An Analytical Solution To The Nonlinear Schrödinger Equation

In many areas of physics,the nonlinear schrodinger equation describes the spread of the wave packet in the weakly nonlinear dispersion medium.It appears in many branch of physics and applied mathematics,which include nonlinear quantum field theory,condensed matter and plasma physics,nonlinear optics,physics,fluid perturbation and phase transition theory,biophysics,etc.The equation of the three-wave resonance interaction is a quadratic coupled non-dispersive partial differential equation set,which describes the time and space evolution of the complex amplitude of the three-wave mode.In the field of nonlinear optics,if the expression of the nonlinear Schrodinger equation is known,then the mathematical expression of the soliton solution can be given.It is the most important application in the study of nonlinear wave propagation for the soliton theory.It has contributed to the understanding of many experiments and has a very important theoretical position.From oceanography to optics,from plasma physics to acoustics,the description of these solutions may be a key step in understanding and predicting rogue waves in a variety of multi-component wave dynamics.In this paper,the expansion method of the（G’/G）is used to study the analytic problem of the nonlinear Schrodinger equation,where G=G（ξ）satisfies a second order linear ordinary differential equation.The expansion method of the（G’/G）can be used to solve the MKDV equation,Boussinesq equation and Hirota--Satsuma equation.When the parameter is taken as a special value,the solitary wave can be obtained from the traveling wave.Traveling wave solutions are expressed by hyperbolic functions,trigonometric functions and rational functions,which can be used to solve many other nonlinear evolution equations.We study the analytical solutions of the high order nonlinear Schrodinger equation by using the extended method of the（G’/G）and discuss the characteristics of some solutions.Here we consider a system of three coupled wave equations which includes as special cases the vector nonlinear Schrodinger equations and the equations describing the resonant interaction of three waves.By using a Lax pair equation and Darboux Dressing Transformation（DDT）,a new solution of the coupling matrix nonlinear integrable evolution partial differential equation are constructed.The soliton-wave analytical solutions of two-wave coupled VNLS equation and three-wave coupled 3WRI equation are obtained by exponentiation based on non-diagonalized matrix.The results obtained in this paper will be helpful to understand the phenomenon of rogue waves.And it is beneficial to the study of wave resonance interaction in a variety of physical conditions.