### Abstract

The vector Maxwell equations of non-linear optics coupled to a single Lorentz oscillator and with instantaneous Kerr non-linearity are investigated by using Lie symmetry group methods. Lagrangian and Hamiltonian formulations of the equations are obtained. The aim of the analysis is to explore the properties of Maxwell's equations in non-linear optics, without resorting to the commonly used non-linear Schrödinger (NLS) equation approximation in which a high frequency carrier wave is modulated on long length and time scales due to non-linear sideband wave interactions. This is important in femto-second pulse propagation in which the NLS approximation is expected to break down. The canonical Hamiltonian description of the equations involves the solution of a polynomial equation for the electric field E, in terms of the canonical variables, with possible multiple real roots for E. In order to circumvent this problem, non-canonical Poisson bracket formulations of the equations are obtained in which the electric field is one of the non-canonical variables. Noether's theorem, and the Lie point symmetries admitted by the equations are used to obtain four conservation laws, including the electromagnetic momentum and energy conservation laws, corresponding to the space and time translation invariance symmetries. The symmetries are used to obtain classical similarity solutions of the equations. The traveling wave similarity solutions for the case of a cubic Kerr non-linearity, are shown to reduce to a single ordinary differential equation for the variable y=E^{2}, where E is the electric field intensity. The differential equation has solutions y=y(ξ), where ξ=z-st is the traveling wave variable and s is the velocity of the wave. These solutions exhibit new phenomena not obtainable by the NLS approximation. The characteristics of the solutions depends on the values of the wave velocity s and the energy integration constant ε. Both smooth periodic traveling waves and non-smooth solutions in which the electric field gradient diverges (i.e. solutions in which |E_{ξ}|→∞ at specific values of E, but where |E| is bounded) are obtained. The traveling wave solutions also include a kink-type solution, with possible important applications in femto-second technology.

Original language | English (US) |
---|---|

Pages (from-to) | 49-80 |

Number of pages | 32 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 191 |

Issue number | 1-2 |

DOIs | |

State | Published - Apr 15 2004 |

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### Keywords

- Non-linear optics
- Similarity solutions
- Traveling waves
- Vector Maxwell's equations

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*191*(1-2), 49-80. https://doi.org/10.1016/j.physd.2003.10.014

**Variational principles, Lie point symmetries, and similarity solutions of the vector Maxwell equations in non-linear optics.** / Webb, Garry; Sørensen, Mads Peter; Brio, Moysey; Zakharian, Aramis R.; Moloney, Jerome V.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 191, no. 1-2, pp. 49-80. https://doi.org/10.1016/j.physd.2003.10.014

}

TY - JOUR

T1 - Variational principles, Lie point symmetries, and similarity solutions of the vector Maxwell equations in non-linear optics

AU - Webb, Garry

AU - Sørensen, Mads Peter

AU - Brio, Moysey

AU - Zakharian, Aramis R.

AU - Moloney, Jerome V

PY - 2004/4/15

Y1 - 2004/4/15

N2 - The vector Maxwell equations of non-linear optics coupled to a single Lorentz oscillator and with instantaneous Kerr non-linearity are investigated by using Lie symmetry group methods. Lagrangian and Hamiltonian formulations of the equations are obtained. The aim of the analysis is to explore the properties of Maxwell's equations in non-linear optics, without resorting to the commonly used non-linear Schrödinger (NLS) equation approximation in which a high frequency carrier wave is modulated on long length and time scales due to non-linear sideband wave interactions. This is important in femto-second pulse propagation in which the NLS approximation is expected to break down. The canonical Hamiltonian description of the equations involves the solution of a polynomial equation for the electric field E, in terms of the canonical variables, with possible multiple real roots for E. In order to circumvent this problem, non-canonical Poisson bracket formulations of the equations are obtained in which the electric field is one of the non-canonical variables. Noether's theorem, and the Lie point symmetries admitted by the equations are used to obtain four conservation laws, including the electromagnetic momentum and energy conservation laws, corresponding to the space and time translation invariance symmetries. The symmetries are used to obtain classical similarity solutions of the equations. The traveling wave similarity solutions for the case of a cubic Kerr non-linearity, are shown to reduce to a single ordinary differential equation for the variable y=E2, where E is the electric field intensity. The differential equation has solutions y=y(ξ), where ξ=z-st is the traveling wave variable and s is the velocity of the wave. These solutions exhibit new phenomena not obtainable by the NLS approximation. The characteristics of the solutions depends on the values of the wave velocity s and the energy integration constant ε. Both smooth periodic traveling waves and non-smooth solutions in which the electric field gradient diverges (i.e. solutions in which |Eξ|→∞ at specific values of E, but where |E| is bounded) are obtained. The traveling wave solutions also include a kink-type solution, with possible important applications in femto-second technology.

AB - The vector Maxwell equations of non-linear optics coupled to a single Lorentz oscillator and with instantaneous Kerr non-linearity are investigated by using Lie symmetry group methods. Lagrangian and Hamiltonian formulations of the equations are obtained. The aim of the analysis is to explore the properties of Maxwell's equations in non-linear optics, without resorting to the commonly used non-linear Schrödinger (NLS) equation approximation in which a high frequency carrier wave is modulated on long length and time scales due to non-linear sideband wave interactions. This is important in femto-second pulse propagation in which the NLS approximation is expected to break down. The canonical Hamiltonian description of the equations involves the solution of a polynomial equation for the electric field E, in terms of the canonical variables, with possible multiple real roots for E. In order to circumvent this problem, non-canonical Poisson bracket formulations of the equations are obtained in which the electric field is one of the non-canonical variables. Noether's theorem, and the Lie point symmetries admitted by the equations are used to obtain four conservation laws, including the electromagnetic momentum and energy conservation laws, corresponding to the space and time translation invariance symmetries. The symmetries are used to obtain classical similarity solutions of the equations. The traveling wave similarity solutions for the case of a cubic Kerr non-linearity, are shown to reduce to a single ordinary differential equation for the variable y=E2, where E is the electric field intensity. The differential equation has solutions y=y(ξ), where ξ=z-st is the traveling wave variable and s is the velocity of the wave. These solutions exhibit new phenomena not obtainable by the NLS approximation. The characteristics of the solutions depends on the values of the wave velocity s and the energy integration constant ε. Both smooth periodic traveling waves and non-smooth solutions in which the electric field gradient diverges (i.e. solutions in which |Eξ|→∞ at specific values of E, but where |E| is bounded) are obtained. The traveling wave solutions also include a kink-type solution, with possible important applications in femto-second technology.

KW - Non-linear optics

KW - Similarity solutions

KW - Traveling waves

KW - Vector Maxwell's equations

UR - http://www.scopus.com/inward/record.url?scp=1842418662&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=1842418662&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2003.10.014

DO - 10.1016/j.physd.2003.10.014

M3 - Article

VL - 191

SP - 49

EP - 80

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-2

ER -