Three New Kinds Of Solution Of Variable Coefcient Nonlinear Schr(?)dinger Equation And Their New Properties

Constant coefcient Nonlinear Schr¨odinger (NLS) equation is one of classic solitonequation which have been studied extensively. It has several kinds of solution, suchas soliton solution and periodic solution which have some interesting properties. SincePeregrine obtained its rational solution in1983, which is now called rogue wave, moreand more people have paid much attention to such kind of solution. However, NLSequation is not a suitable model on describing some phenomenon under inhomogeneousbackground. So, the variable coefcient Nonlinear Schr¨odinger (VCNLS) equation isused as model in many fields of nonlinear sciences which is the object of this paper.There is not a systematical method to solve the VCNLS equation until H. H. Chenand C. S. Liu brought out an idea that one can obtain the solution of the VCNLSequation with an special external potential by building a transformation between theNLS equation and VCNLS equation, then mapping the solution of NLS equation tothe latter. On the basis of this idea, a new transformation which has universality isconstructed in this paper. It has several free functions so that we can design the formof VCNLS equation from the diferent consideration on physics. On the basis of thistransformation, a new integrable nonautonomous NLS equation model is found and itstwo novel solutions are obtained. They are nonautonomous soliton solution and roguewave solution.Inspired by Kovalyov who study on the singular solution, some VCNLS equationwith singular potentials are considered and their novel singular solutions are also givenout in this paper. Last but not least, the application of the three kinds of new solutionsare discussed.