Application Of Riemann–Hilbert Method To Integrable Equations Associated With High Order Spectral Problems

In this thesis,the main aim we are interested in is to study the integrable nonlinear equations associated with higher order matrix spectral problems.There are two parts in this thesis: On the hand,based on Riemann–Hilbert problems and Fokas’ unified transform method,we study the initial-boundary value problems for integrable nonlinear equations on the half-line;On the other hand,under the help of Riemann–Hilbert problems and Deift and Zhou’s steepest descent method,we study the long-time asymptotics of initial value problems for integrable nonlinear equations.In Chapter 2,we study the initial-boundary value problem for the coupled nonlinear Schr¨odinger equation.Firstly,with the aid of Fredholm integral equation,we transform the initial data and the boundary data into,respectively,spectral functions (6))and (6)),to construct a Riemann–Hilbert problem whose jump matrix is characterized with (6))and (6)).Then we show that the solution obtained from the aforementioned Riemann–Hilbert problem satisfies both the coupled nonlinear Schr¨odinger equation and the corresponding initial-boundary conditions.In well-posed initial-boundary value problems,there are unknown boundary data.But in particular occasions,i.e.,when the problem is linearizable,(6))can be given in terms of (6)).Otherwise,when the problem is nonlinearizable,we provide an effective approximation based on perturbation theory.In Chapter 3,we study the initial-boundary value problem for the coupled modified Korteweg–de Vries equation.Different from that in Chapter 2,we“package up” the potentials and write the higher order Lax pair in analogy of a(2×2)-matrices.Then we construct the Riemann–Hilbert problem with the corresponding spectral functions.What’s more,by using Sherman–Morrison formula,we derived the jump matrix of the Riemann–Hilbert problem and the residue conditions agilely.In Chapter 4,by using block matrices,we write higher order matrix in(2 ×2)-form.Firstly,by introducing a -function we perform the first transform of the Riemann–Hilbert problems involved.Because there is no exact approach to defining exactly,we got stuck here.We bypassed this obstacle by replacing it with its long-time approximation.Then,using steepest descending method,after several transforms of the Riemann–Hilbert problem,we obtain a parabolic cylinder equation.Finally,we obtain the long-time asymptotics of the coupled nonlinear Schr¨odinger equation using parabolic cylinder functions.In Chapter 5,similar to that in Chapter 4,we start from the Lax pair of Sasa–Satsuma equation and construct a corresponding Riemann–Hilbert problem.Then we obtain the leading term by means of steepest descent method and parabolic cylinder functions.Different from those in Chapter 4,there are two stationary phase points and as a result the Riemann–Hilbert problems are much more complicated.