Discrete systems and differential-difference equations have become topics of increasing physical and mathematical importance. The variety of physical applications where such models are relevant, and their significant differences from the mathematical theory of partial differential equations, contribute to the extensive recent interest in these topics. Numerous applications of these problems have emerged ranging from nonlinear optics, in the dynamics of guided waves in inhomogeneous optical structures and photonic crystal lattices, to atomic physics, in the dynamics of Bose-Einstein condensate droplets in periodic (optical lattice) potentials and from condensed matter, in Josephson-junction ladders, to biophysics, in various models of the DNA double strand.
One of the prototypical differential-difference models that is both physically relevant and mathematically tractable is the so-called discrete nonlinear Schrodinger (NLS) equation,
i un'(t) + β (un+1 - 2 un + un-1) + γ |un|2 un = 0.where β is the dispersion coefficient, and γ is the nonlinearity coefficient.
The most direct implementation of the discrete NLS equation can be identified in one-dimensional arrays of coupled optical waveguides. These may be multi-core structures created in a slab of a semiconductor material (such as AlGaAs), or virtual ones, induced by a set of laser beams illuminating a photorefractive crystal. In this experimental implementation, there are about forty lattice sites (guiding cores), and the localized modes (discrete solitons) may propagate over twenty diffraction lengths.
Light-induced photonic lattices have recently emerged as another application of the discrete NLS equation. The refractive index of the nonlinear photonic lattices changes periodically due to a grid of strong beams, while a weaker probe beam is used to monitor the localized modes (discrete solitons). A number of promising experimental studies of discrete solitons in light-induced photonic lattices was reported recently in physics literature.
An array of Bose-Einstein condensate droplets trapped in a strong optical lattice with thousands of atoms in each droplet, is another direct physical realization of the discrete NLS equation. In this context, the model can be derived systematically by using the Wannier function expansions. Besides applications to optical waveguides, photonic crystal lattices, and Bose-Einstein condensates trapped in optical lattices, the discrete NLS equation also arises as the envelope wave reduction of the general nonlinear Klein-Gordon lattices.
The physically relevant applications clearly signal the importance of discrete solitons and vortices in two-dimensional discrete lattices. However, most of the above mentioned works are predominantly of experimental or numerical nature, while the mathematical theory of existence and stability of discrete localized structures has not been developed to a similar extent. It is now the time to create new methods of analysis of discrete lattices, based on Lyapunov-Schmidt reductions, central manifold analysis, normal forms, and beyond-all-order persistence analysis.
|
|
|
Manley-Rowe relations | |
|
|
N-Wave Interactions |