Exact solutions to the nonlinear Schrödinger equation

Abstract

A review of a recent method is presented to construct certain exact solutions to the focusing nonlinear Schrödinger 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 multisolitons with any multiplicities, are analytic on the entire $xt$-plane, decay exponentially as $x\to \pm \infty$ 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.

15 pages, to appear in the the Proceedings of IWOTA-2008 as a special volume of Operator Theory: Advances and Applications in the Birkhauser (OT) series