## Soliton triads ensemble in frequency conversion: from inverse scattering theory to experimental observation |

Optics Express, Vol. 19, Issue 14, pp. 13192-13200 (2011)

http://dx.doi.org/10.1364/OE.19.013192

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### Abstract

We consider the spectral theory of three–wave interactions to predict the initiation, formation and dynamics of an ensemble of bright–dark–bright soliton triads in frequency conversion processes. Spatial observation of non–interacting triads ensemble in a KTP crystal confirms theoretical prediction and numerical simulations.

© 2011 OSA

## 1. Introduction

1. D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. interaction in a homogeneous medium,” Rev. Mod. Phys. **51**, 275–309 (1979). [CrossRef]

1. D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. interaction in a homogeneous medium,” Rev. Mod. Phys. **51**, 275–309 (1979). [CrossRef]

11. K. Lamb, “Tidally generated near-resonant internal wave triads at shelf break,” Geophys. Res. Lett. **34**, L18607 (2007). [CrossRef]

13. J. Ibanez and E. Verdaguer, “Soliton collision in general-relativity,” Phys. Rev. Lett. **51**, 1313 (1983). [CrossRef]

17. C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. **102**, 083902 (2009). [CrossRef] [PubMed]

*N*independent degrees of freedom corresponding to

*N*soliton particles are effective. TWI soliton has been predicted in the 70s [1

1. D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. interaction in a homogeneous medium,” Rev. Mod. Phys. **51**, 275–309 (1979). [CrossRef]

18. K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Jpn. **34**, 796–800 (1973). [CrossRef]

19. A. Abdolvand, A. Nazarkin, A. Chugreev, C. Kaminski, and P. Russel, “Solitary pulse generation by backward raman scattering in H-2-filled photonic crystal fibers,” Phys. Rev. Lett. **103**, 183902 (2009). [CrossRef] [PubMed]

20. F. Baronio, M. Conforti, M. Andreana, V. Couderc, C. De Angelis, S. Wabnitz, A. Barthelemy, and A. Degasperis, “Frequency generation and solitonic decay in three wave interactions,” Opt. Express **17**, 13889–13894 (2009). [CrossRef] [PubMed]

21. F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett **104**, 113902 (2010). [CrossRef] [PubMed]

*ω*

_{1}(the signal) and a quasi-plane wave at frequency

*ω*

_{2}(the pump) which mix to generate a beam at the sum frequency (SF)

*ω*

_{3}(the idler), when diffraction is negligible. Depending on the input intensities different nonlinear regimes exist: frequency conversion, single soliton triad generation,

*N*–soliton triads ensemble. Frequency conversion: the signal and pump beams interact and generate an idler beam whose spatial characteristics are associated with the interaction distance in the crystal; signal and pump are depleted. Soliton triad generation: the signal and pump beams interact, generate a stable symbiotic bright-dark-bright triplet moving with a locked spatial nonlinear walk–off [8

8. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. **97**, 093901 (2006). [CrossRef] [PubMed]

21. F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett **104**, 113902 (2010). [CrossRef] [PubMed]

*N*–soliton triads ensemble: the signal and pump input beams generate

*N*soliton triads moving with different spatial nonlinear walk-offs. In this Letter, the focus is on the generation and dynamics of an ensemble of TWI soliton triads. We obtain the results by a complementary use of spectral theory (inverse scattering transform, IST), numerical integration of the TWI equations, and experiments in nonlinear optics.

