Solving The Soliton Equations Associated With Higher Order Matrix Spectral Problem:Riemann-Hilbert Approach

This thesis mainly concerns with the applications of the Riemann-Hilbert ap-proach to integrable nonlinear partial differential equations under Cauchy initial conditions.Based on the Riemann-Hilbert approach,the thesis studies the multi-soliton solutions of some integrable partial differential equations associated with3×3,4×4 and 7×7 matrix spectral problems.Five integrable equations includ-ing the nonlocal two waves action equation,the two component coupled modified Hirota equation,the two component coupled modified Fokas-Lenells equation,the two component coupled complex mKdV equation and the three component coupled Sasa-Satsuma equation are taken into consideration in this thesis.In Chapter 2,the N-soliton solutions for the Cauchy initial problem of the nonlocal two waves action equation are investigated.Starting with a 3×3 matrix spectral problem,the matrix Jost solutions of the spectral equations is analyzed,the corresponding matrix Riemann-Hilbert problem is constructed.Then the regular and nonregular Riemann-Hilbert problems are solved respectively,and the temporal and spatial evolutions of the scattering data are also discussed.Finally,with the help of the algebraic properties of the matrix^±,the exact expressions of the N-soliton solutions for the nonlocal two waves action equation is constructed.In Chapter 3 and 4,the multi-soliton solutions for the two component modi-fied Hirota equation and the two component modified Fokas-Lenells equation are discussed respectively.Though both systems associate with 3×3 matrix spectral problem,there are differences between them.For the two component modified Hirota equation,it admits only one symmetric condition,while the two compo-nent modified Fokas-Lenells equation possess two symmetric conditions,which makes it more difficult to solve the corresponding nonregular Riemann-Hilbert problem.Besides,in the last two chapters,the boundary conditions of the ac-cording matrix Riemann-Hilbert problems are normalized,that is,^±tends to the unit matrix when the spatial variable x tends to infinity.However,when dealing with the two component modified Fokas-Lenells equation,the solutions of the nonregular Riemann-Hilbert problem^±do not tend to the unit matrix,which is fixed by utilizing the asymptotic expression of^±at?0.Finally,the N-soliton solutions of the two component modified Fokas-Lenells equation are constructed.In Chapter 5,the N-soliton solutions of two component modified KdV equa-tion are studied,which associates with a 4×4 matrix spectral problem.With the aid of the matrix Jost solution for the according spectral equation,the matrix Riemann-Hilbert problem for the two component modified KdV equation is de-rived.After a regularization process,the nonregular Riemann-Hilbert problem is transformed into a regular one,which is solved by the Plemelj's formula.Unlike from the former chapters,when constructing the soliton solutions by means of the algebraic properties of the solutions for Riemann-Hilbert problem,the kernel of the matrix⁺(₆))is either 1-dimensional or 2-dimensional,which determines that the corresponding single soliton solutions admit two cases.In Chapter 6,the N-soliton solutions of the three component Sasa-Satsuma equation are investigated,which corresponds to a 7×7 matrix spectral prob-lem.By analyzing the matrix Jost solution of the spectral equation related to x-part,the associated matrix Riemann-Hilbert problem is established.By solv-ing the nonregular Riemann-Hilbert problem,use the algebraic properties of the solutions of Riemann-Hilbert problem to construct the solutions corresponding to the reflectionless case,which is the exact multi-soliton solutions.Unlike from the former chapters,there are two different zero structures for the nonregular Riemann-Hilbert problem,which is more complicated.In each chapter,the graphics of the corresponding single soliton solutions are displayed by the Mathematica.