilized bright solitons in a two-dimensional bose-einstein condensate,” Phys. Rev. Lett. 90(4), 040403 (2003). [CrossRef] [PubMed]
- M. Centurion, M. A. Porter, P. G. Kevrekidis, and D. Psaltis, “Nonlinearity management in optics: experiment, theory, and simulation,” Phys. Rev. Lett. 97(3), 033903 (2006). [CrossRef] [PubMed]
- B. A. Malomed, Soliton Management in Periodic Systems (Springer, New York, 2006).
- Y. V. Kartashov, B. A. Malomed, and L. Torner, “Solitons in nonlinear lattices,” Rev. Mod. Phys. 83(1), 247–306 (2011). [CrossRef]
- H. Sakaguchi and B. A. Malomed, “Matter-wave solitons in nonlinear optical lattices,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 046610 (2005). [CrossRef] [PubMed]
- G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217(1), 31–57 (2006). [CrossRef]
- J. Garnier and F. K. Abdullaev, “Transmission of matter wave solitons through nonlinear traps and barriers,” Phys. Rev. A 74(1), 013604 (2006). [CrossRef]
- D. L. Machacek, E. A. Foreman, Q. E. Hoq, P. G. Kevrekidis, A. Saxena, D. J. Frantzeskakis, and A. R. Bishop, “Statics and dynamics of an inhomogeneously nonlinear lattice,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 74(3), 036602 (2006). [CrossRef] [PubMed]
- Y. Kominis and K. Hizanidis, “Lattice solitons in self-defocusing optical media: analytical solutions of the nonlinear Kronig-Penney model,” Opt. Lett. 31(19), 2888–2890 (2006). [CrossRef] [PubMed]
- J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and P. J. Torres, “Lie symmetries and solitons in nonlinear systems with spatially inhomogeneous nonlinearities,” Phys. Rev. Lett. 98(6), 064102 (2007). [CrossRef] [PubMed]
- P. Niarchou, G. Theocharis, P. G. Kevrekidis, P. Schmelcher, and D. J. Frantzeskakis, “Soliton oscillations in collisionally inhomogeneous attractive Bose-Einstein condensates,” Phys. Rev. A 76(2), 023615 (2007). [CrossRef]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton modes, stability, and drift in optical lattices with spatially modulated nonlinearity,” Opt. Lett. 33(15), 1747–1749 (2008). [CrossRef] [PubMed]
- Y. Sivan, G. Fibich, and B. Ilan, “Drift instability and tunneling of lattice solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(4), 045601 (2008). [CrossRef] [PubMed]
- F. K. Abdullaev, A. Gammal, M. Salerno, and L. Tomio, “Localized modes of binary mixtures of Bose-Einstein condensates in nonlinear optical lattices,” Phys. Rev. A 77(2), 023615 (2008). [CrossRef]
- H. A. Cruz, V. A. Brazhnyi, and V. V. Konotop, “Partial delocalization of two-component condensates in optical lattices,” J. Phys. B 41(3), 035304 (2008). [CrossRef]
- H. Sakaguchi and B. A. Malomed, “Solitons in combined linear and nonlinear lattice potentials,” Phys. Rev. A 81(1), 013624 (2010). [CrossRef]
- V. Pérez-García and R. Pardo, “Localization phenomena in nonlinear Schrödinger equations with spatially inhomogeneous nonlinearities: Theory and applications to Bose-Einstein condensates,” Physica D 238(15), 1352–1361 (2009). [CrossRef]
- B. B. Baizakov, B. A. Malomed, and M. Salerno, “Multidimensional solitons in periodic potentials,” Europhys. Lett. 63(5), 642–648 (2003). [CrossRef]
- D. Neshev, A. Sukhorukov, Y. Kivshar, E. Ostrovskaya, and W. Krolikowski, “Spatial solitons in optically induced gratings,” Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]
- J. Yang and Z. H. Musslimani, “Fundamental and vortex solitons in a two-dimensional optical lattice,” Opt. Lett. 28(21), 2094–2096 (2003). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express 12(13), 2831–2837 (2004). [CrossRef] [PubMed]
- Y. J. He and H. Z. Wang, “(1+1)-dimensional dipole solitons supported by optical lattice,” Opt. Express 14(21), 9832–9837 (2006). [CrossRef] [PubMed]
- O. V. Borovkova, Y. V. Kartashov, B. A. Malomed, and L. Torner, “Algebraic bright and vortex solitons in defocusing media,” Opt. Lett. 36(16), 3088–3090 (2011). [CrossRef] [PubMed]
- O. V. Borovkova, Y. V. Kartashov, L. Torner, and B. A. Malomed, “Bright solitons from defocusing nonlinearities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 84(3), 035602 (2011). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, L. Torner, and B. A. Malomed, “Self-trapping and splitting of bright vector solitons under inhomogeneous defocusing nonlinearities,” Opt. Lett. 36(23), 4587–4589 (2011). [CrossRef] [PubMed]
- W.-P. Zhong, M. Belić, G. Assanto, B. A. Malomed, and T. Huang, “Light bullets in the spatiotemporal nonlinear Schrödinger equation with a variable negative diffraction coefficient,” Phys. Rev. A 84(4), 043801 (2011). [CrossRef]
- B. A. Malomed, “Evolution of nonsoliton and “quasi-classical” wavetrains in nonlinear Schrödinger and Korteweg-de Vries equations with dissipative perturbations,” Physica D 29(1-2), 155–172 (1987). [CrossRef]
- O. Thual and S. Fauve, “Localized structures generated by subcritical instabilities,” J. Phys. France 49(11), 1829–1833 (1988). [CrossRef]
- W. van Saarloos and P. C. Hohenberg, “Pulses and fronts in the complex Ginzburg-Landau equation near a subcritical bifurcation,” Phys. Rev. Lett. 64(7), 749–752 (1990). [CrossRef] [PubMed]
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