## 2. TWI equations and inverse scattering

*k*

_{1},

*k*

_{2},

*k*

_{3}and frequencies

*ω*

_{1},

*ω*

_{2},

*ω*

_{3}, which propagate in a nonlinear optical medium, interact efficiently with each other and exchange energy if the resonance conditions

*k*

_{1}+

*k*

_{2}=

*k*

_{3},

*ω*

_{1}+

*ω*

_{2}=

*ω*

_{3}, are satisfied. The equations describing spatial interaction of beams read [21

21. F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett **104**, 113902 (2010). [CrossRef] [PubMed]

*E*(

_{n}*x, y, z*) are the slowly varying electric field envelopes of the waves at frequencies

*ω*(wavelength

_{j}*λ*),

_{j}*k*=

_{n}*ω*n

_{n}

_{n}*/c*are the wavenumbers, n

*the refractive indexes,*

_{n}*χ*= 2

_{n}*dω*/

_{j}*c*n

*the nonlinear coupling coefficients (*

_{n}*d*is the quadratic nonlinear susceptibility and

*c*is the speed of light),

*ρ*the walk off angles,

_{j}*n*= 1, 2, 3.

*z*is the spatial longitudinal propagation coordinate,

*x*and

*y*are the spatial transverse coordinates. In the case of negligible diffraction, Eqs. (1) can be mapped into adimensional TWI equations: where

*n*= 1, 2, 3

*mod*3,

*δ*=

_{n}*ρ*

_{n}z_{0}/

*x*

_{0},

*ξ*=

*z/z*

_{0},

*s*=

*x/x*

_{0}, with

*z*

_{0},

*x*

_{0}longitudinal and transverse scale lengths, respectively. Diffraction becomes negligible when

*Q*

_{1,3}=

*ϕ*

_{1,2}

*e*

^{−iπ/6},

*ν*

_{1}=

*δ*

_{1},

*ν*

_{2,3}=

*δ*

_{3,2}and (

*σ*

_{1},

*σ*

_{2},

*σ*

_{3}) = (+, –, +). Equations (2) turn out to be:

*s*and the other one in

*ξ*(the Zakharov–Manakov eigenvalue problem [2]). This fact gives a way to set up a nonlinear generalization of the Fourier analysis of solutions of the associated initial value problem, namely the IST. In particular, this generalization leads to decompose a given solution

*Q*

_{1}(

*s*,

*ξ*),

*Q*

_{2}(

*s*,

*ξ*),

*Q*

_{3}(

*s*,

*ξ*) as functions of

*s*at a given fixed

*ξ*in its continuum spectrum component (radiation) and in discrete spectrum component (solitons). This pair of equations (the Lax pair) reads [22

22. A. Degasperis, M. Conforti, F. Baronio, S. Wabnitz, and S. Lombardo, “The three-wave resonant interaction equations: spectral and numerical methods,” Lett. Math. Phys. **96**, 367 (2011). [CrossRef]

*ψ*=

*ψ*(

*s*,

*ξ*,

*λ*) is a 3–dimensional vector solution,

*λ*is the complex spectral variable and

*C*is a constant matrix which depends on the boundary conditions on the

*s*-axis.

*A*and

*B*are constant real and traceless diagonal matrices,

*A*=diag{

*a*

_{1},

*a*

_{2},

*a*

_{3}},

*B*=diag{

*b*

_{1},

*b*

_{2},

*b*

_{3}} whose entries are

*a*= 2

_{n}*ν*–

_{n}*ν*

_{n}_{+1}–

*ν*

_{n}_{+2},

*b*= 2

_{n}*ν*

_{n}_{+1}

*ν*

_{n}_{+2}–

*ν*(

_{n}*ν*

_{n}_{+1}+

*ν*

_{n}_{+2}),

*n*= 1, 2, 3 mod3, while the three wave fields

*Q*enter in the matrices

_{n}*E*(

*s*,

*ξ*) and

*F*(

*s*,

*ξ*) through the expressions: where

*σ*=

*σ*

_{1}

*σ*

_{2}

*σ*

_{3},

*w*= [(

*ν*

_{3}–

*ν*

_{1})(

*ν*

_{2}–

*ν*

_{1})(

*ν*

_{3}–

*ν*

_{2})]

^{−1/2},

*n*= 1, 2, 3 mod3 .

