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

• Received 15 August 2011

DOI:https://doi.org/10.1103/PhysRevE.85.026607