## Similaritons in nonlinear optical systems

Optics Express, Vol. 16, Issue 9, pp. 6352-6360 (2008)

http://dx.doi.org/10.1364/OE.16.006352

Acrobat PDF (283 KB)

### Abstract

By using the lens-type transformation, exact soliton and quasi-soliton similaritons are found in (1+1), (2+1) and (3+1)-dimensional nonlinear Schrödinger equations in the context of nonlinear optical fiber amplifiers and graded-index waveguide amplifiers. The novel analytical and numerical results show that, in addition to the exact solitonic optical waves, quasi-solitonic optical waves with Gaussian, parabolic, vortex and ring soliton profiles can evolve exact self-similarly without any radiation.

© 2008 Optical Society of America

## 1. Introduction

1. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. **84**, 6010–6013 (2000). [CrossRef] [PubMed]

2. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. **25**, 1753–1755 (2000). [CrossRef]

3. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B **19**, 461–468 (2002). [CrossRef]

4. G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, “Self-similar parabolic beam generation and propagation,” Phys. Rev. E **72**, 016609 (2005). [CrossRef]

5. S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E **72**, 016622 (2005). [CrossRef]

6. V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. **85**, 4502 (2000). [CrossRef] [PubMed]

7. V. N. Serkin and A. Hasegawa, “Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain,” JETP Letters , **72**, 89–92 (2000). [CrossRef]

8. V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE J. Sel. Top. Quantum Electron. **8**, 418 (2002). [CrossRef]

9. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. **90**, 113902 (2003). [CrossRef] [PubMed]

10. V. M. Pérez-García, P. J. Torres, and V. V. Konotop, “Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients,” Physica D **221**, 31–36 (2006). [CrossRef]

11. S. A. Ponomarenko and G. P. Agrawal, “Do solitonlike self-similar waves exist in nonlinear Optical media?” Phys. Rev. Lett. **97**, 013901 (2006). [CrossRef] [PubMed]

12. S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express **15**, 2963–2973 (2007). [CrossRef] [PubMed]

13. S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. **32**, 1659–1661 (2007). [CrossRef] [PubMed]

14. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. **98**, 074102 (2007). [CrossRef] [PubMed]

1. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. **84**, 6010–6013 (2000). [CrossRef] [PubMed]

2. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. **25**, 1753–1755 (2000). [CrossRef]

3. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B **19**, 461–468 (2002). [CrossRef]

4. G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, “Self-similar parabolic beam generation and propagation,” Phys. Rev. E **72**, 016609 (2005). [CrossRef]

5. S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E **72**, 016622 (2005). [CrossRef]

6. V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. **85**, 4502 (2000). [CrossRef] [PubMed]

7. V. N. Serkin and A. Hasegawa, “Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain,” JETP Letters , **72**, 89–92 (2000). [CrossRef]

8. V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE J. Sel. Top. Quantum Electron. **8**, 418 (2002). [CrossRef]

9. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. **90**, 113902 (2003). [CrossRef] [PubMed]

10. V. M. Pérez-García, P. J. Torres, and V. V. Konotop, “Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients,” Physica D **221**, 31–36 (2006). [CrossRef]

11. S. A. Ponomarenko and G. P. Agrawal, “Do solitonlike self-similar waves exist in nonlinear Optical media?” Phys. Rev. Lett. **97**, 013901 (2006). [CrossRef] [PubMed]

12. S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express **15**, 2963–2973 (2007). [CrossRef] [PubMed]

13. S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. **32**, 1659–1661 (2007). [CrossRef] [PubMed]

14. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. **98**, 074102 (2007). [CrossRef] [PubMed]

12. S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express **15**, 2963–2973 (2007). [CrossRef] [PubMed]

6. V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. **85**, 4502 (2000). [CrossRef] [PubMed]

8. V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE J. Sel. Top. Quantum Electron. **8**, 418 (2002). [CrossRef]

14. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. **98**, 074102 (2007). [CrossRef] [PubMed]

