Modulation instability and dynamics for the Hermitian symmetric space derivative nonlinear Schrödinger equation

Abstract

In this paper, we analyse modulation instability of the Hermitian symmetric space derivative nonlinear Schrödinger equation. Based on the gauge transformation between the Lax pairs, we derive a classical and a generalized Darboux transformations for the Hermitian symmetric space derivative nonlinear Schrödinger equation and two compact determinant forms of the Darboux transformations by setting a restriction on spectral functions. Employing the two Darboux transformations, dynamical behaviors of all types of solutions of this equation are discussed in detail, such as single-hump soliton solutions, double-hump soliton solutions, eye-shaped rogue wave solutions, four-petaled rogue wave solutions, triangular rogue wave solutions and so on.