Some Applications Of Dbar Problem To Nonlocal Integrable Equations

In this thesis,Dbar method is extended to study four nonlocal integrable equations and one(2+1)-dimensional differential-difference Kadomtsev-Petviashvili equation.Dressing method based on some Bi-Dbar problems and certain symmetry conditions are introduced to investigate the coupled nonlocal nonlinear Schr(?)dinger equation,coupled nonlocal derivative nonlinear Schr(?)dinger equation,coupled shifted nonlocal modified KdV equation.For each coupled nonlocal equation,two different spectral transform matrices are introduced to define two associated Dbar problems.The relations between the coupled nonlocal equation potential and the solution of the Dbar problem are constructed.The spatial transform method is extended to obtain the coupled nonlocal equations and their conservation laws.The general nonlocal reduction of the coupled soliton equations to the nonlocal soliton equations is discussed in detail,the exact solutions are derived.Then the Dbar dressing method is extended to study the coupled nonlocal nonlinear Schr(?)dinger equation in a special case with nonzero boundary condition.A certain distribution for the Dbar problem is introduced to obtain the coupled nonlocal nonlinear Schr(?)dinger equation and the conservation laws.In a special case with nonzero boundary condition,the general nonlocal reduction of the coupled nonlocal nonlinear Schr(?)dinger equations to the nonlocal nonlinear Schr(?)dinger equation is discussed.The exact solutions of the coupled nonlocal nonlinear Schr(?)dinger equation and nonlocal nonlinear Schr(?)dinger equation with nonzero boundary condition are given.In addition,nonlocal Dbar problem is developed to consider the differential-difference Kadomtsev-Petviashvili equation.Then Cauchy type determinant solution and a new type rational solution are given.The initial value problem of differential-difference Kadomtsev-Petviashvili equation is discussed by virtue of the quasi-local Dbar problem.From an impulse initial condition,a new exact solution to the initial value problem of differentialdifference Kadomtsev-Petviashvili equation is obtained.