7. V. N. Serkin and A. Hasegawa, “Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain,” JETP Letters , **72**, 89–92 (2000). [CrossRef]

10. V. M. Pérez-García, P. J. Torres, and V. V. Konotop, “Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients,” Physica D **221**, 31–36 (2006). [CrossRef]

16. C. Hernandez Tenorio, E. Villagran Vargas, V. N. Serkin, M. Agüero Granados, T. L. Belyaeva, R. Peña Moreno, and L. Morales Lara, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: II. Dark solitons,” Quantum Electron. **10**, 929–937 (2005). [CrossRef]

21. S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. **6**, 372–374 (1997). [CrossRef]

22. Y. Ozeki and T. Inoue, “Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression,” Opt. Lett. **31**, 1606–1608 (2006). [CrossRef] [PubMed]

21. S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. **6**, 372–374 (1997). [CrossRef]

22. Y. Ozeki and T. Inoue, “Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression,” Opt. Lett. **31**, 1606–1608 (2006). [CrossRef] [PubMed]

## 2. Model equations

*n*=

*n*

_{0}+

*n*

_{1}

*f*(

*z*)

*x*

^{2}+

*n*

_{2}|

*I*|

^{2}[11

11. S. A. Ponomarenko and G. P. Agrawal, “Do solitonlike self-similar waves exist in nonlinear Optical media?” Phys. Rev. Lett. **97**, 013901 (2006). [CrossRef] [PubMed]

13. S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. **32**, 1659–1661 (2007). [CrossRef] [PubMed]

*u*, propagation distance

*z*, spatial coordinate

*x*and amplification parameter

*g*(

*z*) are respectively normalized by (

*k*

_{0}|

*n*

_{2}|

*L*)

_{D}^{-1/2},

*L*,

_{D}*w*

_{0}and

*L*

^{-1}

_{D}, with the wavenumber

*k*

_{0}=2

*πn*

_{0}/

*λ*at the input wavelength

*λ*, the diffraction length

*L*=

_{D}*k*

_{0}

*w*

^{2}

_{0}and the characteristic transverse scale

*w*

_{0}=(2

*k*

^{2}

_{0}

*n*

_{1})

^{-1/4}. The nonlinear coefficient is σ=sgn(

*n*

_{2})=±1 and the inhomogeneous parameter is

*f*(

*z*), which describes the inhomogeneity of waveguide. Note that when the amplification parameter vanishes, the dynamics and interaction of bright and dark solitons have been studied [15, 16

16. C. Hernandez Tenorio, E. Villagran Vargas, V. N. Serkin, M. Agüero Granados, T. L. Belyaeva, R. Peña Moreno, and L. Morales Lara, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: II. Dark solitons,” Quantum Electron. **10**, 929–937 (2005). [CrossRef]

**85**, 4502 (2000). [CrossRef] [PubMed]

9. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. **90**, 113902 (2003). [CrossRef] [PubMed]

**15**, 2963–2973 (2007). [CrossRef] [PubMed]

*ψ*(

*Z*,

*T*) is the pulse envelope in comoving coordinates,

*β*(

*Z*) is the group velocity dispersion parameter,

*γ*(

*Z*) is the nonlinearity parameter, and

*G*(

*Z*) is the amplification parameter. In fact, the above two equations are almost identical: if we introduce the transformations

*z*→∫

^{Z}

_{0}

*β*(

*z′*)

*dz′*,

*x*→

*T*and

*g*→

*G*/

*β*-(

*β*

_{Z}*γ*-

*γ*

_{Z}*β*)/

*β*

^{2}

*γ*, then Eq. (2) can be transformed to Eq. (1) with

*σ*=sgn(

*βγ*) and

*f*=0.