*ϕ*(

_{n}*s, ξ*

_{0}) is:

- computing the spectral data associated with
*Q*(_{n}*s*,*ξ*_{0}) by integrating the ODE (Eq. (4)), i.e. solving the*direct*problem; - finding the spectral data at a different coordinate
*ξ*≠*ξ*_{0}, which usually reduces to a trivial multiplication for a phase factor; - recovering the solution
*Q*(_{n}*s*,*ξ*) at coordinate*ξ*≠*ξ*_{0}by solving the inverse problem, then*ϕ*(_{n}*s*,*ξ*).

22. A. Degasperis, M. Conforti, F. Baronio, S. Wabnitz, and S. Lombardo, “The three-wave resonant interaction equations: spectral and numerical methods,” Lett. Math. Phys. **96**, 367 (2011). [CrossRef]

## 3. Theoretical analysis

*ϕ*

_{1},

*ϕ*

_{2},

*ϕ*

_{3}in the

*s*–

*ξ*plane, given the initial condition

*ϕ*

_{1}(

*s*, 0) =

*ε*

_{1}sech(

*s*), for the signal,

*ϕ*

_{2}(

*s*, 0) =

*ε*

_{2}[

*h*(

*s*

_{0}–

*s*) +

*h*(

*s*–

*s*

_{∞})] (

*h*(

*s*) is the Heaviside function,

*s*

_{∞}is a large positive constant), for the pump,

*ϕ*

_{3}(

*s*, 0) = 0, for the SF beam. We fix

*δ*

_{1}= −2,

*δ*

_{2}= 0,

*δ*

_{3}= −1

*, s*

_{0}= −6, and we vary

*ε*

_{1},

*ε*

_{2}amplitudes. We assumed a non vanishing pump as

*s*→ +∞ to apply the numerical spectral method for non vanishing boundary conditions [22

22. A. Degasperis, M. Conforti, F. Baronio, S. Wabnitz, and S. Lombardo, “The three-wave resonant interaction equations: spectral and numerical methods,” Lett. Math. Phys. **96**, 367 (2011). [CrossRef]

*ε*

_{1}= 0.5,

*ε*

_{2}= 0.05), the initial data are composed of a continuum spectrum component (radiation) and no discrete spectrum component (solitons). This regime corresponds to the well known optical non-collinear sum-frequency generation. Numerical simulations (Fig. 1) show that signal beam

*ϕ*

_{1}and the pump beam

*ϕ*

_{2}propagate with their own characteristic velocities; as long as the signal beam overtakes the pump beam, a idler beam

*ϕ*

_{3}at the sum-frequency is generated which propagates with its own characteristic walk-off; indeed, the

*V*shape of the idler beam is due to its intrinsic linear walk-off. Pump beam is deeply depleted.

*ε*

_{1}= 1.42,

*ε*

_{2}= 0.7), the initial data are composed of both a continuum spectrum component (radiation) and a discrete spectrum component (solitons). The input waves contain one soliton plus radiation. Figure 2(a) reports the initial envelope conditions, Fig. 2(b) shows the correspondent eignevalue

*λ*= 0.33

*i*and eigenfunctions

*ψ*. Through the spectral data we can predict the initiation of a bright-dark-bright triad and its properties (f.i., nonlinear walk-off, amplitudes, etc.). The relation between eigenvalues and soliton parameters can be found in Ref. [22

_{n}**96**, 367 (2011). [CrossRef]

**104**, 113902 (2010). [CrossRef] [PubMed]

*ξ*= 12. A similar phenomenon of soliton generation could be observed also for a smoother transition between zero and the pump level necessary to support the soliton.

*ε*

_{1}= 2.82,

*ε*

_{2}= 0.7), the initial data contain

*N*= 2 stable soliton triads, with different nonlinear walk-offs, plus radiation. Figure 3(a)–3(c) report the initial conditions, the correspondent eigenvalues

*λ*

_{1}= 0.33

*i*,

*λ*

_{2}=

*i*and eigenfunctions of the discrete spectrum. The stability of soliton triads can be characterized analytically as reported in Ref. [23

23. M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E **74**, 065602 (2006). [CrossRef]

*ε*

_{1}= 4.24,

*ε*

_{2}= 0.75), the initial data contain

*N*= 3 stable soliton triads. The correspondent eigenvalues are

*λ*

_{1}= 0.33

*i*,

*λ*

_{2}=

*i*and

*λ*

_{3}= 1.66

*i*. Figure 4(a)–4(c) show the generation of three soliton bright-dark-bright triads.

