## Solitons in nonlocal nonlinear kerr media with exponential response function |

Optics Express, Vol. 20, Issue 7, pp. 7469-7479 (2012)

http://dx.doi.org/10.1364/OE.20.007469

Acrobat PDF (968 KB)

### Abstract

In this paper, we find some exact analytical solutions including bright soliton solution, dipole-mode soliton solution, double soliton solution and periodic solution when a slit laser beam propagates in Kerr-type nonlinear, nonlocal media with exponential response function. Furthermore, we address the energy flow is a monotonically growing function of *d*_{2} and the Hamiltonian decreases while the energy flow increases. And we also obtain an Airy-like soliton by numerical method.

© 2012 OSA

## 1. Introduction

2. M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today **51**(8), 42–45 (1998). [CrossRef]

3. W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. **6**, 288–294 (2004). [CrossRef]

4. M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. **80**, 4657–4600 (1998). [CrossRef]

5. A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A **56**, 1110–1113 (1997). [CrossRef]

6. M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. **27**, 1460–1462 (2002). [CrossRef]

7. M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. **30**, 415–417 (2005). [CrossRef] [PubMed]

9. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. **36**, 3–8 (1965). [CrossRef]

10. L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose-Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. **85**, 1791–1794 (2000). [CrossRef] [PubMed]

11. V. M. Perez-Garcia, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E **62**, 4300–4308 (2000). [CrossRef]

12. W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E **64**, 016612–016619 (2001). [CrossRef]

13. M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E **68**, 025602–025605 (2003). [CrossRef]

14. S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. **64**, 797–801 (1985). [CrossRef]

16. D. Neshev, G. McCarthy, and W. Królikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. **26**, 1185–1187 (2001). [CrossRef]

17. W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E **70**, 036617–036621 (2004). [CrossRef]

18. J. I. Yakimenko, Y.A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E **71**, 065603 (2005). [CrossRef]

19. Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A **80**, 013838–013844 (2009). [CrossRef]

20. N. I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. **29**, 286–288 (2004). [CrossRef] [PubMed]

21. A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. **96**, 043901 (2006). [CrossRef] [PubMed]

22. S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. **31**, 1100–1102 (2006). [CrossRef] [PubMed]

23. N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E **68**, 036614–036618 (2003). [CrossRef]

24. P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E **73**, 036614–036623 (2006). [CrossRef]

25. M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. **32**, 2490–2492 (2007). [CrossRef] [PubMed]

26. M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express **16**, 3273–3287 (2008). [CrossRef] [PubMed]

12. W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E **64**, 016612–016619 (2001). [CrossRef]

27. J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E **66**, 066615–066627 (2002). [CrossRef]

*q*and nonlinear correction to the refractive index

*n*describing the propagation of a slit laser beam along the

*ξ*axis of a nonlocal focusing Kerr-type medium: where the two variables, namely,

*η*and

*ξ*stand for the transverse and longitudinal coordinates scaled to the the beam width and the diffraction length, respectively, and the parameter

*d*stands for the degree of nonlocality of the nonlinear response. The model (1) for diffusive non-linerities becomes identical to the model for quadratic nonlinearity which is also determined by an exponential response function [23

23. N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E **68**, 036614–036618 (2003). [CrossRef]

*d*→ 0, Eq. (1) describes a local nonlinear response. Contrarily, the case

*d*→ ∞ corresponds to the strongly nonlocal response. The nonlinear contribution to refractive index is given by

*G*(

*η*) = (1/2

*d*

^{1/2})×exp(− |

*η*|/

*d*

^{1/2}) is the response function of the nonlocal medium. As far as we know, such equations have been already studied by the numerical method and the bright, dark, and gray solitons were obtained. It had been proved that these solitons could exist under certain values of the degree of nonlocality of the nonlinear response. Besides, multiple-mode solitons also are found and bound states are stable if they contain fewer than five solitons [28

28. Z. Y. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. **30**, 3171–3173 (2005). [CrossRef] [PubMed]

## 2. The classical Lie-group reduction

29. P. J. Olver, *Applications of Lie Group to Differential Equations* (Spinger,Berlin, 1986). [CrossRef]

30. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. **23**, 171–172 (1973). [CrossRef]

*η*,

*ξ*,

*q*,

*q*

^{*},

*n*) where

*ε*is an infinite parameter. The corresponding of infinitesimal symmetries is the set of vector fields of the form According to the infinitesimal invariance criterion for symmetry, Eq. (1) is invariant under the transformations (2) if and only if