*n*=

*n*

_{0}+

*n*

_{1}

*f*(

*z*)(

*x*

^{2}+

*y*

^{2})+

*n*

_{2}|

*I*|

^{2}. The normalizations are the same with that used in Eq. (1). Finally we consider the (3+1)-dimensional NLSE of the form

*n*=

*n*

_{0}+

*n*

_{1}

*f*(

*z*)(

*x*

^{2}+

*y*

^{2})+

*n*

_{2}|

*I*|

^{2}[23

23. S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. **180**, 377–382 (2000). [CrossRef]

*δ*=sgn(

*β*) with

*β*being the group velocity dispersion parameter,

*τ*the retarded time, while other normalizations are the same as used in Eq. (1).

## 3. Similaritons in (1+1)-dimensional systems

### 3.1. Spatial similaritons

*ℓ*(

*z*). Then the scaling property of the beam power and the invariability of the function form of the beam envelope determine the beam amplitude to be proportional to

*ℓ*

_{z}*x*/

*ℓ*, hence the beam phase contains the quadratic term

*ℓ*

_{z}*x*

^{2}/2

*ℓ*[24

24. C. J. Pethick and H. Smith, *Bose-Einstein Condensation in Dilute Gases* (University of Cambridge, Cambridge, 2001). [CrossRef]

*z*should also be scaled by

*ℓ*(

*z*) accordingly. Thus the beam envelope

*u*can be expressed as follows

*Z*=

*Z*(ℓ) and

*X*=

*x*/

*ℓ*. Further calculations show that when

*Z*=

_{z}*ℓ*

^{-2}and

*ℓ*=-

_{z}*gℓ*, the governing equation for

*U*(

*Z*,

*X*) is obtained as

*K*=(

*g*

^{2}–

*g*–

_{z}*f*)

*ℓ*

^{4}. The transformation utilized in Eq. (5) is known as lens-type transformation [25], which has been widely used in BEC and plasma problems [10

**221**, 31–36 (2006). [CrossRef]

26. A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, “Similarity solutions and collapse in the attractive Gross-Pitaevskii equation,” Phys. Rev. E **62**, 6224 (2000). [CrossRef]

27. G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, “Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions,” Phys. Rev. A **67**, 063610 (2003). [CrossRef]

28. J.-K. Xue, “Controllable compression of bright soliton matter waves”, J. Phys. B: At. mol. Opt. Phys. **38**, 3841 (2005). [CrossRef]

29. L. Wu, Q. Yang, and J.-f. Zhang, “Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schrödinger equation in plasma,” J. Phys. A: Math. Gen. **39**, 11947–11953 (2006). [CrossRef]

30. L. Wu, J.-f. Zhang, and L. Li, “Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential,” New J. Phys. **9**, 69 (2007). [CrossRef]

31. L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, “Controllable exact self-similar evolution of the Bose-Einstein condensate”, New J. Phys. **10**, 023021 (2008). [CrossRef]

*K*vanishes, Eq. (6) turns to the well-known standard, homogeneous NLSE which possesses soliton solutions [32]. Therefore, when parameters

*g*(

*z*) and

*f*(

*z*) satisfy the self-similarity condition

*f*=

*g*

^{2}-

*g*, namely,

_{z}*K*=0, the exact solitonic similariton solution for Eq. (1) can be recovered by lens-type transformation (5) with

*ℓ*=exp[-∫

^{z}

_{0}

*g*(

*z*′)

*dz*′] and

*Z*=∫

^{z}

_{0}exp[2∫

^{z}′

_{0}

*g*(

*z*″)

*dz*″]

*dz*. The exact soliton-like similaritons obtained by this way is equivalent to those presented in [11

**97**, 013901 (2006). [CrossRef] [PubMed]

**15**, 2963–2973 (2007). [CrossRef] [PubMed]

**32**, 1659–1661 (2007). [CrossRef] [PubMed]

*K*is constant, Eq. (6) may possess stationary state solution

*U*(

*Z*,

*X*)=

*S*(

*X*)exp(

*iµZ*), where

*µ*is the propagation constant (Note that here the word ”may” means that for negative

*K*, it is difficult to say whether the stationary state solution exists or not, while for positive

*K*the stationary state solution always exists and it is localized). Then, the corresponding exact similariton solution for Eq. (1) can be written as

*g*(

*z*) and inhomogeneous parameter

*f*(

*z*) satisfy the following self-similarity condition

*g*(

*z*), and iii) the exact solitonic similaritons are just a subclass of the general similaritons (7), since when

*K*=0, nontrivial stationary state solution

*U*(

*Z*,

*X*) takes the exact soliton profile.