*N*bright solitons [1

**51**, 275–309 (1979). [CrossRef]

*δ*

_{1}which reshape, after the interaction of the pump beam, in

*N non-interacting*bright-dark-bright solitons triads [8

8. A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. **97**, 093901 (2006). [CrossRef] [PubMed]

24. A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. **101**, 044101 (2008). [CrossRef] [PubMed]

## 4. Experimental investigation

*ps*pulses at

*λ*= 1064

*nm*. We introduce a Glan polarizer to obtain, after passage of the light through

*P*

_{1}, two independent beams with perpendicular linear polarization states. A half-wave plate placed before the prism serves to adjust the intensity of the two beams. By means of highly reflecting mirrors, beam splitters and lenses the beams are focused and spatially superimposed in the plane of their beam waist with a circular shape of 300

*μ*

*m*and 6

*mm*, full width at half maximum in intensity, for the signal and pump waves respectively. A

*L*= 3

*cm*long KTP crystal

**(**

*d*= 3.29

*pm/V*

**)**cut for type II second harmonic generation is positioned such that its input face corresponds to the plane of superposition of the two input beams.

*°*and 3.6

*°*(in the crystal) with respect to the direction of perfect collinear phase matching for the extraordinary and the ordinary components, respectively (see Fig. 5). The idler second harmonic direction lies in between the input beams directions (−1.16

*°*). With these values of parameters, spatial diffraction, group velocity mismatch and temporal dispersion were negligible. The spatial waves’ patterns at the output of the crystal are imaged with magnification onto a CCD camera and analyzed. We use alternately different filters and polarizers to select either the IR or the green output.

*x*–

*z*plane (

*y*= 0 plane). In the

*x*–

*z*plane (as well as in each plane parallel to

*x*–

*z*), the experimental set-up creates the two-dimensional environement to excite the predicted TWI soliton dynamics.

*x*–

*z*(

*y*= 0) plane as the intensities of the input signal and pump are varied in a suitable range (Figs. 6, 7). Left columns of Figs. 6, 7 show the numerical spatial evolution of the pump beam and the SF beam (which are the beams that best report the TWI soliton evidence) in the ordinary

*x*–

*z*(

*y*= 0) plane; central and right columns report, respectively, the numerical and experimental spatial output profiles of the beams in the

*x*–

*y*(

*z*=

*L*) plane. The numerical and experimental results are reported considering a spatial frame moving with the pump walk-off angle. At intensities

*I*

_{1}= 10

*MW/cm*

^{2},

*I*

_{2}= 0.03

*MW/cm*

^{2}, the signal interacts with the pump and a SF beam at the second harmonic is generated. This regime corresponds to the well-known optical noncollinear second-harmonic frequency conversion (Fig. 6(a), 6(b), 6(c) and Fig. 7(a), 7(b), 7(c)). At intensities

*I*

_{1}= 50

*MW/cm*

^{2},

*I*

_{2}= 0.2

*MW/cm*

^{2}, signal and pumb beams generate a stable bright-dark-bright solitary triplet (Fig. 6(d), 6(e), 6(f) and Fig. 7(d), 7(e), 7(f)). Increasing further input intensities we have observed the generation of two and three non-interacting bright-dark-bright triads. Figures 6(g), 6(h), 6(i) and Fig. 7(g), 7(h), 7(i) show the generation of three non interacting bright-dark-bright triads (

*I*

_{1}= 500

*MW/cm*

^{2},

*I*

_{2}= 5

*MW/cm*

^{2}). Figures 6(i), 7(i) report the first observation, to the best of our knowledge, of an ensemble of non-interacting TWI soliton triads.

