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 × 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) × (n + 1) determinant and n × 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.