The Darboux Transformation And Rogue Waves Of The Kundu-NLS Equation

The Kundu-NLS equation is studied mainly under the transformation of the non-linear Schr¨odinger(NLS)equation by Kundu in the article. Firstly,the Darboux trans-formation of the Kundu-NLS equation is derived and generalized to the matrix of n-foldDarboux transformation, and determinant representation are given. From known so-lution Q, the determinant representation of n-th new solutions of Q[n]is obtained bythe n-fold Darboux transformation.The article is divided into two situations. Firstly, the soliton solutions and positonsolutions are generated from trivial seed solutions. Secondly, the breather solutionsand rogue wave solutions are obtained from periodic seed solutions. More further, thehigher order rogue wave solutions of the Kundu-NLS equation are given. These explicitexpression of the first order, second order and third order rouge wave solutions can alsobe given in this paper.In addition, some free parameters in eigenfunctions are introduced for the higherorder rogue wave solutions of the Kundu-NLS equation, these free parameters in eigen-functions can adjust the patterns of the higher order rogue waves. The characters ofrogue waves can be incorporated, which can be used to describe the optical fiber, pulserogue wave in Bose-Einstein condensates and matter waves and so on.