Explicit Solutions To The Nonlinear Schrodinger Equation

We provide a solution formula that contains a large class of exact solutions to the integrable nonlinear partial differential equation known as the nonlinear Schrodinger (NLS) equation. The NLS equation has important applications in electromagnetic wave propagation in optical fibers and propagation of surface waves in deep waters. The solutions described by our formula include soliton solutions which have important physical applications, e.g., they may be used as carriers of many phone conversations through a single channel. Our solution formula is obtained from a matrix triplet with complex entries and by using matrix  exponentials functions of spatial and temporal variables. We provide sufficient conditions on our matrix triplet guaranteeing the validity of our solutions at all locations at all times. We investigate the behavior of waves representing our solutions using Mathematica animations. We prove the invariance of our solutions under a similarity transformation on our matrix triplet. We also investigate the validity of our solutions as the matrix size of our triplet becomes infinite.