The rogue wave and breather solution of the Gerdjikov-Ivanov equation

The Gerdjikov-Ivanov (GI) system of q and r is defined by a quadratic polynomial spectral problem with 2 x 2 matrix coefficients. Each element of the matrix of n-fold Darboux transformation (DT) for this system is expressed by a ratio of (n + 1) x (n + 1) determinant and n x n determinant of eigenfunctions, which implies the determinant representation of q([n]) and r([n]) generated from known solution q and r. By choosing some special eigenvalues and eigenfunctions according to the reduction conditions q([n]) = -(r([n]))*, the determinant representation of q([n]) provides new solutions of the GI equation. As examples, the breather solutions and rogue wave of the GI are given explicitly by the two-fold DT from a periodic seed with a constant amplitude. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4726510]