Nonlinear Schrödinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions

Details Nonlinear Schrödinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions Nonlinear Schrödinger Equation: Generalized Darboux Transformation and Rogue Wave Solutions

2011-08-14

arxiv.org/abs/1108.2867v2

Nonlinear Sciences

Exactly Solvable and Integrable Systems

17 pages, 4 figures, replaced by revised version

Scientific paper

In this paper, we construct a generalized Darboux transformation for nonlinear Schr\"odinger equation. The associated $N$-fold Darboux transformation is given both in terms of a summation formula and in terms of determinants. As applications, we obtain compact representations for the $N$-th order rogue wave solutions of the focusing nonlinear Schr\"odinger equation and Hirota equation. In particular, the dynamics of the general third order rogue wave is discussed and shown to exhibit interesting structure.

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