Applications Of The Nonlinear Steepest Descent Method In Soliton Equations

This paper mainly discusses applications of nonlinear steepest descent method in soliton equations.In this paper,we study the long time asymptotics of initial value problems of four soliton equations related to matrix spectral problems.These equations include potential WKI equation,coupled modified KdV equation,coherently-coupled nonlinear Schrodinger system and the Hermitian symmetric space derivative nonlinear Schr?dinger equation.In chapter 1,we briefly introduce the development of soliton theory and some important advances in solving asymptotics of soliton equations based on Riemann-Hilbert problem by nonlinear steepest descent method.In chapter 2,we mainly study the long time asymptotics of the initial value problem of the potential WKI equation.Starting from the 2×2 matrix spectral problem associated with the equation,the asymptotic properties of the solution of the spectral problem are obtained as k→0 and k→∞.After spectral analysis and appropriate transformation,the corresponding initial 2×2 matrix RiemannHilbert problem is constructed.Then using the asymptotics of the solution of this Riemann-Hilbert problem when k→0,the expression about the solution of the potential WKI equation can be obtained.By introducing the scalar functionδ(k),a series of equivalent transformations about the initial Riemann-Hilbert problem are carried out and the corresponding jump curve is changed via the direction of adjustment,expansion,reduction and scale transformation.Finally,the asymptotic properties of the parabolic cylinder function are used to calculate the asymptotically principal part of the potential WKI equation.In chapter 3,we mainly discuss the long-time asymptotic behavior of the solution of the coupled modified KdV equation corresponding to the 3× 3 matrix spectral problem.Compared with chapter 2,we use nonlinear steepest descent method to study the higher-order matrix spectral problems.By applying the stationary phase points which are from the jump matrix,we introduce the matrix Riemann-Hilbert problem which the 2×2 matrix-valued function δ(k)satisfies,and it is generally unsolvable.To solve this difficulty,we first explicitly expressed the scalar function detδ(k)by Plemelj formula,by using the asymptotic property of δ(k)itself,the term containing δ(k)is replaced by the term containing detδ(k)and estimate the error between them.At the same time,note that in this chapter the stationary phase points derived from θ(k)are two,thus in the asymptotic analysis,we need to eliminate the interaction of the curves between the two points.Then by scaling operator,the jump curve through the origin is obtained,on the basis of it,a solvable standard Riemann-Hilbert problem is obtained,and then we construct the long-time asymptotics of the initial value problem of the coupled modified KdV equation.In chapter 4 and chapter 5,the long-time asymptotic behavior of the solutions of the coherently-coupled nonlinear Schrodinger system and the Hermitian symmetric space derivative nonlinear Schrodinger equation is considered respectively.These two integrable equations correspond to the 4×4 matrix spectral problems,in these two chapters,we need to introduce two different matrix-valued functions δ1 and δ2,their determinants all satisfy the same scalar Riemann-Hilbert problem,and then use their asymptotic properties to overcome the difficulty that the 2 × 2 matrix Riemann-Hilbert problems which they satisfy are unsolvable.In addition,the partitioning of higher-order matrices in these two chapters is slightly different from that in chapter 3,especially in the end when dealing with the matrix-valued solution of the reduced standard Riemann-Hilbert problem,we need to find a new way to reconsider the relation between every scalar element of the matrix-valued solution and the solution of the scalar Weber equation.