The Application Of Riemann-Hilbert Method In The Long-term Asymptotic Analysis Of Solutions Of Nonlinear Integrable PDEs

In this thesis,we investigate the long-time asymptotic behavior of solutions to the completely integrable nonlinear partial differential equations in 1+1,i.e.,in one spatial and one temporal dimensions.These equations arise widely in mathematical physics,and in order to model realistic applications and understand nonlinear phenomena,it is essential to consider the initial-value with vanishing as well as nonvanishing boundary conditions and initial-boundary value problems.In Chapter 1,we briefly review some of the most significant advainces of the last several decades in the application of Riemann Hilbert method to integrable equations,and then present our main results as well as our future consideration.In Chapter 2,we consider an integrable extended modified Korteweg-de Vries oquation on the line with decaying initial value.We show that the solution of this initial-value problem can be represented in terms of the solution of a 2 × 2 matrix Riomann Hilbert problem formulated in the complex ?-plane with the jump matrices given in terms of two spectral functions a?k?,b?k?obtained from the initial value.By performing the nonlinear steepest descent analysis of the associated matrix Riemann-Hilbert problem.we obtain the explicit leading-order asymptotics of the solution of this initial-value problem as time t goes to infinity in physically interesting region.The main point we are interested in this problem is that there are four stationary phase points and as a result the asymptotic analysis is much more complicated.For a spocial case ? = 0.we present the asymptotic formula of the solution to the extended modified Korteweg-de Vries equation in region P={?x,t?? R²|0<x ? mt^1/5.t ? 3} in terms of the solution of a fourth irder Painlove II equation.In Chapter 3.we present a Riemanu-Hilbert approach for the modified short pulseequation on the line.This approach allows us to give a representation of the solution to the Cauchy problem.which can be efficiently used in studying its long-time behaviour..also to describe the soliton solutions.Different from the analysis in Chapter 2.the modified short pulse equation posses a Wadati-Konno-Ichikawa?WKI?-type Lax pair,which has singularities in the extended complex k-plane at k=0 and at k=?.Thus,in order to establish the analytic properties of solutions near the singular points?w.r.t.the spectral parameter?of the Lax pair equations,it is necessary to transform the Lax pair equations to a certain form.On the other hand,the solution of the modified short pulse equation is constructed from a 2 × 2 matrix Riemann-Hilbert problem in terms of the order O?k?as k ? 0.In Chapter 4,we study the Hirota equation on the quarter plane with the initial and boundary values belonging to the Schwartz space.By using the unified trans-formation approach,one can show that the solution to the Hirota equation can be represented in terms of the solution of a 2 ? 2 matrix Riemann-Hilbert problem for-mulated in the complex k-plane with the four jump matrices given in terms of spectral functions a?k?,b?k?obtained from the initial datum and A?k?,B?k?obtained from the boundary values.Then by combining the ideas of nonlinear steepest descent method,we analyze the long-time asymptotic behavior of the solution to this initial-boundary value problem.Compared with the analysis of the initial-value problem for Hirota e-quation,the Riemann-Hilbert problem relevant to the initial-boundary value problem also has jumps across additional two contours whereas the Riemann-Hilbert problem relevant to the initial-value problem only lias a jump across R.Moreover,the jumps across these additional two lines involve the a new spectral function h?k?.During the asymptotic analysis,one should find an suitable analytic approximation of h?k?.On the other hand,the existence of the additional two jumps leads the contour deforma-tions to be more involved.In Chapter 5,we analyze Gerdjikov-Ivanov type derivative nonlinear Schr???dinger equation formulated on the line with the initial value q?x,0?is given and satisfies the symmetric,nonzero boundary conditions at infinity,that is,q?x,O??q_± as x ?±?,and |q_±|=q0>0.The goal of this chapter is to study the asymptotic behavior of the solution of this initial-value problellm as l??.The main tool is the asymptotic anal-ysis of an associated matrix Riemann Hilbert problem by using the steepest descent method and the so-called g-function mechanism.We show that the solution q?x,t?of this initial-value problem has a different asymptotie behavior in different regions of the xt-plane.In the regions x<??? qand x>2.the solution takes the form of a plane wave.Whereas in the rcegion-2<x<???.the solution takes the form of a modulated elliptic wave.