21. S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. **6**, 372–374 (1997). [CrossRef]

**6**, 372–374 (1997). [CrossRef]

22. Y. Ozeki and T. Inoue, “Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression,” Opt. Lett. **31**, 1606–1608 (2006). [CrossRef] [PubMed]

*S*(

*X*) is chosen to have the same profile with the injected Gaussian beam, only one graded-indexwaveguide amplifier is needed for the exact self-similar propagation of such beam, which means that there will be no radiation. In practice, we can properly choose the parameters

*f*(

*z*) and

*g*(

*z*) according to self-similarity condition (8) to make

*S*(

*X*) asymptotically approach to the Gaussian profile

*A*exp(-

*X*

^{2}/2

*W*

^{2}). We find that when

*A*and

*W*are sufficiently small, then the Kerr nonlinearity can be neglected and when

*K*≈

*W*

^{-4},

*S*(

*X*) do closely approach to Gaussian profile. Numerical calculations of Eq. (6) reveal that even when

*A*=1 and

*W*=0.5, the stationary state solutions

*S*(

*X*) are well described by Gaussian function when

*K*=13.2 for

*σ*=1 and

*K*=18.8 for

*σ*=-1, respectively. The self-similar evolution of this Gaussian beam has been confirmed by numerical simulations of Eq. (1) both for focusing and defocusing nonlinearity, respectively. Specifically, here we choose the gain parameter

*g*(

*z*)=tanh(

*z*). Thus,

*ℓ*=sech(

*z*), which implies that the amplitudes of Gaussian similaritons increase as

*A*/sech(

*z*) while their widths decrease as

*W*sech(

*z*). Figure 1 depicts the excellent agreement between the theoretical predictions and numerical simulations. Further numerical simulations with other amplification parameters and corresponding inhomogeneous parameters lead to similar results.

^{∞}

_{-∞}|

*u*(

*x*,0)|

^{2}

*dx*leads to the negligible Kerr nonlinearity, we find that large initial power results in the negligible diffraction term

*U*when

_{XX}*σ*=-1. In this case, the stationary state solution of Eq. (6) takes the compact parabolic form:

*X*|<

*W*, and

*U*=0 when |

*X*|>

*W*with

*K*=2

*A*

^{2}/

*W*

^{2}. We recall that such parabolic beams can be generated in the planar waveguide [4

4. G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, “Self-similar parabolic beam generation and propagation,” Phys. Rev. E **72**, 016609 (2005). [CrossRef]

*f*(

*z*)=0 and

*g*(

*z*) equals to positive constant

*g*

_{1}. Observe that under such configuration the output beam has a quadratic phase -

*g*

_{1}/6

*x*

^{2}, and its amplitude and width satisfying

*g*should be equal to

*g*

_{1}/3 for the continuity of quadratic phase, meanwhile,

*f*and

*g*should satisfy the similarity condition (8) with

*K*=

*g*

^{2}

_{1}/9. This theoretical prediction is confirmed by the numerical simulation, as shown in Fig. 2.

