2011/09/15 by Shuwei Xu, Jingsong He · 1 citation
Physics and Astronomy · Mathematics · #nlin.SI #math-ph #math.MP #msc:35C08 #msc:37K10 #msc:37K40
paper · pdf · doi:10.1063/1.4726510
8 figures, 17 pages
arxiv created 2011/09/15 · arxiv updated 2015/05/29
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 of 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 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.