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## Broad band solitons in a periodic and nonlinear Maxwell system

Venue: | SIAM J. Appl. Dynam. Syst |

Citations: | 2 - 1 self |

### Citations

678 |
Nonlinear Optics
- Boyd
- 2003
(Show Context)
Citation Context ...near electromagnetic waves in a one-dimensional periodic structure are governed by a nonlinear Maxwell equation: ∂2t ( n2(z)E + χ|E|2E) = ∂2zE. (1.1) Here, χ > 0 is the Kerr nonlinearity coefficient, =-=[3]-=-. We assume a low-contrast, periodic refractive index profile, n(z), with mean n0, given by n(z) = n0 + N(z), n0 > 0, N(z) = N(z + 2pi), 0 < 1; (1.2) n(z) is real-valued; no energy-dissipation has... |

149 |
AUTO: A program for the automatic bifurcation analysis of autonomous systems
- Doedel
- 1981
(Show Context)
Citation Context ...sing a naive continuation algorithm in Matlab, solving with a given value of and using that solution as the initial guess for a larger value of , they were confirmed by our computations using AUTO =-=[7,8]-=-. Though the starting branch may not have an alternating sign structure, sign alternating solutions may still be found at = 1. This makes it challenging to perform numerical continuation with these ... |

98 |
Visible continuum generation in air-silica microstructure optical fibers with anamolous dispersion at 800 nm
- Ranka, Windeler, et al.
- 2000
(Show Context)
Citation Context ...ength scales of many experiments is a negligible effect, [9]. Moreover, there are experimentally realizable regimes in which pulses with spectral content near the zero dispersion point are propagated =-=[15]-=-. In these experiments, a broad band super continuum is generated. The carrier shocking mentioned above is a possible source of such broad band emission. In this paper, we explore, by analytical, asym... |

59 |
Solving ODEs with MATLAB
- Shampine, Gladwell, et al.
- 2003
(Show Context)
Citation Context ...lab’s bvp5c algorithm with absolute tolerance 10−4, relative tolerance 10−8, on the domain [0, 25]. bvp5c is a nonlinear finite difference algorithm for two-point boundary-value problems discussed in =-=[18]-=-. We use the even symmetry of the solutions to impose the boundary condition U ′p(0) = 0, and the artificial boundary condition U ′p(ζmax) + pUp(ζmax) = 0. The results for systems of up to six coupled... |

31 |
Self-induced transparency solitons in nonlinear refractive periodic media,
- Aceves, Wabnitz
- 1989
(Show Context)
Citation Context ...Derivations from the Klein-Fock as well as Gross-Pitaevskii equations have also been obtained; see [13,14,16, 17]. Explicit localized stationary solutions, called gap solitons, for NLCME are given in =-=[1,4]-=- The linear stability of the gap solitons was studied in [5], and a linear, multi-dimensional, analog of NLCME was examined in [2]. However, NLCME is not the correct mathematical description of weakly... |

30 |
Gap solitons,
- Sterke, Sipe
- 1994
(Show Context)
Citation Context ... interest for many years. Early interest arose from the possibility of balancing the band dispersion of the periodic structure with the nonlinearity to form soliton-like structures; see, for example, =-=[6, 10]-=- and references cited therein. While such a heterogeneous medium possesses the same solitonproducing ingredients of dispersion and nonlinearity as found in the well known Korteweg–de Vries (KdV) and n... |

29 |
Nonlinear propagation of light in one-dimensional periodic structures
- Goodman, Weinstein, et al.
- 2002
(Show Context)
Citation Context ...ading order model in numerous contexts. For one-dimensional propagation of electromagnetic waves in nonlinear and periodic media, it was rigorously derived from the anharmonic Maxwell-Lorenz model in =-=[12]-=-. Derivations from the Klein-Fock as well as Gross-Pitaevskii equations have also been obtained; see [13,14,16, 17]. Explicit localized stationary solutions, called gap solitons, for NLCME are given i... |

28 |
Slow Bragg solitons in nonlinear periodic structures
- Christodoulides, Joseph
- 1989
(Show Context)
Citation Context ...Derivations from the Klein-Fock as well as Gross-Pitaevskii equations have also been obtained; see [13,14,16, 17]. Explicit localized stationary solutions, called gap solitons, for NLCME are given in =-=[1,4]-=- The linear stability of the gap solitons was studied in [5], and a linear, multi-dimensional, analog of NLCME was examined in [2]. However, NLCME is not the correct mathematical description of weakly... |

19 |
Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential. Applicable Analysis 2007; 86:1017–1036
- Pelinovsky, Schneider
(Show Context)
Citation Context ...near and periodic media, it was rigorously derived from the anharmonic Maxwell-Lorenz model in [12]. Derivations from the Klein-Fock as well as Gross-Pitaevskii equations have also been obtained; see =-=[13,14,16, 17]-=-. Explicit localized stationary solutions, called gap solitons, for NLCME are given in [1,4] The linear stability of the gap solitons was studied in [5], and a linear, multi-dimensional, analog of NLC... |

