Exact solutions to the nonlinear Schr\"odinger equation

TL;DR

This paper reviews a method for constructing explicit, exact solutions to the focusing nonlinear Schrödinger equation, including multisolitons, using matrix triplets and exponential functions, with applications in nonlinear wave analysis.

Contribution

It introduces a new explicit formula for multisoliton solutions of the nonlinear Schrödinger equation using matrix triplets and matrix exponentials.

Findings

Solutions are analytic on the entire (x,t)-plane.

Solutions decay exponentially as x approaches ± infinity.

Solutions can be expressed as algebraic combinations of exponential, trigonometric, and polynomial functions.

Abstract

A review of a recent method is presented to construct certain exact solutions to the focusing nonlinear Schr\"odinger 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.