## 5. Conclusions

*ω*

_{1}(the signal) and a quasi-plane wave at frequency

*ω*

_{2}(the pump) which mix to generate a beam at the sum frequency (SF)

*ω*

_{3}(the idler), when diffraction is negligible. Depending on the input intensities different nonlinear regimes exist. In this paper, the focus was on the generation and dynamics of an ensemble of TWI soliton triads. We predicted theoretically and demonstrated experimentally non–interacting triads ensemble in a KTP crystal.

## Acknowledgments

## References and links

1. | D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. interaction in a homogeneous medium,” Rev. Mod. Phys. |

2. | V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” JETP Lett. |

3. | V. E. Zakharov, |

4. | A. Hasegawa, |

5. | W. Cheng, Y. Avitzour, Y. Ping, S. Suckewer, N. Fisch, M. Hur, and J. Wurtele, “Reaching the nonlinear regime of raman amplification of ultrashort laser pulses,” Phys. Rev. Lett. |

6. | E. Ibragimov and A. Struthers, “Second harmonic pulse compression in the soliton regime,” Opt. Lett. |

7. | A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett. |

8. | A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. |

9. | M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express |

10. | A. Craik, |

11. | K. Lamb, “Tidally generated near-resonant internal wave triads at shelf break,” Geophys. Res. Lett. |

12. | E. Segre, |

13. | J. Ibanez and E. Verdaguer, “Soliton collision in general-relativity,” Phys. Rev. Lett. |

14. | A. R. Osborne, M. Onorato, M. Serio, and L. Bergamasco, “Soliton creation and destruction, resonant interactions, and inelastic collisions in shallow water waves,” Phys. Rev. Lett. |

15. | B. Damski and W. Zurek, “Soliton creation during a Bose-Einstein Condensation,” Phys. Rev. Lett. |

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

17. | C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. |

18. | K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Jpn. |

19. | A. Abdolvand, A. Nazarkin, A. Chugreev, C. Kaminski, and P. Russel, “Solitary pulse generation by backward raman scattering in H-2-filled photonic crystal fibers,” Phys. Rev. Lett. |

20. | F. Baronio, M. Conforti, M. Andreana, V. Couderc, C. De Angelis, S. Wabnitz, A. Barthelemy, and A. Degasperis, “Frequency generation and solitonic decay in three wave interactions,” Opt. Express |

21. | F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett |

22. | A. Degasperis, M. Conforti, F. Baronio, S. Wabnitz, and S. Lombardo, “The three-wave resonant interaction equations: spectral and numerical methods,” Lett. Math. Phys. |

23. | M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E |

24. | A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 11, 2011

Revised Manuscript: May 9, 2011

Manuscript Accepted: May 26, 2011

Published: June 22, 2011

**Citation**

Fabio Baronio, Marco Andreana, Matteo Conforti, Gabriele Manili, Vincent Couderc, Costantino De Angelis, and Alain Barthélémy, "Soliton triads ensemble in frequency conversion: from inverse scattering theory to experimental observation," Opt. Express **19**, 13192-13200 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-14-13192

