Exact Solutions To The Wadati-konno-ichikawa Equation

In this PhD thesis,we consider the soliton solution,breather solution and rogue wave solution for Wadati-Konno-Ichikawa(WKI)equation,by applying the inverse scattering transformation(1ST),Riemann-Hilbert method and Darboux transformation.In chapter 1,the IST method,Riemann-Hilbert method,Darboux transformation and the WKI equation are introduced in detail.In chapter 2,the N-th soliton solution for WKI equation is solved by IST.The jost solution of WKI equation is special,because it does not go to 1 or 0,but goes to variables about potential,as spectral parameter going to infinity.Using the radio of the second component of jost solution and spectral parameter replaces the second com-ponent of jost solution,simplifying the asymptotic behavior the jost solution,then the Gel'fand-Levitan-Marchenko(GLM)equation is constructed based on the triangular kernel expression of jost solution,and N-th order soliton is obtained by solving GLM equation.In chapter 3,the N-th soliton solution for WKI equation is solved by the Riemann-Hilbert method.The asymptotic behavior of jost solution,obtained in chapter 2,dis-plays that the Riemann-Hilbert problem will be unsolvable,if the jost solution is used directly to construct Riemann-Hilbert problem.By introducing a transformation ma-trix,the suitable jost solution is obtained.Meanwhile,by taking the spectral analysis when spectral parameter goes to 0 and infinity,the solvable Riemann-Hilbert problem is constructed,and the N-th soliton solution of WKI equation can be solved by consid-ering the trivial case of Riemann-Hilbert problem.In chapter 4,the N-th solution for WKI equation is solved by Darboux transfor-mation,which includes the soliton solution,breather solution and rogue wave solution.Owing to the special form of Lax pair for WKI equation,the usual Darboux transforma-tion can not operate it directly.Instead,we consider the Darboux transformation for an equivalent nonlinear integrable equation—modified WKI(m WKI)equation,which is e-quivalent to WKI by the hodograph transformation.By applying the inverse hodograph transformation,the solution for WKI equation is obtained.Furthermore,the elastic col-lisions of N soliton solutions,the analysis of rogue wave solution,the loop soliton and loop rogue wave solution for WKI equation are also considered.