18 |
Block-diagonalization of the symmetric first-order coupled-mode system
- Chugunova, Pelinovsky
(Show Context)
Citation Context ...uations have also been obtained; see [13,14,16, 17]. Explicit localized stationary solutions, called gap solitons, for NLCME are given in [1,4] The linear stability of the gap solitons was studied in =-=[5]-=-, and a linear, multi-dimensional, analog of NLCME was examined in [2]. However, NLCME is not the correct mathematical description of weakly nonlinear and weakly dispersive waves in the nonlinear and ... |

17 |
Nonlinear Photonic Crystals,
- Slusher, Eggleton
- 2003
(Show Context)
Citation Context ... interest for many years. Early interest arose from the possibility of balancing the band dispersion of the periodic structure with the nonlinearity to form soliton-like structures; see, for example, =-=[6, 10]-=- and references cited therein. While such a heterogeneous medium possesses the same solitonproducing ingredients of dispersion and nonlinearity as found in the well known Korteweg–de Vries (KdV) and n... |

16 |
Nonlinear coupled mode dynamics in hyperbolic and parabolic periodically structured spatially extended systems. Asymptotic Analysis 2001
- Schneider, Uecker
(Show Context)
Citation Context ...near and periodic media, it was rigorously derived from the anharmonic Maxwell-Lorenz model in [12]. Derivations from the Klein-Fock as well as Gross-Pitaevskii equations have also been obtained; see =-=[13,14,16, 17]-=-. Explicit localized stationary solutions, called gap solitons, for NLCME are given in [1,4] The linear stability of the gap solitons was studied in [5], and a linear, multi-dimensional, analog of NLC... |

15 | Modeling of wave resonances in low-contrast photonic crystals
- Agueev, Pelinovsky
- 2005
(Show Context)
Citation Context ...d stationary solutions, called gap solitons, for NLCME are given in [1,4] The linear stability of the gap solitons was studied in [5], and a linear, multi-dimensional, analog of NLCME was examined in =-=[2]-=-. However, NLCME is not the correct mathematical description of weakly nonlinear and weakly dispersive waves in the nonlinear and periodic Maxwell equation (1.1), (1.2). The deficiency of the NLCME sy... |

15 | Stopping light on a defect
- Goodman, Slusher, et al.
(Show Context)
Citation Context ...in periodic optical fiber gratings. Such states have been shown to propagate at a fraction of the speed of light and have been proposed in schemes for optical storage and buffering; see, for example, =-=[11]-=-. In the simplest setting, nonlinear electromagnetic waves in a one-dimensional periodic structure are governed by a nonlinear Maxwell equation: ∂2t ( n2(z)E + χ|E|2E) = ∂2zE. (1.1) Here, χ > 0 is the... |

10 |
Existence and stability of modulating pulse solutions in Maxwell's equations describing nonlinear optics
- Schneider, Uecker
- 2003
(Show Context)
Citation Context ...near and periodic media, it was rigorously derived from the anharmonic Maxwell-Lorenz model in [12]. Derivations from the Klein-Fock as well as Gross-Pitaevskii equations have also been obtained; see =-=[13,14,16, 17]-=-. Explicit localized stationary solutions, called gap solitons, for NLCME are given in [1,4] The linear stability of the gap solitons was studied in [5], and a linear, multi-dimensional, analog of NLC... |

2 |
Nonlinear propagation in superstructure bragg gratings
- Eggleton, Sterke, et al.
- 1996
(Show Context)
Citation Context ...ersion which, as in the anharmonic Maxwell-Lorentz model [12], takes off resonance the higher harmonics. However, chromatic dispersion on the length scales of many experiments is a negligible effect, =-=[9]-=-. Moreover, there are experimentally realizable regimes in which pulses with spectral content near the zero dispersion point are propagated [15]. In these experiments, a broad band super continuum is ... |

2 | Moving gap solitons in periodic potentials
- Pelinovsky, Schneider
- 1739
(Show Context)
Citation Context |

2 |
Gap solitons in a medium with third-harmonic generation
- Tasgal, Band, et al.
- 2005
(Show Context)
Citation Context ... Numerically Computed Gap Solitons Using our observations from the Rayleigh-Ritz approximation, we are motivated to solve the xNLS, (3.32), directly for existence of the gap solitons. We note that in =-=[20]-=-, the authors explored the related problem of broad band solitons of xNLCME truncated to two modes. 24 0 0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 γ h̃1 h̃2 h̃3 h̃4 h̃5 h̃6 Figure 4: Two–param... |

1 | Coherent structures and carrier shocks in the nonlinear maxwell equations
- Simpson, Weinstein
(Show Context)
Citation Context ...t, other models, such as the aforementioned anharmonic Maxwell-Lorenz system and the Gross-Pitaevskii equation, remain dispersive in the = 0 limit; this precludes infinitely many resonant modes. In =-=[19]-=-, nonlocal equations derived from nonlinear geometrical optics and an equivalent system of infinitely many coupled PDEs, which take into account the infinitely many resonances, were systematically stu... |