A vector Geng–Li model: New nonlinear phenomena and breathers on periodic background waves

Abstract

Based on an introduced (n+2)×(n+2) matrix spectral problem, a vector Geng–Li model is proposed, which is a vector generalization of the Geng–Li model. Employing the Riccati equations derived from the spectral problems and the gauge transformations between the spectral problems, a general N-fold Darboux transformation of the vector Geng–Li model is constructed. As applications of the Darboux transformations, some exact solutions for the Geng–Li model are obtained, which reveal two novel nonlinear phenomena. The first one is that all the solutions of the Geng–Li equation found in this paper have the characteristics of square-root functions. The second one is the interaction of two waves: as time goes on, the amplitudes of the two waves get higher and the distances between the two waves get narrower. And then, a general approach which deals with periodic seed solutions is given by using algebraic curves and Baker–Akhiezer functions, from which breathers of the Geng–Li equation on periodic background waves are obtained.