Fractional Noether´s Theorem and Fractional Optical Solitons with Ortigueira´s and Grünwald-Letnikov´s Derivatives

Abstract

This paper shows that the centered fractional derivatives introduced by Manuel Duarte Ortigueira in 2006 are useful in the description of optical solitons. It is shown that we can construct a fractional extension of the nonlinear Schrödinger (NLS) equation which incorporates Ortigueira´s derivatives and has soliton solutions. It is also shown that this fractional NLS equation has a Lagrangian density and can be derived from a variational principle. Finally, it is shown that fractional extensions of Noether´s theorem can be formulated for fractional generalizations of the NLS equation, using either Ortigueira´s centered derivatives, or Grünwald-Letnikov´s derivatives. And from these extensions of Noether´s theorem we determine the conserved quantities associated to the invariances of the action integral under infinitesimal transformations.