*Pr*

^{(2)}

*V*is the second prolongation of

*V*where

*ζ*,

^{ξ}*ζ*and

^{ηη}*δ*are given explicitly in terms of invariants

^{ηη}*χ*,

*τ*,

*ζ*and

*n*and their derivatives, Solving Eq. (4) by substituting of Eqs. (5) and (6) into it and collecting together the coefficients of like-derivative terms of

*q*and

*n*, then setting all of them to zero, we get a system of linear partial differential equations from which we can find

*χ*,

*τ*,

*ζ*,

*ζ*

^{*}and

*δ*, where a, b,

*c*

_{0}and

*c*

_{1}are constants.

*ψ*, similarity solutions

*q*and

*n*by integrating the following characteristic equations: Taking the constant

*b*is zero or not in Eq. (7), we can obtain two different types of similarity reductions of Eq. (8). When

*b*is zero, Eq. (7) becomes By solving the characteristic equations Eq. (8), We have where the integral invariant

*u*(

*ψ*) stands for a normalized real function. The quantity

*u*= d

_{ψ}*u*/d

*ψ*. In the other case, we will study Eq. (7) with

*b*≠ 0. Now the substitution of Eq. (7) into Eq. (8) arrives at where

*c*

_{3}is an integral constant. Making use of Eqs. (1) and (13), we obtain

## 3. Analytic solutions and numerical solution

*u*and

*Q*.

### 3.1. Soliton solutions

#### 3.1.1. Bright soliton

*c*

_{0},

*c*

_{1}are arbitrary constants and

*c*

_{2}is zero. As Figs. 1(a) and 1(b) show, it is evident that we can find out that

*u*and

*Q*are ground-state bright solitons. We can recall the properties of ground-state solitons, namely, the width of a ground-state soliton increases while its peak amplitude decreases with increasing degree of nonlocality

*d*according to Ref. [31

31. W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E **63**, 016610–016615 (2000). [CrossRef]

*U*is a monotonically growing function of

*d*

_{2}. As

*d*

_{2}→ 0 the soliton broadens drastically while its energy flow vanishes. The soliton is stable in the entire domain of its existence and achieve the absolute minimum of Hamiltonian

*H*for a fixed energy flow

*U*[Fig. 1(d)]. And, the analytical solution we get is different from the steady-state analytical solution because the soliton will evolve along line

*c*

_{0}= 0, the solution is reduced to which is equal to the steady-state analytical solution.

#### 3.1.2. Dipole-mode soliton

*π*phase difference between solitons leads to a local decrease of refractive index in the overlap region and results in repulsion. By comparison, the whole intensity distribution in the transverse direction decides the refractive-index change in the overlap region in nonlocal media. And under appropriate conditions the nonlocality can cause an increase in refractive index and attraction between solitons. Thus, the proper choice of separation between solitons forms bound state. In fact, we can find the bright and dipole-mode solitons we obtain are similar to the approximate analytical solutions in quadratic nonlinear materials with exponential response function [23

23. N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E **68**, 036614–036618 (2003). [CrossRef]

#### 3.1.3. Double soliton

*c*

_{2}= 0. Obviously, as the Figs. 3(a), 3(b) and 3(c) show, the solution we obtain is double soliton. The center of double soliton is located in

*η*= 0 when the light is incident on the media, namely

*ξ*= 0. With the light propagates along the

*ξ*axis, although the center of the solition moves right, the peak amplitude is still invariant.

### 3.2. Period solutions

#### 3.2.1. *sn* type period solution

*α*= 2

*m*

^{4}+

*m*

^{2}+ 2 and

*sn*is the usual Jacobi elliptic sine function and

*m*is the modular of the function

*sn*. And it is a periodic solution when 0 <

*m*< 1 [see Fig. 4(a)]. The substitution of Eq. (23) into Eq.(10), then arrives at

*d*

_{2}.

#### 3.2.2. *sn*^{−}^{1} type divergent solution

*c*

_{0},

*c*

_{1}are arbitrary constants and

*c*

_{2}= 0.

