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 $2 \times 2$ matrix coefficients. Each element of the matrix of n-fold Darboux transformation of this system is expressed by a ratio of $(n+1)\times (n+1)$ determinant and $n\times n$ 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 some new solutions of the GI equation. As examples, the breather solutions and rogue wave of the GI is given explicitly by two-fold DT from a periodic "seed" with a constant amplitude.