The discrete nonlinear Schrodinger equation: A survey of recent results

In this paper we review a number of recent developments in the study of the Discrete Nonlinear Schrodinger (DNLS) equation. Results concerning ground and excited states, their construction, stability and bifurcations are pres ented in one and two spatial dimensions. Combinations of such steady states lead to the study of coherent structure bound states. A special case of su ch structures axe the so-called twisted modes and their two-dimensional dis crete vortex generalization. The ideas oil such multi-coherent structures a nd their interactions are also useful in treating finite system effects thr ough the image method. The statistical mechanics of the system is also anal yzed and the partition function calculated in one spatial dimension using t he transfer integral method. Finally, a number of open problems and future directions in the field are proposed.