#### 3.2.3. *cn**−**dn* type period solution

### 3.3. Airy-like solution

*c*

_{3}= 0, Eq. (14) can be given Here, we can search for stationary soliton solutions of Eqs. (15) and (32) numerically. We get an Airy-like solution when

*ρ*= −8,

*κ*= −0.01,

*λ*= 3.1 and

*d*= 0.1. As Fig. 5(a) displays, the amplitude of the light field has a strong Airy tail. It sharply increases when

*ψ*< 0, while it possesses the maximum at

*ψ*= 0. Then it oscillates around

*u*= 0 and decays. However, the amplitude of the refractive index oscillates above

*u*= 0 and decays. At last, it tends to a constant which is greater than zero.

## 4. Conclusions

## Acknowledgments

## References and links

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

2. | M. Segev and G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today |

3. | W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. |

4. | M. Mitchell, M. Segev, and D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. |

5. | A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, and M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A |

6. | M. Peccianti, K. A. Brzdakiewicz, and G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. |

7. | M. Peccianti, C. Conti, and G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. |

8. | A. G. Litvak, V. A. Mironov, G. M. Fraiman, and A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. |

9. | J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. |

10. | L. Santos, G. V. Shlyapnikov, P. Zoller, and M. Lewenstein, “Bose-Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. |

11. | V. M. Perez-Garcia, V. V. Konotop, and J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E |

12. | W. Królikowski, O. Bang, J. J. Rasmussen, and J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E |

13. | M. Peccianti, C. Conti, and G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E |

14. | S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. |

15. | O. Bang, W. Królikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E |

16. | D. Neshev, G. McCarthy, and W. Królikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. |

17. | W. Królikowski, O. Bang, and J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E |

18. | J. I. Yakimenko, Y.A. Zaliznyak, and Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E |

19. | Y. Y. Lin, R. K. Lee, and B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A |

20. | N. I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J. J. Rasmussen, and P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. |

21. | A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, and W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. |

22. | S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, and O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. |

23. | N. I. Nikolov, D. Neshev, O. Bang, and W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E |

24. | P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, and S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E |

25. | M. Bache, O. Bang, J. Moses, and F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. |

26. | M. Bache, O. Bang, W. Królikowski, J. Moses, and F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express |

27. | J. Wyller, W. Królikowski, O. Bang, and J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E |

28. | Z. Y. Xu, Y. V. Kartashov, and L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. |

29. | P. J. Olver, |

30. | A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. |

31. | W. Królikowski and O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 7, 2011

Revised Manuscript: February 16, 2012

Manuscript Accepted: February 27, 2012

Published: March 19, 2012

**Citation**

Jian Jia and Ji Lin, "Solitons in nonlocal nonlinear kerr media with exponential response function," Opt. Express **20**, 7469-7479 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-7-7469

