Integrable Boundary Conditions For Higher-order Nonlinear Schr(?)dinger Equation

In this paper,we mainly learned a series of methods and concepts for the initialboundary value problem of nonlinear integrable partial differential equations,including using the inverse scattering methods to solve the nonlinear Schr(?)dinger equation and obtaining multi-soliton solutions;sklyanin’s integrable boundary theory and its application on nonlinear Schr(?)dinger equation with boundary conditions;and using dressing the boundary methods to solve the soliton solution of the nonlinear Schr(?)dinger equation with boundary conditions on the half-line.Based on this,we combine the idea of integrable boundary theory with the notion of the hierarchy of nonlinear Schr(?)dinger equations,integrable boundary conditions for higher-order nonlinear Schr(?)dinger equations are derived by using recursion operator of the hierarchy of Schr(?)dinger equations.We find that there are different cases for odd order equation and even order equation.We obtain,for even order nonlinear Schr(?)dinger equations,a class of integrable boundary conditions;using the method of dressing the boundary,we construct soliton solutions of the equations on the half-line subject to integrable boundary conditions.As to half-line problems for odd order nonlinear Schr(?)dinger equations,the formalism of integrable boundary only leads to a real reduction of the model: all the equations are reduced to real equations.The study of the integrable boundary problems for higher-order nonlinear Schr(?)dinger equations is an extension of integrable boundary theory,which lays a foundation for the study of general nonlinear boundary problems and the development of related analytical methods.