### 3.2. temporal similaritons

*g*satisfies the following equation

*g*=

*G*/

*β*-(

*β*-

_{Z}γ*γ*)/

_{Z}β*β*

^{2}

*γ*is a function of

*Z*instead of

*z*, so strictly speaking, Eq. (9) should take the form

*K*=0, the exact optical similaritons are solitonic similaritons. One can check that the exact solitonic similariton is equivalent to those found in [6

**85**, 4502 (2000). [CrossRef] [PubMed]

**90**, 113902 (2003). [CrossRef] [PubMed]

**15**, 2963–2973 (2007). [CrossRef] [PubMed]

*K*≠0, the exact optical similaritons are quasi-soliton similaritons [21

**6**, 372–374 (1997). [CrossRef]

**31**, 1606–1608 (2006). [CrossRef] [PubMed]

**6**, 372–374 (1997). [CrossRef]

*β*=1, the self-similarity condition for exact quasi-soliton similaritons is

*c*

_{1}and

*c*

_{2}are constant with

## 4. Similaritons in (2+1)-dimensional systems

*X*,

*Y*)=(

*x*,

*y*)/

*ℓ*,

*Z*=

_{z}*ℓ*

^{-2},

*ℓ*=-

_{z}*gℓ*.

*g*in Eq. (8), it has nothing to do with the amplification parameter anymore: now it is just a sign to identify −

*ℓ*/

_{z}*ℓ*and this means the power of optical similariton is constant, and ii) the change of the amplitude and width of optical beam is determined by

*ℓ*, and

*ℓ*is controlled by

*f*(

*z*). Similar to the discussion in (1+1)-dimensional case, we can choose inhomogeneous parameter

*f*(

*z*) to let the Gaussian or parabolic beam propagate self-similarly in two-dimensional graded-index waveguide. Furthermore, we find that such waveguides also support the exact self-similar evolution of other soliton structures such as ring solitons and vortex rings. Figures 3 and 4 show the examples of the exact self-similar evolution of an initial optical vortex with topological charge 1 and an initial ring soliton in two-dimensional graded-index waveguides. In both cases, the numerical simulations (solid lines) agree well with theoretical predications (marked by cross) which imply that the amplitudes of the vortex and ring solitons are inversely proportional to

*ℓ*while their widths are proportional to

*ℓ*.

## 5. Similaritons in (3+1)-dimensional systems

*X*,

*Y*,

*T*)=(

*x*,

*y*,

*τ*)/

*ℓ*,

*ℓ*=

_{z}*gℓ*and

*Z*=

_{z}*ℓ*

^{-2}. Substituting this in Eq. (4) and performing some manipulations, it can be converted to

*K*and

_{XY}*K*are non-negative constants, Eq. (16) possesses stationary state solutions. This implies that the optical bullets described by Eq. (4) can evolve exact self-similarly under the following exact self-similarity condition, which is obtained by solving Eq. (17)

_{T}*c*

_{1}and

*c*

_{2}are constants satisfying

^{z}

_{0}-

*g*(

*z*′)

*dz*′). This means that when the amplification parameter

*g*(

*t*) is positive/negative, the amplitude of exact self-similar optical bullet decreases/increases and its width increases/decreases, which is in contrast to optical similaritons in (1+1)-dimensional NLSE.

## 6. Conclusion

## Acknowledgments

## References and links

1. | M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-Similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. |

2. | V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high-power parabolic pulses in optical fiber amplifiers,” Opt. Lett. |

3. | V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers,” J. Opt. Soc. Am. B |

4. | G. Chang, H. G. Winful, A. Galvanauskas, and T. B. Norris, “Self-similar parabolic beam generation and propagation,” Phys. Rev. E |

5. | S. Chen, L. Yi, D.-S. Guo, and P. Lu, “Self-similar evolutions of parabolic, Hermite-Gaussian, and hybrid optical pulses: Universality and diversity,” Phys. Rev. E |

6. | V. N. Serkin and A. Hasegawa, “Novel soliton solutions of the nonlinear Schrödinger equation model,” Phys. Rev. Lett. |

7. | V. N. Serkin and A. Hasegawa, “Soliton management in the nonlinear Schrödinger equation model with varying dispersion, nonlinearity, and gain,” JETP Letters , |

8. | V. N. Serkin and A. Hasegawa, “Exactly integrable nonlinear Schrödinger equation models with varying dispersion, nonlinearity and gain: Application for soliton dispersion managements,” IEEE J. Sel. Top. Quantum Electron. |

