Darboux Transformation and Symbolic Computation on Multi-Soliton and Periodic Solutions for Multi-Component Nonlinear Schr¨odinger Equations in an Isotropic Medium

Abstract The Darboux transformation is applied to a multi-component nonlinear Schr¨odinger system, which governs the propagation of polarized optical waves in an isotropic medium. Based on the Lax pair associated with this integrable system, the formula for the n -times iterative Darboux transformation is constructed in the form of block matrices. The purely algebraic iterative algorithm is carried out via symbolic computation, and two different kinds of solutions of practical interest, i. e., bright multi-soliton solutions and periodic solutions, are also presented according to the zero and nonzero backgrounds.