Fujioka J, Espinosa A, Rodríguez R F, Malomed B A
Departamento de Física Química, Instituto de Física, Universidad Nacional Autónoma de México, Mexico, DF 04510, Mexico.
Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel.
Chaos. 2014 Sep;24(3):033121. doi: 10.1063/1.4892616.
It was recently found [Fujioka et al., Phys. Lett. A 374, 1126 (2010)] that the propagation of solitary waves can be described by a fractional extension of the nonlinear Schrödinger (NLS) equation which involves a temporal fractional derivative (TFD) of order α > 2. In the present paper, we show that there is also another fractional extension of the NLS equation which contains a TFD with α < 2, and in this case, the new equation describes the propagation of radiating solitons. We show that the emission of the radiation (when α < 2) is explained by resonances at various frequencies between the pulses and the linear modes of the system. It is found that the new fractional NLS equation can be derived from a suitable Lagrangian density, and a fractional Noether's theorem can be applied to it, thus predicting the conservation of the Hamiltonian, momentum and energy.
