Darboux Transformation and Explicit Solutions for Two New Differential-integral Equations

A Darboux transformation for two new differential-integral equations in the Manakov hierarchy is given by using the gauge transformation between the corresponding matrix spectral problems. As an application of the Darboux transformation, soliton solutions of the two equations are obtained. 1. INTRODUCTION In the past few years, there are extensive development in the area of nonlinear mathematical physics, especially in the work of integrable systems. Fairly satisfied understanding has been got for nonlinear integrable models. To investigate the soliton equations numerically, which associated with the integrable systems closely, the problem that how to solve partial differential equations and get the explicit solutions of them becomes an important research topic in the meaning of theory and practical application. Especially the explicit solutions provide a powerful tool to study the variety of properties of the soliton equations. Fortunately, quite a few methods of finding solutions of such soliton equations are well established, such as the inverse scattering transformation [1],[2], the Hirota's method [3], the Bäcklund and Darboux transformations [4]-[7], the algebra-geometric method [8], and so on. Among the various approaches, the Darboux transformation is known to be a powerful tool for finding explicit solutions of soliton equations [9]-[14]. The main aim of the present paper is to construct a Darboux transformation and explicit solutions for the following two differential-integral equations