9. | V. I. Kruglov, A. C. Peacock, and J. D. Harvey, “Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients,” Phys. Rev. Lett. |

10. | V. M. Pérez-García, P. J. Torres, and V. V. Konotop, “Similarity transformations for nonlinear Schrödinger equations with time-dependent coefficients,” Physica D |

11. | S. A. Ponomarenko and G. P. Agrawal, “Do solitonlike self-similar waves exist in nonlinear Optical media?” Phys. Rev. Lett. |

12. | S. A. Ponomarenko and G. P. Agrawal, “Interactions of chirped and chirp-free similaritons in optical fiber amplifiers,” Opt. Express |

13. | S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. |

14. | V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, “Nonautonomous solitons in external potentials,” Phys. Rev. Lett. |

15. | T. C. Hernandez, V. E. Villargan, V. N. Serkin, G. M. Agüero, T. L. Belyaeva, M. R. Peña, and L. L. Morales, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: I. Bright solitons,” Quantum Electron. |

16. | C. Hernandez Tenorio, E. Villagran Vargas, V. N. Serkin, M. Agüero Granados, T. L. Belyaeva, R. Peña Moreno, and L. Morales Lara, “Dynamics of solitons in the model of nonlinear Schrödinger equation with an external harmonic potential: II. Dark solitons,” Quantum Electron. |

17. | Y. R. Shen, |

18. | P. G. Drazin and R. S. Jonson, |

19. | F. Calogero and A. Degasperis, |

20. | G. L. Lamb, |

21. | S. Kumar and A. Hasegawa, “Quasi-soliton propagation in dispersion-managed optical fibers,” Opt. Lett. |

22. | Y. Ozeki and T. Inoue, “Stationary rescaled pulse in dispersion-decreasing fiber for pedestal-free pulse compression,” Opt. Lett. |

23. | S. Raghavan and G. P. Agrawal, “Spatiotemporal solitons in inhomogeneous nonlinear media,” Opt. Commun. |

24. | C. J. Pethick and H. Smith, |

25. | C. Sulem and P. L. Sulem, |

26. | A. V. Rybin, G. G. Varzugin, M. Lindberg, J. Timonen, and R. K. Bullough, “Similarity solutions and collapse in the attractive Gross-Pitaevskii equation,” Phys. Rev. E |

27. | G. Theocharis, Z. Rapti, P. G. Kevrekidis, D. J. Frantzeskakis, and V. V. Konotop, “Modulational instability of Gross-Pitaevskii-type equations in 1+1 dimensions,” Phys. Rev. A |

28. | J.-K. Xue, “Controllable compression of bright soliton matter waves”, J. Phys. B: At. mol. Opt. Phys. |

29. | L. Wu, Q. Yang, and J.-f. Zhang, “Bright solitons on a continuous wave background for the inhomogeneous nonlinear Schrödinger equation in plasma,” J. Phys. A: Math. Gen. |

30. | L. Wu, J.-f. Zhang, and L. Li, “Modulational instability and bright solitary wave solution for Bose-Einstein condensates with time-dependent scattering length and harmonic potential,” New J. Phys. |

31. | L. Wu, L. Li, G. Chen, Q. Tian, and J.-f. Zhang, “Controllable exact self-similar evolution of the Bose-Einstein condensate”, New J. Phys. |

32. | V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Sov. Phys. JETP |

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(190.4370) Nonlinear optics : Nonlinear optics, fibers

(230.7370) Optical devices : Waveguides

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: February 1, 2008

Revised Manuscript: March 12, 2008

Manuscript Accepted: March 28, 2008

Published: April 21, 2008

**Citation**

Lei Wu, Jie-Fang Zhang, Lu Li, Qing Tian, and K. Porsezian, "Similaritons in nonlinear optical systems," Opt. Express **16**, 6352-6360 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-9-6352

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### References

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