## Variational Principles, Lie Point Symmetries, and Similarity Solutions of the Vector Maxwell Equations in Non-linear Optics

Publication: Research - peer-review › Journal article – Annual report year: 2004

The vector Maxwell equations of nonlinear optics coupled to a single
Lorentz oscillator and with instantaneous Kerr nonlinearity
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 nonlinear
optics, without resorting to the commonly used nonlinear Schr\"odinger
(NLS) equation approximation in which a high frequency carrier wave is
modulated on long length and time scales due to nonlinear 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
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 nonlinearity, 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 $y=y(\xi)$, where
$\xi=z-st$ 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 $\epsilon$.
Both smooth periodic traveling waves and non-smooth solutions in which the
electric field gradient diverges (i.e. solutions in which $|E_\xi|\to\infty$
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 |
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Journal | Physica D: Nonlinear Phenomena |

Volume | 191 |

Issue number | 1-2 |

Pages (from-to) | 49-80 |

ISSN | 0167-2789 |

DOIs | |

State | Published - 2004 |

Citations | Web of Science® Times Cited: 5 |
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### Keywords

- similarity solutions, Nonlinear optics, traveling waves, vector Maxwell's equations

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