## Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity |

JOSA B, Vol. 30, Issue 5, pp. 1276-1283 (2013)

http://dx.doi.org/10.1364/JOSAB.30.001276

Enhanced HTML Acrobat PDF (840 KB)

### Abstract

Using multivariate self-similarity transformation, we construct explicit spatial bright and dark solitary wave solutions of the generalized nonlinear Schrödinger equation with spatially Bessel-modulated nonlinearity and an external potential. Special kinds of explicit solutions, such as periodically breathing bright and dark solitary waves, are discussed in detail. The stability of these solutions is verified by means of direct numerical simulation.

© 2013 Optical Society of America

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 17, 2013

Revised Manuscript: March 15, 2013

Manuscript Accepted: March 15, 2013

Published: April 19, 2013

**Citation**

Wei-Ping Zhong, Milivoj R. Belić, and Tingwen Huang, "Solitary waves in the nonlinear Schrödinger equation with spatially modulated Bessel nonlinearity," J. Opt. Soc. Am. B **30**, 1276-1283 (2013)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-30-5-1276

Sort: Year | Journal | Reset

### References

- A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford University, 1995).
- F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, “Theory of Bose-Einstein condensation in trapped gases,” Rev. Mod. Phys. 71, 463–512 (1999). [CrossRef]
- L. P. Pitaevski and S. Stringari, Bose-Einstein Condensation (Oxford University, 2003).
- B. B. Baizakov, G. Filatrella, B. A. Malomed, and M. Salerno, “Double parametric resonance for matter-wave solitons in a time-modulated trap,” Phys. Rev. E 71, 036619 (2005). [CrossRef]
- C. Sulem, and P. L. Sulem, The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse (Springer-Verlag, 1999).
- B. A. Malomed, Soliton Manegement in Periodic Systems (Springer, 2006).
- Y. S. Kivshar, and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003).
- V. N. Serkin, and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. 85, 4502–4505 (2000). [CrossRef]
- W. P. Zhong, R. H. Xie, M. Belić, N. Petrović, and G. Chen, “Exact spatial soliton solutions of the two-dimensional generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. A 78, 023821 (2008). [CrossRef]
- M. Belić, N. Petrović, W. P. Zhong, R. H. Xie, and G. Chen, “Analytical light bullet solutions to the generalized (1+3)-dimensional nonlinear Schrödinger equation,” Phys. Rev. Lett. 101, 123904 (2008). [CrossRef]
- H. H. Chen, and C. S. Liu, “Solitons in nonuniform media,” Phys. Rev. Lett. 37, 693–697 (1976). [CrossRef]
- H. H. Chen, and C. S. Liu, “Nonlinear wave and soliton propagation in media with arbitrary inhomogeneities,” Phys. Fluids 21, 377–380 (1978). [CrossRef]
- S. P. Burtsev, and I. R. Gabitov, “Alternative integrable equations of nonlinear optics,” Phys. Rev. A 49, 2065–2070 (1994). [CrossRef]
- F. Calogero, and A. Degasperis, “Coupled nonlinear evolution equations solvable via the inverse spectral transform, and solitons that come back: the boomeron,” Lett. Nuovo Cimento 16, 425–433 (1976). [CrossRef]
- V. V. Konotop, “Soliton on a disordered lattice,” Phys. Rev. E 47, 1423–1426 (1993). [CrossRef]
- V. V. Konotop, O. A. Chubykalo, and L. Vazquez, “Dynamics and interaction of solitons on an integrable inhomogeneous lattice,” Phys. Rev. E 48, 563–568 (1993). [CrossRef]
- V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. 98, 074102 (2007). [CrossRef]
- V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous matter-wave solitons near the Feshbach resonance,” Phys. Rev. A 81, 023610 (2010). [CrossRef]
- A. T. Avelar, D. Bazeia, and W. B. Cardoso, “Solitons with cubic and quintic nonlinearities modulated in space and time,” Phys. Rev. E 79, 025602 (2009). [CrossRef]
