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

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)
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