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### References

- Y. S. Kivshar, G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, London, 2003).
- M. Segev, G. I. Stegeman, “Self-Trapping of Optical Beams: Spatial solitons,” Phys. Today 51(8), 42–45 (1998). [CrossRef]
- W. Królikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, D. Edmundson, “Modulational instability, solitonsand beam propagation in spatially nonlocal nonlinear media,” J. Opt. B. 6, 288–294 (2004). [CrossRef]
- M. Mitchell, M. Segev, D. N. Christodoulides, “Observation of Multihump Multimode Solitons,” Phys. Rev. Lett. 80, 4657–4600 (1998). [CrossRef]
- A. V. Mamaev, A. A. Zozulya, V. K. Mezentsev, D. Z. Anderson, M. Saffman, “Bound dipole solitary solutions in anisotropic nonlocal self-focusing media,” Phys. Rev. A 56, 1110–1113 (1997). [CrossRef]
- M. Peccianti, K. A. Brzdakiewicz, G. Assanto, “Nonlocal spatial soliton interactions in nematic liquid crystals,” Opt. Lett. 27, 1460–1462 (2002). [CrossRef]
- M. Peccianti, C. Conti, G. Assanto, “Interplay between nonlocality and nonlinearity in nematic liquid crystals,” Opt. Lett. 30, 415–417 (2005). [CrossRef] [PubMed]
- A. G. Litvak, V. A. Mironov, G. M. Fraiman, A. D. Yunakovskii, “Direct measurement of the attenuation length of extensive air showers,” Sov. J. Plasma Phys. 1, 31–33 (1975).
- J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, “Long-transient effects in lasers with inserted liquid samples,” J. Appl. Phys. 36, 3–8 (1965). [CrossRef]
- L. Santos, G. V. Shlyapnikov, P. Zoller, M. Lewenstein, “Bose-Einstein condensation in trapped dipolar gases,” Phys. Rev. Lett. 85, 1791–1794 (2000). [CrossRef] [PubMed]
- V. M. Perez-Garcia, V. V. Konotop, J. J. García-Ripoll, “Dynamics of quasicollapse in nonlinear Schrödinger systems with nonlocal interactions,” Phys. Rev. E 62, 4300–4308 (2000). [CrossRef]
- W. Królikowski, O. Bang, J. J. Rasmussen, J. Wyller, “Modulational instability in nonlocal nonlinear Kerr media,” Phys. Rev. E 64, 016612–016619 (2001). [CrossRef]
- M. Peccianti, C. Conti, G. Assanto, “Optical modulational instability in a nonlocal medium,” Phys. Rev. E 68, 025602–025605 (2003). [CrossRef]
- S. K. Turitsyn, “Spatial dispersion of nonlinearity and stability of multidimensional solitons,” Theor. Math. Phys. 64, 797–801 (1985). [CrossRef]
- O. Bang, W. Królikowski, J. Wyller, J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66, 046619–046623 (2002). [CrossRef]
- D. Neshev, G. McCarthy, W. Królikowski, “Dipole-mode vector solitons in anisotropic nonlocal self-focusing media,” Opt. Lett. 26, 1185–1187 (2001). [CrossRef]
- W. Królikowski, O. Bang, J. Wyller, “Nonlocal incoherent solitons,” Phys. Rev. E 70, 036617–036621 (2004). [CrossRef]
- J. I. Yakimenko, Y.A. Zaliznyak, Y. Kivshar, “Stable vortex solitons in nonlocal self-focusing nonlinear media,” Phys. Rev. E 71, 065603 (2005). [CrossRef]
- Y. Y. Lin, R. K. Lee, B. A. Malomed, “Bragg solitons in nonlocal nonlinear media,” Phys. Rev. A 80, 013838–013844 (2009). [CrossRef]
- N. I. Nikolov, D. Neshev, W. Królikowski, O. Bang, J. J. Rasmussen, P. L. Christiansen, “Attraction of nonlocal dark optical solitons,” Opt. Lett. 29, 286–288 (2004). [CrossRef] [PubMed]
- A. Dreischuh, D. N. Neshev, D. E. Petersen, O. Bang, W. Królikowski, “Observation of attraction between dark solitons,” Phys. Rev. Lett. 96, 043901 (2006). [CrossRef] [PubMed]
- S. Lopez-Aguayo, A. S. Desyatnikov, Y. S. Kivshar, S. Skupin, W. Królikowski, O. Bang, “Stable rotating dipole solitons in nonlocal optical media,” Opt. Lett. 31, 1100–1102 (2006). [CrossRef] [PubMed]
- N. I. Nikolov, D. Neshev, O. Bang, W.Z. Królikowski, “Quadratic solitons as nonlocal solitons,” Phys. Rev. E 68, 036614–036618 (2003). [CrossRef]
- P. V. Larsen, M. P. Sørensen, O. Bang, W. Z. Królikowski, S. Trillo, “Nonlocal description of X waves in quadratic nonlinear materials,” Phys. Rev. E 73, 036614–036623 (2006). [CrossRef]
- M. Bache, O. Bang, J. Moses, F. W. Wise, “Nonlocal explanation of stationary and nonstationary regimes in cascaded soliton pulse compression,” Opt. Lett. 32, 2490–2492 (2007). [CrossRef] [PubMed]
- M. Bache, O. Bang, W. Królikowski, J. Moses, F. W. Wise, “Limits to compression with cascaded quadratic soliton compressors,” Opt. Express 16, 3273–3287 (2008). [CrossRef] [PubMed]
- J. Wyller, W. Królikowski, O. Bang, J. J. Rasmussen, “Generic features of modulational instability in nonlocal Kerr media,” Phys. Rev. E 66, 066615–066627 (2002). [CrossRef]
- Z. Y. Xu, Y. V. Kartashov, L. Torner, “Upper threshold for stability of multipole-mode solitons in nonlocal nonlinear media,” Opt. Lett. 30, 3171–3173 (2005). [CrossRef] [PubMed]
- P. J. Olver, Applications of Lie Group to Differential Equations (Spinger,Berlin, 1986). [CrossRef]
- A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. II. Normal dispersion,” Appl. Phys. Lett. 23, 171–172 (1973). [CrossRef]
- W. Królikowski, O. Bang, “Solitons in nonlocal nonlinear media: Exact solutions,” Phys. Rev. E 63, 016610–016615 (2000). [CrossRef]

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