Mark J. Ablowitz - Department of Applied Mathematics, University of Colorado, Boulder, CO 80309-0526, United States (email)
Abstract: Nonlinear wave equations are central to the study of nonlinear optics and fluid dynamics. Notably, recent research has shown that solitons can be generated in mode-locked lasers. An interesting application of these lasers is the development of optical clocks which have the potential to be considerably more accurate than atomic clocks. Another important area of research in nonlinear optics is lattice dynamics where localized solitary wave or solitons can be obtained in periodic and irregular lattice systems. In honeycomb lattices, discrete and continuous nonlinear Dirac systems can be derived in certain parameter regimes; the Dirac systems describe conical diffraction, a phenomena observed in recent experiments. In water and internal waves the classical equations are reformulated as a system of coupled equations where the free surface equations are formulated as nonlocal equation and the depth variable is eliminated. These systems reduce to interesting asymptotic equations in suitable limits. A numerical method, termed spectral renormalization, is used to find solitary waves in nonlinear optics, water waves and multi-fluid systems.
Keywords: Mode-locked lasers, honeycomb lattice, nonlinear Dirac system, spectral renormalization.
Received: September 2009; Revised: December 2009; Published: December 2010.