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### References

- D. J. Kaup, A. Reiman, and A. Bers, “Space-time evolution of nonlinear three-wave interactions. I. interaction in a homogeneous medium,” Rev. Mod. Phys. 51, 275–309 (1979). [CrossRef]
- V. E. Zakharov and S. V. Manakov, “Resonant interaction of wave packets in nonlinear media,” JETP Lett. 18, 243–245 (1973).
- V. E. Zakharov, What is Integrability? (Springer-Verlag, 1991).
- A. Hasegawa, Plasma Instabilities and Nonlinear Effects (Springer-Verlag, 2001).
- W. Cheng, Y. Avitzour, Y. Ping, S. Suckewer, N. Fisch, M. Hur, and J. Wurtele, “Reaching the nonlinear regime of raman amplification of ultrashort laser pulses,” Phys. Rev. Lett. 94, 045003 (2005). [CrossRef] [PubMed]
- E. Ibragimov and A. Struthers, “Second harmonic pulse compression in the soliton regime,” Opt. Lett. 21, 1582–1584 (1996). [CrossRef] [PubMed]
- A. Picozzi and M. Haelterman, “Spontaneous formation of symbiotic solitary waves in a backward quasi-phase-matched parametric oscillator,” Opt. Lett. 23, 1808–1810 (1998). [CrossRef]
- A. Degasperis, M. Conforti, F. Baronio, and S. Wabnitz, “Stable control of pulse speed in parametric three-wave solitons,” Phys. Rev. Lett. 97, 093901 (2006). [CrossRef] [PubMed]
- M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Parametric frequency conversion of short optical pulses controlled by a CW background,” Opt. Express 15, 12246–12251 (2007). [CrossRef] [PubMed]
- A. Craik, Wave Interactions and Fluid Flows (Cambridge Univ. Press, 1985).
- K. Lamb, “Tidally generated near-resonant internal wave triads at shelf break,” Geophys. Res. Lett. 34, L18607 (2007). [CrossRef]
- E. Segre, Collected Papers of Enrico Fermi (University of Chicago Press, 1965).
- J. Ibanez and E. Verdaguer, “Soliton collision in general-relativity,” Phys. Rev. Lett. 51, 1313 (1983). [CrossRef]
- A. R. Osborne, M. Onorato, M. Serio, and L. Bergamasco, “Soliton creation and destruction, resonant interactions, and inelastic collisions in shallow water waves,” Phys. Rev. Lett. 81, 3559 (1998). [CrossRef]
- B. Damski and W. Zurek, “Soliton creation during a Bose-Einstein Condensation,” Phys. Rev. Lett. 104, 160404 (2010). [CrossRef] [PubMed]
- Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic Press, 2003).
- C. Conti, A. Fratalocchi, M. Peccianti, G. Ruocco, and S. Trillo, “Observation of a gradient catastrophe generating solitons,” Phys. Rev. Lett. 102, 083902 (2009). [CrossRef] [PubMed]
- K. Nozaki and T. Taniuti, “Propagation of solitary pulses in interactions of plasma waves,” J. Phys. Soc. Jpn. 34, 796–800 (1973). [CrossRef]
- A. Abdolvand, A. Nazarkin, A. Chugreev, C. Kaminski, and P. Russel, “Solitary pulse generation by backward raman scattering in H-2-filled photonic crystal fibers,” Phys. Rev. Lett. 103, 183902 (2009). [CrossRef] [PubMed]
- F. Baronio, M. Conforti, M. Andreana, V. Couderc, C. De Angelis, S. Wabnitz, A. Barthelemy, and A. Degasperis, “Frequency generation and solitonic decay in three wave interactions,” Opt. Express 17, 13889–13894 (2009). [CrossRef] [PubMed]
- F. Baronio, M. Conforti, C. De Angelis, A. Degasperis, M. Andreana, V. Couderc, and A. Barthelemy, “Velocity-locked solitary waves in quadratic media,” Phys. Rev. Lett 104, 113902 (2010). [CrossRef] [PubMed]
- A. Degasperis, M. Conforti, F. Baronio, S. Wabnitz, and S. Lombardo, “The three-wave resonant interaction equations: spectral and numerical methods,” Lett. Math. Phys. 96, 367 (2011). [CrossRef]
- M. Conforti, F. Baronio, A. Degasperis, and S. Wabnitz, “Inelastic scattering and interactions of three-wave parametric solitons,” Phys. Rev. E 74, 065602 (2006). [CrossRef]
- A. Fratalocchi, C. Conti, G. Ruocco, and S. Trillo “Free-energy transition in a gas of noninteracting nonlinear wave particles,” Phys. Rev. Lett. 101, 044101 (2008). [CrossRef] [PubMed]

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