- W. B. Cardoso, A. T. Avelar, and D. Bazeia, “Modulation of breathers in cigar- shaped Bose–Einstein condensates,” Phys. Lett. A 374, 2640–2645 (2010). [CrossRef]
- W. B. Cardoso, A. T. Avelar, D. Bazeia, and M. S. Hussein, “Solitons of two-component Bose–Einstein condensates modulated in space and time,” Phys. Lett. A 374, 2356–2360 (2010). [CrossRef]
- J. Belmonte-Beitia, and G. F. Calvo, “Exact solutions for the quintic nonlinear Schrödinger equation with time and space modulated nonlinearities and potentials,” Phys. Lett. A 373, 448–453 (2009). [CrossRef]
- 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, 064102 (2007). [CrossRef]
- J. Belmonte-Beitia, V. M. Pérez-García, V. Vekslerchik, and V. V. Konotop, “Localized nonlinear waves in systems with time- and space-modulated nonlinearities,” Phys. Rev. Lett. 100, 164102 (2008). [CrossRef]
- X. G. He, D. Zhao, L. Li, and H. G. Luo, “Engineering integrable nonautonomous nonlinear Schrödinger equations,” Phys. Rev. E 79, 056610 (2009). [CrossRef]
- F. Kh. Abdullaev, A. M. Kamchatnov, V. V. Konotop, and V. A. Brazhnyi, “Adiabatic dynamics of periodic waves in Bose-Einstein condensates with time dependent atomic scattering length,” Phys. Rev. Lett. 90, 230402 (2003). [CrossRef]
- G. Fibich, Y. Sivan, and M. I. Weinstein, “Bound states of nonlinear Schrödinger equations with a periodic nonlinear microstructure,” Physica D 217, 31–57 (2006). [CrossRef]
- 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, 043801 (2011). [CrossRef]
- W. P. Zhong and M. Belić, “Traveling wave and soliton solutions of coupled nonlinear Schrödinger equations with variable coefficients,” Phys. Rev. E 82, 047601 (2010). [CrossRef]
- W. P. Zhong, M. Belić, Y. Lu, and T. Huang, “Traveling and solitary wave solutions to the one-dimensional Gross-Pitaevskii equation,” Phys. Rev. E 81, 016605 (2010). [CrossRef]
- Z. Bouchal, “Nondifracting optical beams,” Czech. J. Phys. 53, 537–578 (2003). [CrossRef]
- J. Durnin, “Exact solutions for nondiffracting beams,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef]
- D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005). [CrossRef]
- P. Johannisson, D. Anderson, M. Lisak, and M. Marklund, “Nonlinear Bessel beams,” Opt. Commun. 222, 107–115(2003). [CrossRef]
- F. Ye, Y. V. Kartashov, B. Hu, and L. Torner, “Light bullets in Bessel optical lattices with spatially modulated nonlinearity,” Opt. Express 17, 11328 (2009). [CrossRef]
- W.-P. Zhong, M. Belić, and T. Huang, “Three-dimensional Bessel light bullets in self-focusing Kerr media,” Phys. Rev. 82, 033834 (2010). [CrossRef]
- W. J. Tomlinson, R. J. Hawkins, A. M. Weiner, J. P. Heritage, and R. N. Thurston, “Dark optical solitons with finite-width background pulses,” J. Opt. Soc. Am. B 6, 329–334 (1989). [CrossRef]
- Y. S. Kivshar and X. Yang, “Dark solitons on background of finite extent,” Opt. Commun. 107, 93–98 (1994). [CrossRef]
- V. M. Perez-Garcia, P. J. Torres, and V. V. Konotop, “Similarity transformation for NLS equations with time-dependent coefficients,” Phys. D 221, 31–36 (2006). [CrossRef]
- B. Yang, W. P. Zhong, and M. Belić, “Self-similar HG spatial solitons in 2-dimensional nonlocal nonlinear media,” Commun. Theor. Phys. 53, 937–942 (2010). [CrossRef]
- W. P. Zhong, M. Belić, and G. Assanto, “Localized nonlinear wavepackets in with radial-azimuthal modulated nonlinearity and an external potential,” Phys. Scripta 84, 055001 (2011). [CrossRef]
- C. Q. Dai, R. P. Chen, and Y. Y. Wang, “Spatiotemporal self-similar solutions for the nonautonomous (3+1)D cubic-quintic GP equation,” Chin. Phys. B 21, 030508 (2012). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.