Solitary Wave Solutions to the Discrete Nonlinear Schrodinger Equation

Our goal is to find a large class of exact solutions to the integrable nonlinear
partial dierential-dierence equation known as the discrete nonlinear
Schodinger (NLS) equation. This equation describes the propagation of electromagnetic
waves in a nonlinear lattice crystal. It belongs to a special class of
nonlinear equations known as integrable evolution equations and hence it has
exact solutions. Using a recently developed technique yielding exact solutions
to integrable evolution equations, we obtain a solution formula for the discrete
NLS equation in terms of three constant matrices of complex valued entries.
Our solutions contain so-called matrix exponentials and include the so-called
soliton solutions. We prove that our solutions remain unchanged when our matrix
triplet undergoes a similarity transformation. We analyze our solutions and
relate the mathematical parameters in our matrix triplets to the physical properties
of waves representing our solutions. We animate our solutions by using
the software Mathematica.