Exact solutions to the nonlinear Schrödinger equation. (English) Zbl 1190.37078

Aktosun, Tuncay; Busse, Theresa; Demontis, Francesco; van der Mee, Cornelis

Ball, Joseph A. (ed.) et al., Topics in operator theory. Volume 2: Systems and mathematical physics. Proceedings of the 19th international workshop on operator theory and applications (IWOTA), College of William and Mary, Williamsburg, VA, USA, July 22–26, 2008. A tribute to Israel Gohberg on the occasion of his 80th birthday. Basel: Birkhäuser (ISBN 978-3-0346-0160-3/hbk; 978-3-0346-0163-4/set; 978-3-0346-0161-0/ebook). Operator Theory: Advances and Applications 203, 1-12 (2010).

Summary: 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.
For the entire collection see [Zbl 1181.47003].

MSC:

37K15	Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
35Q51	Soliton equations
35Q55	NLS equations (nonlinear Schrödinger equations)

Keywords:

nonlinear Schrödinger equation exact solutions; explicit solutions; focusing NLS equation; NLS equation with cubic nonlinearity; inverse scattering transform