Exact solutions to the nonlinear Schrodinger equation

A review of a recent method is presented to construct certain exact solutions to the focusing nonlinear Schrodinger equation on the line with a cubic nonlinearity. With motivation by the inverse scattering transform and help from the state-space method, an explicit formula is obtained to express such exact solutions in a compact form in terms of a matrix triplet and by using matrix exponentials. Such solutions consist of rnultisolitons with any multiplicities, are analytic on the entire xt-plane, decay exponentially as x -> +/-infinity at each fixed t, and can alternatively be written explicitly as algebraic combinations of exponential, trigonometric, and polynomial functions of the spatial and temporal coordinates x and t. Various equivalent forms of the matrix triplet are presented yielding the same exact solution.